3.75 \(\int \frac {\sinh ^2(c+d x)}{a+b \tanh ^3(c+d x)} \, dx\)
Optimal. Leaf size=384 \[ \frac {b \left (2 a^2+b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{3 d \left (a^2-b^2\right )^2}-\frac {a^{2/3} \sqrt [3]{b} \left (3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 d \left (a^2-b^2\right )^2}+\frac {a^{2/3} \sqrt [3]{b} \left (3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 d \left (a^2-b^2\right )^2}+\frac {a^{2/3} \sqrt [3]{b} \left (-3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} d \left (a^2-b^2\right )^2}+\frac {1}{4 d (a+b) (1-\tanh (c+d x))}-\frac {1}{4 d (a-b) (\tanh (c+d x)+1)}+\frac {(a-2 b) \log (1-\tanh (c+d x))}{4 d (a+b)^2}-\frac {(a+2 b) \log (\tanh (c+d x)+1)}{4 d (a-b)^2} \]
[Out]
1/4*(a-2*b)*ln(1-tanh(d*x+c))/(a+b)^2/d-1/4*(a+2*b)*ln(1+tanh(d*x+c))/(a-b)^2/d+1/3*a^(2/3)*b^(1/3)*(a^2+3*a^(
2/3)*b^(4/3)+2*b^2)*ln(a^(1/3)+b^(1/3)*tanh(d*x+c))/(a^2-b^2)^2/d-1/6*a^(2/3)*b^(1/3)*(a^2+3*a^(2/3)*b^(4/3)+2
*b^2)*ln(a^(2/3)-a^(1/3)*b^(1/3)*tanh(d*x+c)+b^(2/3)*tanh(d*x+c)^2)/(a^2-b^2)^2/d+1/3*b*(2*a^2+b^2)*ln(a+b*tan
h(d*x+c)^3)/(a^2-b^2)^2/d+1/3*a^(2/3)*b^(1/3)*(a^2-3*a^(2/3)*b^(4/3)+2*b^2)*arctan(1/3*(a^(1/3)-2*b^(1/3)*tanh
(d*x+c))/a^(1/3)*3^(1/2))/(a^2-b^2)^2/d*3^(1/2)+1/4/(a+b)/d/(1-tanh(d*x+c))-1/4/(a-b)/d/(1+tanh(d*x+c))
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Rubi [A] time = 0.63, antiderivative size = 384, normalized size of antiderivative = 1.00,
number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used
= {3663, 6725, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac {a^{2/3} \sqrt [3]{b} \left (3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 d \left (a^2-b^2\right )^2}+\frac {b \left (2 a^2+b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{3 d \left (a^2-b^2\right )^2}+\frac {a^{2/3} \sqrt [3]{b} \left (3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 d \left (a^2-b^2\right )^2}+\frac {a^{2/3} \sqrt [3]{b} \left (-3 a^{2/3} b^{4/3}+a^2+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} d \left (a^2-b^2\right )^2}+\frac {1}{4 d (a+b) (1-\tanh (c+d x))}-\frac {1}{4 d (a-b) (\tanh (c+d x)+1)}+\frac {(a-2 b) \log (1-\tanh (c+d x))}{4 d (a+b)^2}-\frac {(a+2 b) \log (\tanh (c+d x)+1)}{4 d (a-b)^2} \]
Antiderivative was successfully verified.
