3.2.37 \(\int \frac {\coth ^3(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx\) [137]
Optimal. Leaf size=262 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 (a-b)^{3/2} d}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 (a+b)^{3/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a+b) d (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a-b) d (1+\text {sech}(c+d x))} \]
[Out]
1/4*b*arctanh((a+b*sech(d*x+c))^(1/2)/(a-b)^(1/2))/(a-b)^(3/2)/d-1/4*b*arctanh((a+b*sech(d*x+c))^(1/2)/(a+b)^(
1/2))/(a+b)^(3/2)/d+2*arctanh((a+b*sech(d*x+c))^(1/2)/a^(1/2))/d/a^(1/2)-arctanh((a+b*sech(d*x+c))^(1/2)/(a-b)
^(1/2))/d/(a-b)^(1/2)-arctanh((a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2))/d/(a+b)^(1/2)-1/4*(a+b*sech(d*x+c))^(1/2)/(
a+b)/d/(1-sech(d*x+c))-1/4*(a+b*sech(d*x+c))^(1/2)/(a-b)/d/(1+sech(d*x+c))
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Rubi [A]
time = 0.22, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3970, 912,
1252, 212, 205, 213} \begin {gather*} -\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 d (a+b) (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 d (a-b) (\text {sech}(c+d x)+1)}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 d (a-b)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{d \sqrt {a-b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{d \sqrt {a+b}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 d (a+b)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Coth[c + d*x]^3/Sqrt[a + b*Sech[c + d*x]],x]
[Out]
(2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]]/(S
qrt[a - b]*d) + (b*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(3/2)*d) - (b*ArcTanh[Sqrt[a + b
*Sech[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(3/2)*d) - ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b
]*d) - Sqrt[a + b*Sech[c + d*x]]/(4*(a + b)*d*(1 - Sech[c + d*x])) - Sqrt[a + b*Sech[c + d*x]]/(4*(a - b)*d*(1
+ Sech[c + d*x]))
Rule 205
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 213
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rule 912
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{q = De
nominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 + a*e^2)/e^2 - 2*c*
d*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*
g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]
Rule 1252
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d
+ e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((Intege
rQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])
Rule 3970
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[-(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[(b^2 - x^2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
Rubi steps
\begin {align*} \int \frac {\coth ^3(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx &=-\frac {b^4 \text {Subst}\left (\int \frac {1}{x \sqrt {a+x} \left (b^2-x^2\right )^2} \, dx,x,b \text {sech}(c+d x)\right )}{d}\\ &=-\frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {1}{\left (-a+x^2\right ) \left (-a^2+b^2+2 a x^2-x^4\right )^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}\\ &=-\frac {\left (2 b^4\right ) \text {Subst}\left (\int \left (-\frac {1}{b^4 \left (a-x^2\right )}+\frac {1}{4 b^3 \left (a+b-x^2\right )^2}+\frac {1}{2 b^4 \left (a+b-x^2\right )}-\frac {1}{4 b^3 \left (-a+b+x^2\right )^2}-\frac {1}{2 b^4 \left (-a+b+x^2\right )}\right ) \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}+\frac {\text {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}+\frac {2 \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}-\frac {b \text {Subst}\left (\int \frac {1}{\left (a+b-x^2\right )^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{2 d}+\frac {b \text {Subst}\left (\int \frac {1}{\left (-a+b+x^2\right )^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{2 d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a+b) d (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a-b) d (1+\text {sech}(c+d x))}-\frac {b \text {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{4 (a-b) d}-\frac {b \text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{4 (a+b) d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 (a-b)^{3/2} d}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 (a+b)^{3/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a+b) d (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a-b) d (1+\text {sech}(c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 5.42, size = 505, normalized size = 1.93 \begin {gather*} \frac {\frac {-2 a^2 \coth ^2(c+d x)+2 b^2 \text {csch}^2(c+d x)}{a^2-b^2}+\frac {\sqrt {b+a \cosh (c+d x)} \left (a^{3/2} \sqrt {-a+b} \left (2 a^2+a b-3 b^2\right ) \tanh ^{-1}\left (\frac {a-a \cosh (c+d x)+\sqrt {a \cosh (c+d x)} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a} \sqrt {a+b}}\right )-(a+b) \left (-2 i \sqrt {a} (-a+b)^{3/2} \tanh ^{-1}\left (\frac {a-a \cosh (c+d x)-i \sqrt {-a \cosh (c+d x)} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a} \sqrt {a+b}}\right ) \sqrt {-a^2 \cosh ^2(c+d x)}+\sqrt {a+b} \left (a^{3/2} (2 a-3 b) \text {ArcTan}\left (\frac {a+a \cosh (c+d x)-\sqrt {a \cosh (c+d x)} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a} \sqrt {-a+b}}\right ) \cosh (c+d x)+2 (a-b) \left (4 \sqrt {-a+b} \text {ArcTan}\left (\frac {\sqrt {b+a \cosh (c+d x)}}{\sqrt {-a \cosh (c+d x)}}\right )-i \sqrt {a} \text {ArcTan}\left (\frac {a+a \cosh (c+d x)+i \sqrt {-a \cosh (c+d x)} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a} \sqrt {-a+b}}\right )\right ) \sqrt {-a^2 \cosh ^2(c+d x)}\right )\right ) \text {sech}(c+d x)\right )}{a (-a+b)^{3/2} (a+b)^{3/2} \sqrt {a \cosh (c+d x)}}}{4 d \sqrt {a+b \text {sech}(c+d x)}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[Coth[c + d*x]^3/Sqrt[a + b*Sech[c + d*x]],x]
[Out]
((-2*a^2*Coth[c + d*x]^2 + 2*b^2*Csch[c + d*x]^2)/(a^2 - b^2) + (Sqrt[b + a*Cosh[c + d*x]]*(a^(3/2)*Sqrt[-a +
b]*(2*a^2 + a*b - 3*b^2)*ArcTanh[(a - a*Cosh[c + d*x] + Sqrt[a*Cosh[c + d*x]]*Sqrt[b + a*Cosh[c + d*x]])/(Sqrt
[a]*Sqrt[a + b])] - (a + b)*((-2*I)*Sqrt[a]*(-a + b)^(3/2)*ArcTanh[(a - a*Cosh[c + d*x] - I*Sqrt[-(a*Cosh[c +
d*x])]*Sqrt[b + a*Cosh[c + d*x]])/(Sqrt[a]*Sqrt[a + b])]*Sqrt[-(a^2*Cosh[c + d*x]^2)] + Sqrt[a + b]*(a^(3/2)*(
2*a - 3*b)*ArcTan[(a + a*Cosh[c + d*x] - Sqrt[a*Cosh[c + d*x]]*Sqrt[b + a*Cosh[c + d*x]])/(Sqrt[a]*Sqrt[-a + b
])]*Cosh[c + d*x] + 2*(a - b)*(4*Sqrt[-a + b]*ArcTan[Sqrt[b + a*Cosh[c + d*x]]/Sqrt[-(a*Cosh[c + d*x])]] - I*S
qrt[a]*ArcTan[(a + a*Cosh[c + d*x] + I*Sqrt[-(a*Cosh[c + d*x])]*Sqrt[b + a*Cosh[c + d*x]])/(Sqrt[a]*Sqrt[-a +
b])])*Sqrt[-(a^2*Cosh[c + d*x]^2)]))*Sech[c + d*x]))/(a*(-a + b)^(3/2)*(a + b)^(3/2)*Sqrt[a*Cosh[c + d*x]]))/(
4*d*Sqrt[a + b*Sech[c + d*x]])
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Maple [F]
time = 3.45, size = 0, normalized size = 0.00 \[\int \frac {\coth ^{3}\left (d x +c \right )}{\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(d*x+c)^3/(a+b*sech(d*x+c))^(1/2),x)
[Out]
int(coth(d*x+c)^3/(a+b*sech(d*x+c))^(1/2),x)
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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(coth(d*x+c)^3/(a+b*sech(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
integrate(coth(d*x + c)^3/sqrt(b*sech(d*x + c) + a), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1870 vs.
\(2 (218) = 436\).
