3.1.73 \(\int \frac {\sinh ^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx\) [73]

Optimal. Leaf size=491 \[ -\frac {a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{4/3} b^{2/3}-b^2\right ) \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \left (a^{4/3}+a^{2/3} b^{2/3}+b^{4/3}\right )^3 d}-\frac {3 a (a-5 b) \log (1-\tanh (c+d x))}{16 (a+b)^3 d}+\frac {3 a (a+5 b) \log (1+\tanh (c+d x))}{16 (a-b)^3 d}-\frac {a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac {a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 \left (a^2-b^2\right )^3 d}-\frac {a^2 b \left (a^2+2 b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{\left (a^2-b^2\right )^3 d}+\frac {1}{16 (a+b) d (1-\tanh (c+d x))^2}-\frac {5 a-b}{16 (a+b)^2 d (1-\tanh (c+d x))}-\frac {1}{16 (a-b) d (1+\tanh (c+d x))^2}+\frac {5 a+b}{16 (a-b)^2 d (1+\tanh (c+d x))} \]

[Out]

-3/16*a*(a-5*b)*ln(1-tanh(d*x+c))/(a+b)^3/d+3/16*a*(a+5*b)*ln(1+tanh(d*x+c))/(a-b)^3/d-1/3*a^(2/3)*b^(1/3)*(a^
4+7*a^2*b^2+b^4+3*a^(2/3)*b^(4/3)*(2*a^2+b^2))*ln(a^(1/3)+b^(1/3)*tanh(d*x+c))/(a^2-b^2)^3/d+1/6*a^(2/3)*b^(1/
3)*(a^4+7*a^2*b^2+b^4+3*a^(2/3)*b^(4/3)*(2*a^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)^3/d-a^2*b*(a^2+2*b^2)*ln(a+b*tanh(d*x+c)^3)/(a^2-b^2)^3/d-1/3*a^(2/3)*b^(1/3)*(a^2+3*a^(4/3)*b^
(2/3)-b^2)*arctan(1/3*(a^(1/3)-2*b^(1/3)*tanh(d*x+c))/a^(1/3)*3^(1/2))/(a^(4/3)+a^(2/3)*b^(2/3)+b^(4/3))^3/d*3
^(1/2)+1/16/(a+b)/d/(1-tanh(d*x+c))^2+1/16*(-5*a+b)/(a+b)^2/d/(1-tanh(d*x+c))-1/16/(a-b)/d/(1+tanh(d*x+c))^2+1
/16*(5*a+b)/(a-b)^2/d/(1+tanh(d*x+c))

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Rubi [A]
time = 0.65, antiderivative size = 491, 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 = {3744, 6857, 1885, 1874, 31, 648, 631, 210, 642, 266} \begin {gather*} -\frac {a^2 b \left (a^2+2 b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{d \left (a^2-b^2\right )^3}-\frac {a^{2/3} \sqrt [3]{b} \left (3 a^{4/3} b^{2/3}+a^2-b^2\right ) \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} d \left (a^{2/3} b^{2/3}+a^{4/3}+b^{4/3}\right )^3}+\frac {a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )+b^4\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 )^3}-\frac {a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )+b^4\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 d \left (a^2-b^2\right )^3}-\frac {5 a-b}{16 d (a+b)^2 (1-\tanh (c+d x))}+\frac {5 a+b}{16 d (a-b)^2 (\tanh (c+d x)+1)}+\frac {1}{16 d (a+b) (1-\tanh (c+d x))^2}-\frac {1}{16 d (a-b) (\tanh (c+d x)+1)^2}-\frac {3 a (a-5 b) \log (1-\tanh (c+d x))}{16 d (a+b)^3}+\frac {3 a (a+5 b) \log (\tanh (c+d x)+1)}{16 d (a-b)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^4/(a + b*Tanh[c + d*x]^3),x]

[Out]