[In]
Int[Sinh[c + d*x]^2/(a + b*Tanh[c + d*x]^3),x]
[Out]
(a^(2/3)*b^(1/3)*(a^2 - 3*a^(2/3)*b^(4/3) + 2*b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tanh[c + d*x])/(Sqrt[3]*a^(1/3)
)])/(Sqrt[3]*(a^2 - b^2)^2*d) + ((a - 2*b)*Log[1 - Tanh[c + d*x]])/(4*(a + b)^2*d) - ((a + 2*b)*Log[1 + Tanh[c
+ d*x]])/(4*(a - b)^2*d) + (a^(2/3)*b^(1/3)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*Log[a^(1/3) + b^(1/3)*Tanh[c +
d*x]])/(3*(a^2 - b^2)^2*d) - (a^(2/3)*b^(1/3)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*
Tanh[c + d*x] + b^(2/3)*Tanh[c + d*x]^2])/(6*(a^2 - b^2)^2*d) + (b*(2*a^2 + b^2)*Log[a + b*Tanh[c + d*x]^3])/(
3*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - Tanh[c + d*x])) - 1/(4*(a - b)*d*(1 + Tanh[c + d*x]))
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 260
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]
Rule 617
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 634
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]
Rule 1860
Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]
Rule 1871
Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] || !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
Rule 3663
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff^(m + 1))/f, Subst[Int[(x^m*(a + b*(ff*x)^n)^p)/(c^2 + ff^2*x^2
)^(m/2 + 1), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2]
Rule 6725
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]
Rubi steps
\begin {align*} \int \frac {\sinh ^2(c+d x)}{a+b \tanh ^3(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{\left (1-x^2\right )^2 \left (a+b x^3\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{4 (a+b) (-1+x)^2}+\frac {a-2 b}{4 (a+b)^2 (-1+x)}+\frac {1}{4 (a-b) (1+x)^2}+\frac {-a-2 b}{4 (a-b)^2 (1+x)}+\frac {b \left (3 a^2 b-a \left (a^2+2 b^2\right ) x+b \left (2 a^2+b^2\right ) x^2\right )}{\left (a^2-b^2\right )^2 \left (a+b x^3\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {(a-2 b) \log (1-\tanh (c+d x))}{4 (a+b)^2 d}-\frac {(a+2 b) \log (1+\tanh (c+d x))}{4 (a-b)^2 d}+\frac {1}{4 (a+b) d (1-\tanh (c+d x))}-\frac {1}{4 (a-b) d (1+\tanh (c+d x))}+\frac {b \operatorname {Subst}\left (\int \frac {3 a^2 b-a \left (a^2+2 b^2\right ) x+b \left (2 a^2+b^2\right ) x^2}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{\left (a^2-b^2\right )^2 d}\\ &=\frac {(a-2 b) \log (1-\tanh (c+d x))}{4 (a+b)^2 d}-\frac {(a+2 b) \log (1+\tanh (c+d x))}{4 (a-b)^2 d}+\frac {1}{4 (a+b) d (1-\tanh (c+d x))}-\frac {1}{4 (a-b) d (1+\tanh (c+d x))}+\frac {b \operatorname {Subst}\left (\int \frac {3 a^2 b-a \left (a^2+2 b^2\right ) x}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{\left (a^2-b^2\right )^2 d}+\frac {\left (b^2 \left (2 a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{\left (a^2-b^2\right )^2 d}\\ &=\frac {(a-2 b) \log (1-\tanh (c+d x))}{4 (a+b)^2 d}-\frac {(a+2 b) \log (1+\tanh (c+d x))}{4 (a-b)^2 d}+\frac {b \left (2 a^2+b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\tanh (c+d x))}-\frac {1}{4 (a-b) d (1+\tanh (c+d x))}+\frac {b^{2/3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{a} \left (6 a^2 b^{4/3}-a^{4/3} \left (a^2+2 b^2\right )\right )+\sqrt [3]{b} \left (-3 a^2 b^{4/3}-a^{4/3} \left (a^2+2 b^2\right )\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tanh (c+d x)\right )}{3 a^{2/3} \left (a^2-b^2\right )^2 d}+\frac {\left (a^{2/3} b^{2/3} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}\\ &=\frac {(a-2 b) \log (1-\tanh (c+d x))}{4 (a+b)^2 d}-\frac {(a+2 b) \log (1+\tanh (c+d x))}{4 (a-b)^2 d}+\frac {a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {b \left (2 a^2+b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\tanh (c+d x))}-\frac {1}{4 (a-b) d (1+\tanh (c+d x))}-\frac {\left (a b^{2/3} \left (a^2-3 a^{2/3} b^{4/3}+2 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tanh (c+d x)\right )}{2 \left (a^2-b^2\right )^2 d}-\frac {\left (a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tanh (c+d x)\right )}{6 \left (a^2-b^2\right )^2 d}\\ &=\frac {(a-2 b) \log (1-\tanh (c+d x))}{4 (a+b)^2 d}-\frac {(a+2 b) \log (1+\tanh (c+d x))}{4 (a-b)^2 d}+\frac {a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}-\frac {a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 \left (a^2-b^2\right )^2 d}+\frac {b \left (2 a^2+b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\tanh (c+d x))}-\frac {1}{4 (a-b) d (1+\tanh (c+d x))}-\frac {\left (a^{2/3} \sqrt [3]{b} \left (a^2-3 a^{2/3} b^{4/3}+2 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt [3]{a}}\right )}{\left (a^2-b^2\right )^2 d}\\ &=\frac {a^{2/3} \sqrt [3]{b} \left (a^2-3 a^{2/3} b^{4/3}+2 b^2\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} \left (a^2-b^2\right )^2 d}+\frac {(a-2 b) \log (1-\tanh (c+d x))}{4 (a+b)^2 d}-\frac {(a+2 b) \log (1+\tanh (c+d x))}{4 (a-b)^2 d}+\frac {a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}-\frac {a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{2/3} b^{4/3}+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 \left (a^2-b^2\right )^2 d}+\frac {b \left (2 a^2+b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\tanh (c+d x))}-\frac {1}{4 (a-b) d (1+\tanh (c+d x))}\\ \end {align*}
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Mathematica [C] time = 3.81, size = 423, normalized size = 1.10 \[ -\frac {4 b \text {RootSum}\left [\text {$\#$1}^3 a+\text {$\#$1}^3 b+3 \text {$\#$1}^2 a-3 \text {$\#$1}^2 b+3 \text {$\#$1} a+3 \text {$\#$1} b+a-b\& ,\frac {-4 \text {$\#$1}^2 a^2 \log \left (e^{2 (c+d x)}-\text {$\#$1}\right )+8 \text {$\#$1}^2 a^2 c+8 \text {$\#$1}^2 a^2 d x+4 \text {$\#$1}^2 a b \log \left (e^{2 (c+d x)}-\text {$\#$1}\right )-8 \text {$\#$1}^2 a b c-8 \text {$\#$1}^2 a b d x-\text {$\#$1}^2 b^2 \log \left (e^{2 (c+d x)}-\text {$\#$1}\right )+2 \text {$\#$1}^2 b^2 c+2 \text {$\#$1}^2 b^2 d x-2 a^2 \log \left (e^{2 (c+d x)}-\text {$\#$1}\right )-2 \text {$\#$1} a^2 \log \left (e^{2 (c+d x)}-\text {$\#$1}\right )+4 \text {$\#$1} a^2 c+4 \text {$\#$1} a^2 d x-b^2 \log \left (e^{2 (c+d x)}-\text {$\#$1}\right )+2 \text {$\#$1} b^2 \log \left (e^{2 (c+d x)}-\text {$\#$1}\right )-4 \text {$\#$1} b^2 c-4 \text {$\#$1} b^2 d x+4 a^2 c+4 a^2 d x+2 b^2 c+2 b^2 d x}{\text {$\#$1}^2 a-\text {$\#$1}^2 b+2 \text {$\#$1} a+2 \text {$\#$1} b+a-b}\& \right ]+6 \left (a^2-3 a b+2 b^2\right ) (c+d x)-3 a (a+b) \sinh (2 (c+d x))+3 b (a+b) \cosh (2 (c+d x))}{12 d (a-b) (a+b)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[Sinh[c + d*x]^2/(a + b*Tanh[c + d*x]^3),x]
[Out]
-1/12*(6*(a^2 - 3*a*b + 2*b^2)*(c + d*x) + 3*b*(a + b)*Cosh[2*(c + d*x)] + 4*b*RootSum[a - b + 3*a*#1 + 3*b*#1
+ 3*a*#1^2 - 3*b*#1^2 + a*#1^3 + b*#1^3 & , (4*a^2*c + 2*b^2*c + 4*a^2*d*x + 2*b^2*d*x - 2*a^2*Log[E^(2*(c +
d*x)) - #1] - b^2*Log[E^(2*(c + d*x)) - #1] + 4*a^2*c*#1 - 4*b^2*c*#1 + 4*a^2*d*x*#1 - 4*b^2*d*x*#1 - 2*a^2*Lo
g[E^(2*(c + d*x)) - #1]*#1 + 2*b^2*Log[E^(2*(c + d*x)) - #1]*#1 + 8*a^2*c*#1^2 - 8*a*b*c*#1^2 + 2*b^2*c*#1^2 +
8*a^2*d*x*#1^2 - 8*a*b*d*x*#1^2 + 2*b^2*d*x*#1^2 - 4*a^2*Log[E^(2*(c + d*x)) - #1]*#1^2 + 4*a*b*Log[E^(2*(c +
d*x)) - #1]*#1^2 - b^2*Log[E^(2*(c + d*x)) - #1]*#1^2)/(a - b + 2*a*#1 + 2*b*#1 + a*#1^2 - b*#1^2) & ] - 3*a*
(a + b)*Sinh[2*(c + d*x)])/((a - b)*(a + b)^2*d)
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fricas [C] time = 1.41, size = 10695, normalized size = 27.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)^2/(a+b*tanh(d*x+c)^3),x, algorithm="fricas")
[Out]
-1/72*(36*(a^3 + 4*a^2*b + 5*a*b^2 + 2*b^3)*d*x*cosh(d*x + c)^2 - 9*(a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)^
4 - 36*(a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^3 - 9*(a^3 - a^2*b - a*b^2 + b^3)*sinh(d*x + c)
^4 + 9*a^3 + 9*a^2*b - 9*a*b^2 - 9*b^3 - 4*((a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2 + 2*(a^4 - 2*a^2*b^2 + b
^4)*d*cosh(d*x + c)*sinh(d*x + c) + (a^4 - 2*a^2*b^2 + b^4)*d*sinh(d*x + c)^2)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2
+ b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((
a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*
d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3
) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2
*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b
^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*log(1/
18*(a^5 + 2*a^4*b - 2*a^3*b^2 - 4*a^2*b^3 + a*b^4 + 2*b^5)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*
b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*
d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*
b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*
b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4
*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^
2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 + a^3 + 2*a^2*b + 2*
a*b^2 + 4*b^3 - 1/3*(a^4 + 3*a^3*b + 13*a^2*b^2 + 6*a*b^3 + 4*b^4)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) -
(2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*
a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3
+ 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18
*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3
)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((
a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d + (a^3 + 8*a*b^2
)*cosh(d*x + c)^2 + 2*(a^3 + 8*a*b^2)*cosh(d*x + c)*sinh(d*x + c) + (a^3 + 8*a*b^2)*sinh(d*x + c)^2) + 18*(2*(
a^3 + 4*a^2*b + 5*a*b^2 + 2*b^3)*d*x - 3*(a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 2*(18*
(2*a^2*b + b^3)*cosh(d*x + c)^2 + 36*(2*a^2*b + b^3)*cosh(d*x + c)*sinh(d*x + c) + 18*(2*a^2*b + b^3)*sinh(d*x
+ c)^2 - ((a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2 + 2*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)*sinh(d*x + c)
+ (a^4 - 2*a^2*b^2 + b^4)*d*sinh(d*x + c)^2)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a
^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2
)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 -
2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/
((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^
2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1
/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d)) - 3*sqrt(1/3)*((a^4 - 2*a^2*b^2 + b^4)*
d*cosh(d*x + c)^2 + 2*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)*sinh(d*x + c) + (a^4 - 2*a^2*b^2 + b^4)*d*sinh(d
*x + c)^2)*sqrt((288*a^4*b^2 + 720*a^2*b^4 - 36*b^6 - (a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*((b^2/(a
^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18
*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3
)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((
a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^
2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4
*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2
*b^2*d + b^4*d))^2*d^2 + 12*(2*a^6*b - 3*a^4*b^3 + b^7)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b +
b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2
+ b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(
a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b +
b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d
- 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^
4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d)/((a^8 - 4*a^6*b^2 + 6*a^4*
b^4 - 4*a^2*b^6 + b^8)*d^2)))*log(-1/36*(a^6 + 3*a^5*b - 6*a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 + 2*b^6)*((b^2/(a^4*d
^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*
a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/
(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2
- b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d
+ b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3
) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2
*d + b^4*d))^2*d^2 + a^4 - 3*a^3*b + 10*a^2*b^2 - 15*a*b^3 - 2*b^4 + 1/6*(a^5 + 4*a^4*b + 16*a^3*b^2 + 19*a^2*
b^3 + 10*a*b^4 + 4*b^5)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b
^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d
+ b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^
3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*
d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*
b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) +
6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d + (a^4 + a^3*b + 8*a^2*b^2 + 8*a*b^3)*cosh(d*x + c)^2 + 2*(
a^4 + a^3*b + 8*a^2*b^2 + 8*a*b^3)*cosh(d*x + c)*sinh(d*x + c) + (a^4 + a^3*b + 8*a^2*b^2 + 8*a*b^3)*sinh(d*x