-((a^(2/3)*b^(1/3)*(a^2 + 3*a^(4/3)*b^(2/3) - b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tanh[c + d*x])/(Sqrt[3]*a^(1/3)
)])/(Sqrt[3]*(a^(4/3) + a^(2/3)*b^(2/3) + b^(4/3))^3*d)) - (3*a*(a - 5*b)*Log[1 - Tanh[c + d*x]])/(16*(a + b)^
3*d) + (3*a*(a + 5*b)*Log[1 + Tanh[c + d*x]])/(16*(a - b)^3*d) - (a^(2/3)*b^(1/3)*(a^4 + 7*a^2*b^2 + b^4 + 3*a
^(2/3)*b^(4/3)*(2*a^2 + b^2))*Log[a^(1/3) + b^(1/3)*Tanh[c + d*x]])/(3*(a^2 - b^2)^3*d) + (a^(2/3)*b^(1/3)*(a^
4 + 7*a^2*b^2 + b^4 + 3*a^(2/3)*b^(4/3)*(2*a^2 + b^2))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tanh[c + d*x] + b^(2/3)*T
anh[c + d*x]^2])/(6*(a^2 - b^2)^3*d) - (a^2*b*(a^2 + 2*b^2)*Log[a + b*Tanh[c + d*x]^3])/((a^2 - b^2)^3*d) + 1/
(16*(a + b)*d*(1 - Tanh[c + d*x])^2) - (5*a - b)/(16*(a + b)^2*d*(1 - Tanh[c + d*x])) - 1/(16*(a - b)*d*(1 + T
anh[c + d*x])^2) + (5*a + b)/(16*(a - b)^2*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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 266

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 631

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 642

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 648

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 1874

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 1885

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 3744

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 6857

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 ^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^3 \left (a+b x^3\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{8 (a+b) (-1+x)^3}+\frac {-5 a+b}{16 (a+b)^2 (-1+x)^2}-\frac {3 a (a-5 b)}{16 (a+b)^3 (-1+x)}+\frac {1}{8 (a-b) (1+x)^3}+\frac {-5 a-b}{16 (a-b)^2 (1+x)^2}+\frac {3 a (a+5 b)}{16 (a-b)^3 (1+x)}+\frac {a b \left (-3 a b \left (2 a^2+b^2\right )+\left (a^4+7 a^2 b^2+b^4\right ) x-3 a b \left (a^2+2 b^2\right ) x^2\right )}{\left (a^2-b^2\right )^3 \left (a+b x^3\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {3 a (a-5 b) \log (1-\tanh (c+d x))}{16 (a+b)^3 d}+\frac {3 a (a+5 b) \log (1+\tanh (c+d x))}{16 (a-b)^3 d}+\frac {1}{16 (a+b) d (1-\tanh (c+d x))^2}-\frac {5 a-b}{16 (a+b)^2 d (1-\tanh (c+d x))}-\frac {1}{16 (a-b) d (1+\tanh (c+d x))^2}+\frac {5 a+b}{16 (a-b)^2 d (1+\tanh (c+d x))}+\frac {(a b) \text {Subst}\left (\int \frac {-3 a b \left (2 a^2+b^2\right )+\left (a^4+7 a^2 b^2+b^4\right ) x-3 a b \left (a^2+2 b^2\right ) x^2}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{\left (a^2-b^2\right )^3 d}\\ &=-\frac {3 a (a-5 b) \log (1-\tanh (c+d x))}{16 (a+b)^3 d}+\frac {3 a (a+5 b) \log (1+\tanh (c+d x))}{16 (a-b)^3 d}+\frac {1}{16 (a+b) d (1-\tanh (c+d x))^2}-\frac {5 a-b}{16 (a+b)^2 d (1-\tanh (c+d x))}-\frac {1}{16 (a-b) d (1+\tanh (c+d x))^2}+\frac {5 a+b}{16 (a-b)^2 d (1+\tanh (c+d x))}+\frac {(a b) \text {Subst}\left (\int \frac {-3 a b \left (2 a^2+b^2\right )+\left (a^4+7 a^2 b^2+b^4\right ) x}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{\left (a^2-b^2\right )^3 d}-\frac {\left (3 a^2 b^2 \left (a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{\left (a^2-b^2\right )^3 d}\\ &=-\frac {3 a (a-5 b) \log (1-\tanh (c+d x))}{16 (a+b)^3 d}+\frac {3 a (a+5 b) \log (1+\tanh (c+d x))}{16 (a-b)^3 d}-\frac {a^2 b \left (a^2+2 b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{\left (a^2-b^2\right )^3 d}+\frac {1}{16 (a+b) d (1-\tanh (c+d x))^2}-\frac {5 a-b}{16 (a+b)^2 d (1-\tanh (c+d x))}-\frac {1}{16 (a-b) d (1+\tanh (c+d x))^2}+\frac {5 a+b}{16 (a-b)^2 d (1+\tanh (c+d x))}+\frac {\left (\sqrt [3]{a} b^{2/3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{a} \left (-6 a b^{4/3} \left (2 a^2+b^2\right )+\sqrt [3]{a} \left (a^4+7 a^2 b^2+b^4\right )\right )+\sqrt [3]{b} \left (3 a b^{4/3} \left (2 a^2+b^2\right )+\sqrt [3]{a} \left (a^4+7 a^2 b^2+b^4\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 \left (a^2-b^2\right )^3 d}-\frac {\left (a^{2/3} b^{2/3} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\right )\right ) \text {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 )^3 d}\\ &=-\frac {3 a (a-5 b) \log (1-\tanh (c+d x))}{16 (a+b)^3 d}+\frac {3 a (a+5 b) \log (1+\tanh (c+d x))}{16 (a-b)^3 d}-\frac {a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}-\frac {a^2 b \left (a^2+2 b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{\left (a^2-b^2\right )^3 d}+\frac {1}{16 (a+b) d (1-\tanh (c+d x))^2}-\frac {5 a-b}{16 (a+b)^2 d (1-\tanh (c+d x))}-\frac {1}{16 (a-b) d (1+\tanh (c+d x))^2}+\frac {5 a+b}{16 (a-b)^2 d (1+\tanh (c+d x))}+\frac {\left (a b^{2/3} \left (a^2+3 a^{4/3} b^{2/3}-b^2\right )\right ) \text {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^{4/3}+a^{2/3} b^{2/3}+b^{4/3}\right )^3 d}+\frac {\left (a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\right )\right ) \text {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 )^3 d}\\ &=-\frac {3 a (a-5 b) \log (1-\tanh (c+d x))}{16 (a+b)^3 d}+\frac {3 a (a+5 b) \log (1+\tanh (c+d x))}{16 (a-b)^3 d}-\frac {a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac {a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\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 )^3 d}-\frac {a^2 b \left (a^2+2 b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{\left (a^2-b^2\right )^3 d}+\frac {1}{16 (a+b) d (1-\tanh (c+d x))^2}-\frac {5 a-b}{16 (a+b)^2 d (1-\tanh (c+d x))}-\frac {1}{16 (a-b) d (1+\tanh (c+d x))^2}+\frac {5 a+b}{16 (a-b)^2 d (1+\tanh (c+d x))}+\frac {\left (a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{4/3} b^{2/3}-b^2\right )\right ) \text {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^{4/3}+a^{2/3} b^{2/3}+b^{4/3}\right )^3 d}\\ &=-\frac {a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{4/3} b^{2/3}-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^{4/3}+a^{2/3} b^{2/3}+b^{4/3}\right )^3 d}-\frac {3 a (a-5 b) \log (1-\tanh (c+d x))}{16 (a+b)^3 d}+\frac {3 a (a+5 b) \log (1+\tanh (c+d x))}{16 (a-b)^3 d}-\frac {a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac {a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\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 )^3 d}-\frac {a^2 b \left (a^2+2 b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{\left (a^2-b^2\right )^3 d}+\frac {1}{16 (a+b) d (1-\tanh (c+d x))^2}-\frac {5 a-b}{16 (a+b)^2 d (1-\tanh (c+d x))}-\frac {1}{16 (a-b) d (1+\tanh (c+d x))^2}+\frac {5 a+b}{16 (a-b)^2 d (1+\tanh (c+d x))}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 3.21, size = 645, normalized size = 1.31 \begin {gather*} \frac {-32 a b \text {RootSum}\left [a-b+3 a \text {$\#$1}+3 b \text {$\#$1}+3 a \text {$\#$1}^2-3 b \text {$\#$1}^2+a \text {$\#$1}^3+b \text {$\#$1}^3\&,\frac {-6 a^3 c-12 a b^2 c-6 a^3 d x-12 a b^2 d x+3 a^3 \log \left (e^{2 (c+d x)}-\text {$\#$1}\right )+6 a b^2 \log \left (e^{2 (c+d x)}-\text {$\#$1}\right )-8 a^3 c \text {$\#$1}+4 a^2 b c \text {$\#$1}+8 a b^2 c \text {$\#$1}-4 b^3 c \text {$\#$1}-8 a^3 d x \text {$\#$1}+4 a^2 b d x \text {$\#$1}+8 a b^2 d x \text {$\#$1}-4 b^3 d x \text {$\#$1}+4 a^3 \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}-2 a^2 b \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}-4 a b^2 \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}+2 b^3 \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}-10 a^3 c \text {$\#$1}^2+20 a^2 b c \text {$\#$1}^2-20 a b^2 c \text {$\#$1}^2+4 b^3 c \text {$\#$1}^2-10 a^3 d x \text {$\#$1}^2+20 a^2 b d x \text {$\#$1}^2-20 a b^2 d x \text {$\#$1}^2+4 b^3 d x \text {$\#$1}^2+5 a^3 \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}^2-10 a^2 b \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}^2+10 a b^2 \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}^2-2 b^3 \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}^2}{a-b+2 a \text {$\#$1}+2 b \text {$\#$1}+a \text {$\#$1}^2-b \text {$\#$1}^2}\&\right ]+3 \left (4 b \left (5 a^3+5 a^2 b+a b^2+b^3\right ) \cosh (2 (c+d x))-(a-b) b (a+b)^2 \cosh (4 (c+d x))-8 a \left (a^3+a^2 b+2 a b^2+2 b^3\right ) \sinh (2 (c+d x))+a (a-b) \left (12 \left (a^2-6 a b+5 b^2\right ) (c+d x)+(a+b)^2 \sinh (4 (c+d x))\right )\right )}{96 (a-b)^2 (a+b)^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^4/(a + b*Tanh[c + d*x]^3),x]