+ c)^2 + 1/12*sqrt(1/3)*((a^6 + 3*a^5*b - 6*a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 + 2*b^6)*((b^2/(a^4*d^2 - 2*a^2*b^2*
d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^
2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*
b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^
(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/2
7*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 +
8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d^
2 + 6*(a^5 - 2*a^4*b - 2*a^3*b^2 + 4*a^2*b^3 + a*b^4 - 2*b^5)*d)*sqrt((288*a^4*b^2 + 720*a^2*b^4 - 36*b^6 - (a
^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a
^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2
)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 -
2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/
((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^
2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1
/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 + 12*(2*a^6*b - 3*a^4*b^3 + b^7)*
((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1
)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2
*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*
a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d -
2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d
^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d
- 2*a^2*b^2*d + b^4*d))*d)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2))) - 2*(18*(2*a^2*b + b^3)*co
sh(d*x + c)^2 + 36*(2*a^2*b + b^3)*cosh(d*x + c)*sinh(d*x + c) + 18*(2*a^2*b + b^3)*sinh(d*x + c)^2 - ((a^4 -
2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2 + 2*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)*sinh(d*x + c) + (a^4 - 2*a^2*b^
2 + b^4)*d*sinh(d*x + c)^2)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d
+ b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b
^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^
4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*
b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1
/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1
) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d)) + 3*sqrt(1/3)*((a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2
+ 2*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)*sinh(d*x + c) + (a^4 - 2*a^2*b^2 + b^4)*d*sinh(d*x + c)^2)*sqrt((2
88*a^4*b^2 + 720*a^2*b^4 - 36*b^6 - (a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*((b^2/(a^4*d^2 - 2*a^2*b^2
*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b
^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2
*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))
^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/
27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2
+ 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2
*d^2 + 12*(2*a^6*b - 3*a^4*b^3 + b^7)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2
*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d
- 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^
2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^
2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^
4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*s
qrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 +
b^8)*d^2)))*log(-1/36*(a^6 + 3*a^5*b - 6*a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 + 2*b^6)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2
+ b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/(
(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2
*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/
3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(
2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*
b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2
+ a^4 - 3*a^3*b + 10*a^2*b^2 - 15*a*b^3 - 2*b^4 + 1/6*(a^5 + 4*a^4*b + 16*a^3*b^2 + 19*a^2*b^3 + 10*a*b^4 + 4
*b^5)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(
3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*
(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8
*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^
4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2
*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/