[Out]

(-32*a*b*RootSum[a - b + 3*a*#1 + 3*b*#1 + 3*a*#1^2 - 3*b*#1^2 + a*#1^3 + b*#1^3 & , (-6*a^3*c - 12*a*b^2*c -
6*a^3*d*x - 12*a*b^2*d*x + 3*a^3*Log[E^(2*(c + d*x)) - #1] + 6*a*b^2*Log[E^(2*(c + d*x)) - #1] - 8*a^3*c*#1 +
4*a^2*b*c*#1 + 8*a*b^2*c*#1 - 4*b^3*c*#1 - 8*a^3*d*x*#1 + 4*a^2*b*d*x*#1 + 8*a*b^2*d*x*#1 - 4*b^3*d*x*#1 + 4*a
^3*Log[E^(2*(c + d*x)) - #1]*#1 - 2*a^2*b*Log[E^(2*(c + d*x)) - #1]*#1 - 4*a*b^2*Log[E^(2*(c + d*x)) - #1]*#1
+ 2*b^3*Log[E^(2*(c + d*x)) - #1]*#1 - 10*a^3*c*#1^2 + 20*a^2*b*c*#1^2 - 20*a*b^2*c*#1^2 + 4*b^3*c*#1^2 - 10*a
^3*d*x*#1^2 + 20*a^2*b*d*x*#1^2 - 20*a*b^2*d*x*#1^2 + 4*b^3*d*x*#1^2 + 5*a^3*Log[E^(2*(c + d*x)) - #1]*#1^2 -
10*a^2*b*Log[E^(2*(c + d*x)) - #1]*#1^2 + 10*a*b^2*Log[E^(2*(c + d*x)) - #1]*#1^2 - 2*b^3*Log[E^(2*(c + d*x))
- #1]*#1^2)/(a - b + 2*a*#1 + 2*b*#1 + a*#1^2 - b*#1^2) & ] + 3*(4*b*(5*a^3 + 5*a^2*b + a*b^2 + b^3)*Cosh[2*(c
+ d*x)] - (a - b)*b*(a + b)^2*Cosh[4*(c + d*x)] - 8*a*(a^3 + a^2*b + 2*a*b^2 + 2*b^3)*Sinh[2*(c + d*x)] + a*(
a - b)*(12*(a^2 - 6*a*b + 5*b^2)*(c + d*x) + (a + b)^2*Sinh[4*(c + d*x)])))/(96*(a - b)^2*(a + b)^3*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 4.42, size = 452, normalized size = 0.92 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^4/(a+b*tanh(d*x+c)^3),x,method=_RETURNVERBOSE)

[Out]

1/d*(-8/(32*a-32*b)/(tanh(1/2*d*x+1/2*c)+1)^4+32/(64*a-64*b)/(tanh(1/2*d*x+1/2*c)+1)^3-1/8*(-a-5*b)/(a-b)^2/(t
anh(1/2*d*x+1/2*c)+1)^2-1/8*(3*a+3*b)/(a-b)^2/(tanh(1/2*d*x+1/2*c)+1)+3/8*a*(a+5*b)/(a-b)^3*ln(tanh(1/2*d*x+1/
2*c)+1)-1/3*a*b/(a-b)^3/(a+b)^3*sum((3*a^2*(a^2+2*b^2)*_R^5+3*a*b*(-2*a^2-b^2)*_R^4+2*(4*a^4+13*a^2*b^2+b^4)*_
R^3+12*a*b*(a^2+2*b^2)*_R^2+(a^4-8*a^2*b^2-2*b^4)*_R+6*a^3*b+3*a*b^3)/(_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))+8/(32*a+32*b)/(tanh(1/2*d*x+1/2*c)-1)^4+32/
(64*a+64*b)/(tanh(1/2*d*x+1/2*c)-1)^3-1/8*(a-5*b)/(a+b)^2/(tanh(1/2*d*x+1/2*c)-1)^2-1/8*(3*a-3*b)/(a+b)^2/(tan
h(1/2*d*x+1/2*c)-1)-3/8*a*(a-5*b)/(a+b)^3*ln(tanh(1/2*d*x+1/2*c)-1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4/(a+b*tanh(d*x+c)^3),x, algorithm="maxima")

[Out]

-6*a^4*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^6 - 3*a^4*b^2 + 3*a^2
*b^4 - b^6) - (d*x + c)/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*d)) - 12*a^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^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6) - (d*x + c)/((a^6 - 3*a^4*b^2
+ 3*a^2*b^4 - b^6)*d)) + 10*a^4*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^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5) - 20*a^3*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^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5) + 20*a^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^5 + a^4*b - 2*a^3*b^2
- 2*a^2*b^3 + a*b^4 + b^5) - 4*a*b^4*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^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5) + 8*a^4*b*i
ntegrate(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^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5) - 4*a^3*b^2*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^5 + a^4*b - 2*a^3*b^2 -
2*a^2*b^3 + a*b^4 + b^5) - 8*a^2*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^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5) + 4*a*b^4*i
ntegrate(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^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5) - 1/64*(a^4 + 2*a^3*b - 2*a*b^3 - b^4 - 24*(a^4*d
*e^(4*c) - 7*a^3*b*d*e^(4*c) + 11*a^2*b^2*d*e^(4*c) - 5*a*b^3*d*e^(4*c))*x*e^(4*d*x) - (a^4*e^(8*c) - 2*a^2*b^
2*e^(8*c) + b^4*e^(8*c))*e^(8*d*x) + 4*(2*a^4*e^(6*c) - 3*a^3*b*e^(6*c) - a^2*b^2*e^(6*c) + 3*a*b^3*e^(6*c) -
b^4*e^(6*c))*e^(6*d*x) - 4*(2*a^4*e^(2*c) + 7*a^3*b*e^(2*c) + 9*a^2*b^2*e^(2*c) + 5*a*b^3*e^(2*c) + b^4*e^(2*c
))*e^(2*d*x))*e^(-4*d*x)/(a^5*d*e^(4*c) + a^4*b*d*e^(4*c) - 2*a^3*b^2*d*e^(4*c) - 2*a^2*b^3*d*e^(4*c) + a*b^4*
d*e^(4*c) + b^5*d*e^(4*c))

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Fricas [C] Result contains complex when optimal does not.
time = 1.49, size = 17123, normalized size = 34.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4/(a+b*tanh(d*x+c)^3),x, algorithm="fricas")

[Out]