(a^4*d - 2*a^2*b^2*d + b^4*d))*d + (a^4 + a^3*b + 8*a^2*b^2 + 8*a*b^3)*cosh(d*x + c)^2 + 2*(a^4 + a^3*b + 8*a^
2*b^2 + 8*a*b^3)*cosh(d*x + c)*sinh(d*x + c) + (a^4 + a^3*b + 8*a^2*b^2 + 8*a*b^3)*sinh(d*x + c)^2 - 1/12*sqrt
(1/3)*((a^6 + 3*a^5*b - 6*a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 + 2*b^6)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (
2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^
2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 +
1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(
2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^
3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^
2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d^2 + 6*(a^5 - 2*a^4
*b - 2*a^3*b^2 + 4*a^2*b^3 + a*b^4 - 2*b^5)*d)*sqrt((288*a^4*b^2 + 720*a^2*b^4 - 36*b^6 - (a^8 - 4*a^6*b^2 + 6
*a^4*b^4 - 4*a^2*b^6 + b^8)*((b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d
+ b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b
^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^
4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*
b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1
/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1
) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 + 12*(2*a^6*b - 3*a^4*b^3 + b^7)*((b^2/(a^4*d^2 - 2
*a^2*b^2*d^2 + b^4*d^2) - (2*a^2*b + b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(-1/18*(2*a^2*b
+ b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d
- 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)
^4*d^3))^(1/3) - 9*(-1/18*(2*a^2*b + b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*
d)) + 1/27*(2*a^2*b + b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 1/54*b/(a^4*d^3 - 2*a^2*b^2*d^3 + b^4*d^3) + 1/
54*(a^2 + 8*b^2)*a^2*b/((a^2 - b^2)^4*d^3))^(1/3)*(I*sqrt(3) + 1) + 6*(2*a^2*b + b^3)/(a^4*d - 2*a^2*b^2*d + b
^4*d))*d)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2))) + 36*(2*(a^3 + 4*a^2*b + 5*a*b^2 + 2*b^3)*d*
x*cosh(d*x + c) - (a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)^3)*sinh(d*x + c))/((a^4 - 2*a^2*b^2 + b^4)*d*cosh(
d*x + c)^2 + 2*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)*sinh(d*x + c) + (a^4 - 2*a^2*b^2 + b^4)*d*sinh(d*x + c)
^2)
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giac [A] time = 0.37, size = 223, normalized size = 0.58 \[ -\frac {\frac {12 \, {\left (a + 2 \, b\right )} d x}{a^{2} - 2 \, a b + b^{2}} - \frac {3 \, {\left (2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} - a + b\right )} e^{\left (-2 \, d x\right )}}{a^{2} e^{\left (2 \, c\right )} - 2 \, a b e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}} - \frac {8 \, {\left (2 \, a^{2} b + b^{3}\right )} \log \left ({\left | a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a - b \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac {3 \, e^{\left (2 \, d x + 10 \, c\right )}}{a e^{\left (8 \, c\right )} + b e^{\left (8 \, c\right )}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)^2/(a+b*tanh(d*x+c)^3),x, algorithm="giac")
[Out]
-1/24*(12*(a + 2*b)*d*x/(a^2 - 2*a*b + b^2) - 3*(2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) - a + b)*e^(-2*d*x)
/(a^2*e^(2*c) - 2*a*b*e^(2*c) + b^2*e^(2*c)) - 8*(2*a^2*b + b^3)*log(abs(a*e^(6*d*x + 6*c) + b*e^(6*d*x + 6*c)
+ 3*a*e^(4*d*x + 4*c) - 3*b*e^(4*d*x + 4*c) + 3*a*e^(2*d*x + 2*c) + 3*b*e^(2*d*x + 2*c) + a - b))/(a^4 - 2*a^
2*b^2 + b^4) - 3*e^(2*d*x + 10*c)/(a*e^(8*c) + b*e^(8*c)))/d
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maple [C] time = 0.46, size = 356, normalized size = 0.93 \[ \frac {b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (a \left (2 a^{2}+b^{2}\right ) \textit {\_R}^{5}-3 \textit {\_R}^{4} a^{2} b +6 a \left (a^{2}+b^{2}\right ) \textit {\_R}^{3}+4 b \left (2 a^{2}+b^{2}\right ) \textit {\_R}^{2}-3 a \,b^{2} \textit {\_R} +3 a^{2} b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d \left (a -b \right )^{2} \left (a +b \right )^{2}}+\frac {4}{d \left (8 a +8 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {8}{d \left (16 a +16 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a}{2 d \left (a +b \right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b}{d \left (a +b \right )^{2}}-\frac {4}{d \left (8 a -8 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {8}{d \left (16 a -16 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a}{2 d \left (a -b \right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b}{d \left (a -b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sinh(d*x+c)^2/(a+b*tanh(d*x+c)^3),x)