1/192*(3*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(d*x + c)^8 + 24*(a^5 - a^4*b - 2*a^3*b^2 + 2
*a^2*b^3 + a*b^4 - b^5)*cosh(d*x + c)*sinh(d*x + c)^7 + 3*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*
sinh(d*x + c)^8 + 72*(a^5 + 8*a^4*b + 18*a^3*b^2 + 16*a^2*b^3 + 5*a*b^4)*d*x*cosh(d*x + c)^4 - 12*(2*a^5 - 5*a
^4*b + 2*a^3*b^2 + 4*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^6 - 12*(2*a^5 - 5*a^4*b + 2*a^3*b^2 + 4*a^2*b^3 -
4*a*b^4 + b^5 - 7*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 24*(7
*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(d*x + c)^3 - 3*(2*a^5 - 5*a^4*b + 2*a^3*b^2 + 4*a^2*
b^3 - 4*a*b^4 + b^5)*cosh(d*x + c))*sinh(d*x + c)^5 - 3*a^5 - 3*a^4*b + 6*a^3*b^2 + 6*a^2*b^3 - 3*a*b^4 - 3*b^
5 + 6*(35*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(d*x + c)^4 + 12*(a^5 + 8*a^4*b + 18*a^3*b^2
+ 16*a^2*b^3 + 5*a*b^4)*d*x - 30*(2*a^5 - 5*a^4*b + 2*a^3*b^2 + 4*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^2)*s
inh(d*x + c)^4 + 24*(7*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(d*x + c)^5 + 12*(a^5 + 8*a^4*b
+ 18*a^3*b^2 + 16*a^2*b^3 + 5*a*b^4)*d*x*cosh(d*x + c) - 10*(2*a^5 - 5*a^4*b + 2*a^3*b^2 + 4*a^2*b^3 - 4*a*b^
4 + b^5)*cosh(d*x + c)^3)*sinh(d*x + c)^3 + 12*(2*a^5 + 5*a^4*b + 2*a^3*b^2 - 4*a^2*b^3 - 4*a*b^4 - b^5)*cosh(
d*x + c)^2 + 32*((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*d*cosh(d*x + c)^4 + 4*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)
*d*cosh(d*x + c)^3*sinh(d*x + c) + 6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*d*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4
*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*d*s
inh(d*x + c)^4)*(6*(1/2)^(2/3)*(a^2*b^2/(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2) - 3*(a^4*b + 2*a^2
*b^3)^2/(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*d)^2)*(-I*sqrt(3) + 1)/(27*(a^4*b + 2*a^2*b^3)*a^2*b^2/((a^6*
d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)*(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*d)) - a^2*b/(a^6*d^3 -
3*a^4*b^2*d^3 + 3*a^2*b^4*d^3 - b^6*d^3) - 54*(a^4*b + 2*a^2*b^3)^3/(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*
d)^3 - (a^6 + 24*a^4*b^2 + 3*a^2*b^4 - b^6)*a^2*b/((a^2 - b^2)^6*d^3))^(1/3) - (1/2)^(1/3)*(27*(a^4*b + 2*a^2*
b^3)*a^2*b^2/((a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)*(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*d))
- a^2*b/(a^6*d^3 - 3*a^4*b^2*d^3 + 3*a^2*b^4*d^3 - b^6*d^3) - 54*(a^4*b + 2*a^2*b^3)^3/(a^6*d - 3*a^4*b^2*d +
3*a^2*b^4*d - b^6*d)^3 - (a^6 + 24*a^4*b^2 + 3*a^2*b^4 - b^6)*a^2*b/((a^2 - b^2)^6*d^3))^(1/3)*(I*sqrt(3) + 1
) - 6*(a^4*b + 2*a^2*b^3)/(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*d))*log(-a^7 - 4*a^6*b - 14*a^5*b^2 - 40*a^
4*b^3 - 11*a^3*b^4 - 10*a^2*b^5 - a*b^6 - 1/2*(a^9 + 5*a^8*b - a^7*b^2 - 14*a^6*b^3 - 3*a^5*b^4 + 12*a^4*b^5 +
5*a^3*b^6 - 2*a^2*b^7 - 2*a*b^8 - b^9)*(6*(1/2)^(2/3)*(a^2*b^2/(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6
*d^2) - 3*(a^4*b + 2*a^2*b^3)^2/(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*d)^2)*(-I*sqrt(3) + 1)/(27*(a^4*b + 2
*a^2*b^3)*a^2*b^2/((a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)*(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^
6*d)) - a^2*b/(a^6*d^3 - 3*a^4*b^2*d^3 + 3*a^2*b^4*d^3 - b^6*d^3) - 54*(a^4*b + 2*a^2*b^3)^3/(a^6*d - 3*a^4*b^
2*d + 3*a^2*b^4*d - b^6*d)^3 - (a^6 + 24*a^4*b^2 + 3*a^2*b^4 - b^6)*a^2*b/((a^2 - b^2)^6*d^3))^(1/3) - (1/2)^(
1/3)*(27*(a^4*b + 2*a^2*b^3)*a^2*b^2/((a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)*(a^6*d - 3*a^4*b^2*d
+ 3*a^2*b^4*d - b^6*d)) - a^2*b/(a^6*d^3 - 3*a^4*b^2*d^3 + 3*a^2*b^4*d^3 - b^6*d^3) - 54*(a^4*b + 2*a^2*b^3)^
3/(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*d)^3 - (a^6 + 24*a^4*b^2 + 3*a^2*b^4 - b^6)*a^2*b/((a^2 - b^2)^6*d^
3))^(1/3)*(I*sqrt(3) + 1) - 6*(a^4*b + 2*a^2*b^3)/(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*d))^2*d^2 - (a^8 +
5*a^7*b + 45*a^6*b^2 + 45*a^5*b^3 + 96*a^4*b^4 + 30*a^3*b^5 + 20*a^2*b^6 + a*b^7)*(6*(1/2)^(2/3)*(a^2*b^2/(a^6
*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2) - 3*(a^4*b + 2*a^2*b^3)^2/(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d -
b^6*d)^2)*(-I*sqrt(3) + 1)/(27*(a^4*b + 2*a^2*b^3)*a^2*b^2/((a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^
2)*(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*d)) - a^2*b/(a^6*d^3 - 3*a^4*b^2*d^3 + 3*a^2*b^4*d^3 - b^6*d^3) -
54*(a^4*b + 2*a^2*b^3)^3/(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*d)^3 - (a^6 + 24*a^4*b^2 + 3*a^2*b^4 - b^6)*
a^2*b/((a^2 - b^2)^6*d^3))^(1/3) - (1/2)^(1/3)*(27*(a^4*b + 2*a^2*b^3)*a^2*b^2/((a^6*d^2 - 3*a^4*b^2*d^2 + 3*a
^2*b^4*d^2 - b^6*d^2)*(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*d)) - a^2*b/(a^6*d^3 - 3*a^4*b^2*d^3 + 3*a^2*b^
4*d^3 - b^6*d^3) - 54*(a^4*b + 2*a^2*b^3)^3/(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*d)^3 - (a^6 + 24*a^4*b^2
+ 3*a^2*b^4 - b^6)*a^2*b/((a^2 - b^2)^6*d^3))^(1/3)*(I*sqrt(3) + 1) - 6*(a^4*b + 2*a^2*b^3)/(a^6*d - 3*a^4*b^2
*d + 3*a^2*b^4*d - b^6*d))*d - (a^7 + 24*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(d*x + c)^2 - 2*(a^7 + 24*a^5*b^2 +
3*a^3*b^4 - a*b^6)*cosh(d*x + c)*sinh(d*x + c) - (a^7 + 24*a^5*b^2 + 3*a^3*b^4 - a*b^6)*sinh(d*x + c)^2) + 12*
(7*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**4/(a+b*tanh(d*x+c)**3),x)

[Out]

Timed out

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Giac [A]
time = 0.67, size = 338, normalized size = 0.69 \begin {gather*} \frac {\frac {24 \, {\left (a^{2} + 5 \, a b\right )} {\left (d x + c\right )}}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {{\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 90 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} - 2 \, a b + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {64 \, {\left (a^{4} b + 2 \, a^{2} 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^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )}}{a^{2} + 2 \, a b + b^{2}}}{64 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4/(a+b*tanh(d*x+c)^3),x, algorithm="giac")

[Out]

1/64*(24*(a^2 + 5*a*b)*(d*x + c)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) - (18*a^2*e^(4*d*x + 4*c) + 90*a*b*e^(4*d*x +
4*c) - 8*a^2*e^(2*d*x + 2*c) + 4*a*b*e^(2*d*x + 2*c) + 4*b^2*e^(2*d*x + 2*c) + a^2 - 2*a*b + b^2)*e^(-4*d*x -
4*c)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) - 64*(a^4*b + 2*a^2*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^6 - 3*a^4*
b^2 + 3*a^2*b^4 - b^6) + (a*e^(4*d*x + 4*c) + b*e^(4*d*x + 4*c) - 8*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c))/(
a^2 + 2*a*b + b^2))/d

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Mupad [B]
time = 4.36, size = 2500, normalized size = 5.09 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^4/(a + b*tanh(c + d*x)^3),x)

[Out]

symsum(log(- root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*
z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k)*((96*(a^2*b^10*d + 20*a^3*b^9*d - 89*a^4*b^8*d + 270*a^
5*b^7*d - 417*a^6*b^6*d + 408*a^7*b^5*d - 190*a^8*b^4*d + 58*a^9*b^3*d - 7*a^10*b^2*d - a^2*b^10*d*exp(2*root(
81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2
*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k))*exp(2*d*x) - 52*a^3*b^9*d*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3
*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k
))*exp(2*d*x) + 59*a^4*b^8*d*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*
z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k))*exp(2*d*x) - 218*a^5*b^7*d*exp(2
*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4
*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k))*exp(2*d*x) + 241*a^6*b^6*d*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*
b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*
b, z, k))*exp(2*d*x) + 220*a^7*b^5*d*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*
a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k))*exp(2*d*x) - 298*a^8*b^4
*d*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2
- 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k))*exp(2*d*x) + 50*a^9*b^3*d*exp(2*root(81*a^4*b^2*d^3*z^3 -
81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z
- a^2*b, z, k))*exp(2*d*x) - a^10*b^2*d*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 -
27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k))*exp(2*d*x)))/((a + b
)*(a^2 - b^2)*(a*b^2 - a^2*b - a^3 + b^3)*(3*a*b^2 + 3*a^2*b + a^3 + b^3)*(a*b^4 + a^4*b + a^5 + b^5 - 2*a^2*b
^3 - 2*a^3*b^2)) - (288*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a
^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k)*(a^7*b*d^2 - 8*a^2*b^6*d^2 + 16*a^3*b^5*d^2
- 41*a^4*b^4*d^2 + 37*a^5*b^3*d^2 - 5*a^6*b^2*d^2 + 18*a^2*b^6*d^2*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*
d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z
, k))*exp(2*d*x) + 14*a^3*b^5*d^2*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6
*d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k))*exp(2*d*x) + 79*a^4*b^4*d^2
*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2 -
81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k))*exp(2*d*x) + 81*a^5*b^3*d^2*exp(2*root(81*a^4*b^2*d^3*z^3 -
81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z
- a^2*b, z, k))*exp(2*d*x) - a^6*b^2*d^2*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3
- 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k))*exp(2*d*x) + a^7*b*
d^2*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2
- 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k))*exp(2*d*x)))/((a + b)^2*(a - b)*(a*b^2 - a^2*b - a^3 + b^
3)*(3*a*b^2 + 3*a^2*b + a^3 + b^3))) - (32*(22*a^4*b^7 - 4*a^3*b^8 - 68*a^5*b^6 + 85*a^6*b^5 - 56*a^7*b^4 + 10
*a^8*b^3 + 2*a^9*b^2 + 6*a^3*b^8*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*
d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k))*exp(2*d*x) - 10*a^4*b^7*exp(
2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^
4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k))*exp(2*d*x) + 54*a^5*b^6*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^
4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b,
z, k))*exp(2*d*x) - 101*a^6*b^5*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*
d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k))*exp(2*d*x) + 56*a^7*b^4*exp(
2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^
4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k))*exp(2*d*x) - 12*a^8*b^3*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^
4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b,
z, k))*exp(2*d*x) + 4*a^9*b^2*exp(2*root(81*a^4*b^2*d^3*z^3 - 81*a^2*b^4*d^3*z^3 + 27*b^6*d^3*z^3 - 27*a^6*d^
3*z^3 - 162*a^2*b^3*d^2*z^2 - 81*a^4*b*d^2*z^2 - 27*a^2*b^2*d*z - a^2*b, z, k))*exp(2*d*x)))/((a + b)*(3*a*b^2
+ 3*a^2*b + a^3 + b^3)*(a*b^4 + a^4*b + a^5 + ...

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