[Out]
1/3/d*b/(a-b)^2/(a+b)^2*sum((a*(2*a^2+b^2)*_R^5-3*_R^4*a^2*b+6*a*(a^2+b^2)*_R^3+4*b*(2*a^2+b^2)*_R^2-3*a*b^2*_
R+3*a^2*b)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^
2*a+a))+4/d/(8*a+8*b)/(tanh(1/2*d*x+1/2*c)-1)^2+8/d/(16*a+16*b)/(tanh(1/2*d*x+1/2*c)-1)+1/2/d/(a+b)^2*ln(tanh(
1/2*d*x+1/2*c)-1)*a-1/d/(a+b)^2*ln(tanh(1/2*d*x+1/2*c)-1)*b-4/d/(8*a-8*b)/(tanh(1/2*d*x+1/2*c)+1)^2+8/d/(16*a-
16*b)/(tanh(1/2*d*x+1/2*c)+1)-1/2/d/(a-b)^2*ln(tanh(1/2*d*x+1/2*c)+1)*a-1/d/(a-b)^2*ln(tanh(1/2*d*x+1/2*c)+1)*
b
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 4 \, a^{2} b {\left (\frac {-{\left (a - b\right )} \int \frac {1}{{\left (a e^{\left (6 \, c\right )} + b e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 3 \, {\left (a e^{\left (4 \, c\right )} - b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 3 \, {\left (a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + a - b}\,{d x} + x}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac {d x + c}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d}\right )} + 2 \, b^{3} {\left (\frac {-{\left (a - b\right )} \int \frac {1}{{\left (a e^{\left (6 \, c\right )} + b e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 3 \, {\left (a e^{\left (4 \, c\right )} - b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 3 \, {\left (a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + a - b}\,{d x} + x}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac {d x + c}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d}\right )} - \frac {0 \, }{a^{3} + a^{2} b - a b^{2} - b^{3}} + \frac {0 \, }{a^{3} + a^{2} b - a b^{2} - b^{3}} - \frac {0 \, }{a^{3} + a^{2} b - a b^{2} - b^{3}} - \frac {0 \, }{a^{3} + a^{2} b - a b^{2} - b^{3}} + \frac {0 \, }{a^{3} + a^{2} b - a b^{2} - b^{3}} - \frac {{\left (4 \, {\left (a^{2} d e^{\left (2 \, c\right )} - 3 \, a b d e^{\left (2 \, c\right )} + 2 \, b^{2} d e^{\left (2 \, c\right )}\right )} x e^{\left (2 \, d x\right )} + a^{2} + 2 \, a b + b^{2} - {\left (a^{2} e^{\left (4 \, c\right )} - b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )}\right )} e^{\left (-2 \, d x\right )}}{8 \, {\left (a^{3} d e^{\left (2 \, c\right )} + a^{2} b d e^{\left (2 \, c\right )} - a b^{2} d e^{\left (2 \, c\right )} - b^{3} d e^{\left (2 \, c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)^2/(a+b*tanh(d*x+c)^3),x, algorithm="maxima")
[Out]
4*a^2*b*(integrate(((a + b)*e^(4*d*x + 4*c) + 3*(a - b)*e^(2*d*x + 2*c) + 3*a + 3*b)*e^(2*d*x + 2*c)/((a + b)*
e^(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c) + a - b), x)/(a^4 - 2*a^2*b^2 + b^4) -
(d*x + c)/((a^4 - 2*a^2*b^2 + b^4)*d)) + 2*b^3*(integrate(((a + b)*e^(4*d*x + 4*c) + 3*(a - b)*e^(2*d*x + 2*c
) + 3*a + 3*b)*e^(2*d*x + 2*c)/((a + b)*e^(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c
) + a - b), x)/(a^4 - 2*a^2*b^2 + b^4) - (d*x + c)/((a^4 - 2*a^2*b^2 + b^4)*d)) - 8*a^2*b*integrate(e^(4*d*x +
4*c)/((a + b)*e^(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c) + a - b), x)/(a^3 + a^2
*b - a*b^2 - b^3) + 8*a*b^2*integrate(e^(4*d*x + 4*c)/((a + b)*e^(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3
*(a + b)*e^(2*d*x + 2*c) + a - b), x)/(a^3 + a^2*b - a*b^2 - b^3) - 2*b^3*integrate(e^(4*d*x + 4*c)/((a + b)*e
^(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c) + a - b), x)/(a^3 + a^2*b - a*b^2 - b^3
) - 4*a^2*b*integrate(e^(2*d*x + 2*c)/((a + b)*e^(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*
x + 2*c) + a - b), x)/(a^3 + a^2*b - a*b^2 - b^3) + 4*b^3*integrate(e^(2*d*x + 2*c)/((a + b)*e^(6*d*x + 6*c) +
3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c) + a - b), x)/(a^3 + a^2*b - a*b^2 - b^3) - 1/8*(4*(a^2*
d*e^(2*c) - 3*a*b*d*e^(2*c) + 2*b^2*d*e^(2*c))*x*e^(2*d*x) + a^2 + 2*a*b + b^2 - (a^2*e^(4*c) - b^2*e^(4*c))*e
^(4*d*x))*e^(-2*d*x)/(a^3*d*e^(2*c) + a^2*b*d*e^(2*c) - a*b^2*d*e^(2*c) - b^3*d*e^(2*c))
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mupad [B] time = 2.81, size = 2100, normalized size = 5.47 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sinh(c + d*x)^2/(a + b*tanh(c + d*x)^3),x)
[Out]
symsum(log(root(54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^2*b*d^2*z^2 + 27*b^3*d^2*z^2 - 9*b
^2*d*z + b, z, k)*((2304*root(54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^2*b*d^2*z^2 + 27*b^3
*d^2*z^2 - 9*b^2*d*z + b, z, k)*(146*a^5*b^5*d^2 - 133*a^4*b^6*d^2 - 24*a^3*b^7*d^2 - 12*a^6*b^4*d^2 + 22*a^7*
b^3*d^2 + a^8*b^2*d^2 + 32*a^3*b^7*d^2*exp(2*root(54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^
2*b*d^2*z^2 + 27*b^3*d^2*z^2 - 9*b^2*d*z + b, z, k))*exp(2*d*x) + 577*a^4*b^6*d^2*exp(2*root(54*a^2*b^2*d^3*z^
3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^2*b*d^2*z^2 + 27*b^3*d^2*z^2 - 9*b^2*d*z + b, z, k))*exp(2*d*x) + 5
48*a^5*b^5*d^2*exp(2*root(54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^2*b*d^2*z^2 + 27*b^3*d^2
*z^2 - 9*b^2*d*z + b, z, k))*exp(2*d*x) + 70*a^6*b^4*d^2*exp(2*root(54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*a
^4*d^3*z^3 + 54*a^2*b*d^2*z^2 + 27*b^3*d^2*z^2 - 9*b^2*d*z + b, z, k))*exp(2*d*x) + 68*a^7*b^3*d^2*exp(2*root(
54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^2*b*d^2*z^2 + 27*b^3*d^2*z^2 - 9*b^2*d*z + b, z, k
))*exp(2*d*x) + a^8*b^2*d^2*exp(2*root(54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^2*b*d^2*z^2
+ 27*b^3*d^2*z^2 - 9*b^2*d*z + b, z, k))*exp(2*d*x)))/((a + b)^8*(a - b)^2*(a^2 - 2*a*b + b^2)) + (1536*(24*a
^3*b^8*d + 105*a^4*b^7*d - 156*a^5*b^6*d + 51*a^6*b^5*d - 30*a^7*b^4*d + 6*a^8*b^3*d - 32*a^3*b^8*d*exp(2*root
(54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^2*b*d^2*z^2 + 27*b^3*d^2*z^2 - 9*b^2*d*z + b, z,
k))*exp(2*d*x) - 509*a^4*b^7*d*exp(2*root(54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^2*b*d^2*
z^2 + 27*b^3*d^2*z^2 - 9*b^2*d*z + b, z, k))*exp(2*d*x) - 350*a^5*b^6*d*exp(2*root(54*a^2*b^2*d^3*z^3 - 27*b^4
*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^2*b*d^2*z^2 + 27*b^3*d^2*z^2 - 9*b^2*d*z + b, z, k))*exp(2*d*x) + 64*a^6*b^5*
d*exp(2*root(54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^2*b*d^2*z^2 + 27*b^3*d^2*z^2 - 9*b^2*
d*z + b, z, k))*exp(2*d*x) - 50*a^7*b^4*d*exp(2*root(54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54
*a^2*b*d^2*z^2 + 27*b^3*d^2*z^2 - 9*b^2*d*z + b, z, k))*exp(2*d*x) + 13*a^8*b^3*d*exp(2*root(54*a^2*b^2*d^3*z^
3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^2*b*d^2*z^2 + 27*b^3*d^2*z^2 - 9*b^2*d*z + b, z, k))*exp(2*d*x)))/(
(a + b)^3*(a^2 - b^2)*(a - b)*(a^2 - 2*a*b + b^2)*(3*a*b^2 + 3*a^2*b + a^3 + b^3)^2)) + (256*(72*a^5*b^5 - 45*
a^4*b^6 - 24*a^3*b^7 - 9*a^6*b^4 + 6*a^7*b^3 + 32*a^3*b^7*exp(2*root(54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*
a^4*d^3*z^3 + 54*a^2*b*d^2*z^2 + 27*b^3*d^2*z^2 - 9*b^2*d*z + b, z, k))*exp(2*d*x) + 393*a^4*b^6*exp(2*root(54
*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^2*b*d^2*z^2 + 27*b^3*d^2*z^2 - 9*b^2*d*z + b, z, k))
*exp(2*d*x) + 86*a^5*b^5*exp(2*root(54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^2*b*d^2*z^2 +
27*b^3*d^2*z^2 - 9*b^2*d*z + b, z, k))*exp(2*d*x) + 57*a^6*b^4*exp(2*root(54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3
- 27*a^4*d^3*z^3 + 54*a^2*b*d^2*z^2 + 27*b^3*d^2*z^2 - 9*b^2*d*z + b, z, k))*exp(2*d*x) + 8*a^7*b^3*exp(2*root
(54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^2*b*d^2*z^2 + 27*b^3*d^2*z^2 - 9*b^2*d*z + b, z,
k))*exp(2*d*x)))/((a + b)^2*(a^2 - b^2)*(a - b)*(a^2 - 2*a*b + b^2)*(2*a*b + a^2 + b^2)^2*(3*a*b^2 + 3*a^2*b +
a^3 + b^3)))*root(54*a^2*b^2*d^3*z^3 - 27*b^4*d^3*z^3 - 27*a^4*d^3*z^3 + 54*a^2*b*d^2*z^2 + 27*b^3*d^2*z^2 -
9*b^2*d*z + b, z, k), k, 1, 3) - (x*(a + 2*b))/(2*(a - b)^2) + exp(2*c + 2*d*x)/(8*d*(a + b)) - exp(- 2*c - 2*
d*x)/(8*d*(a - b))
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)**2/(a+b*tanh(d*x+c)**3),x)
[Out]
Timed out
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