time = 4.40, size = 20300, normalized size = 77.48 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)^3/(a+b*sech(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
[1/16*(((4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cosh(d*x + c)^4 + 4*(4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*co
sh(d*x + c)*sinh(d*x + c)^3 + (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*sinh(d*x + c)^4 + 4*a^4 + 3*a^3*b - 6*a^
2*b^2 - 5*a*b^3 - 2*(4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cosh(d*x + c)^2 - 2*(4*a^4 + 3*a^3*b - 6*a^2*b^2 -
5*a*b^3 - 3*(4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((4*a^4 + 3*a^3*b -
6*a^2*b^2 - 5*a*b^3)*cosh(d*x + c)^3 - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*s
qrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cosh(d*x + c)^4 + (8*a^2 - 8*a*b + b^2)*sinh(d*x + c)^4 + 4*(4*a*b - 3*
b^2)*cosh(d*x + c)^3 + 4*(4*a*b - 3*b^2 + (8*a^2 - 8*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(8*a^2 - 8*
a*b + 3*b^2)*cosh(d*x + c)^2 + 2*(3*(8*a^2 - 8*a*b + b^2)*cosh(d*x + c)^2 + 8*a^2 - 8*a*b + 3*b^2 + 6*(4*a*b -
3*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + 8*a^2 - 8*a*b + b^2 - 4*((2*a - b)*cosh(d*x + c)^4 + (2*a - b)*sinh(d
*x + c)^4 + 2*b*cosh(d*x + c)^3 + 2*(2*(2*a - b)*cosh(d*x + c) + b)*sinh(d*x + c)^3 + 2*(2*a - b)*cosh(d*x + c
)^2 + 2*(3*(2*a - b)*cosh(d*x + c)^2 + 3*b*cosh(d*x + c) + 2*a - b)*sinh(d*x + c)^2 + 2*b*cosh(d*x + c) + 2*(2
*(2*a - b)*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)^2 + 2*(2*a - b)*cosh(d*x + c) + b)*sinh(d*x + c) + 2*a - b)*sqr
t(a - b)*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c)) + 4*(4*a*b - 3*b^2)*cosh(d*x + c) + 4*((8*a^2 - 8*a*b + b^2
)*cosh(d*x + c)^3 + 3*(4*a*b - 3*b^2)*cosh(d*x + c)^2 + 4*a*b - 3*b^2 + (8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c))
*sinh(d*x + c))/(cosh(d*x + c)^4 + 4*(cosh(d*x + c) + 1)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 4*cosh(d*x + c)^3
+ 6*(cosh(d*x + c)^2 + 2*cosh(d*x + c) + 1)*sinh(d*x + c)^2 + 6*cosh(d*x + c)^2 + 4*(cosh(d*x + c)^3 + 3*cosh
(d*x + c)^2 + 3*cosh(d*x + c) + 1)*sinh(d*x + c) + 4*cosh(d*x + c) + 1)) + ((4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a
*b^3)*cosh(d*x + c)^4 + 4*(4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (4*a^4 - 3*a
^3*b - 6*a^2*b^2 + 5*a*b^3)*sinh(d*x + c)^4 + 4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - 2*(4*a^4 - 3*a^3*b - 6*a
^2*b^2 + 5*a*b^3)*cosh(d*x + c)^2 - 2*(4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - 3*(4*a^4 - 3*a^3*b - 6*a^2*b^2
+ 5*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^3 - (4*
a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos
h(d*x + c)^4 + (8*a^2 + 8*a*b + b^2)*sinh(d*x + c)^4 + 4*(4*a*b + 3*b^2)*cosh(d*x + c)^3 + 4*(4*a*b + 3*b^2 +
(8*a^2 + 8*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(8*a^2 + 8*a*b + 3*b^2)*cosh(d*x + c)^2 + 2*(3*(8*a^2
+ 8*a*b + b^2)*cosh(d*x + c)^2 + 8*a^2 + 8*a*b + 3*b^2 + 6*(4*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + 8
*a^2 + 8*a*b + b^2 - 4*((2*a + b)*cosh(d*x + c)^4 + (2*a + b)*sinh(d*x + c)^4 + 2*b*cosh(d*x + c)^3 + 2*(2*(2*
a + b)*cosh(d*x + c) + b)*sinh(d*x + c)^3 + 2*(2*a + b)*cosh(d*x + c)^2 + 2*(3*(2*a + b)*cosh(d*x + c)^2 + 3*b
*cosh(d*x + c) + 2*a + b)*sinh(d*x + c)^2 + 2*b*cosh(d*x + c) + 2*(2*(2*a + b)*cosh(d*x + c)^3 + 3*b*cosh(d*x
+ c)^2 + 2*(2*a + b)*cosh(d*x + c) + b)*sinh(d*x + c) + 2*a + b)*sqrt(a + b)*sqrt((a*cosh(d*x + c) + b)/cosh(d
*x + c)) + 4*(4*a*b + 3*b^2)*cosh(d*x + c) + 4*((8*a^2 + 8*a*b + b^2)*cosh(d*x + c)^3 + 3*(4*a*b + 3*b^2)*cosh
(d*x + c)^2 + 4*a*b + 3*b^2 + (8*a^2 + 8*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c))/(cosh(d*x + c)^4 + 4*(cosh
(d*x + c) - 1)*sinh(d*x + c)^3 + sinh(d*x + c)^4 - 4*cosh(d*x + c)^3 + 6*(cosh(d*x + c)^2 - 2*cosh(d*x + c) +
1)*sinh(d*x + c)^2 + 6*cosh(d*x + c)^2 + 4*(cosh(d*x + c)^3 - 3*cosh(d*x + c)^2 + 3*cosh(d*x + c) - 1)*sinh(d*
x + c) - 4*cosh(d*x + c) + 1)) + 8*((a^4 - 2*a^2*b^2 + b^4)*cosh(d*x + c)^4 + 4*(a^4 - 2*a^2*b^2 + b^4)*cosh(d
*x + c)*sinh(d*x + c)^3 + (a^4 - 2*a^2*b^2 + b^4)*sinh(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2
+ b^4)*cosh(d*x + c)^2 - 2*(a^4 - 2*a^2*b^2 + b^4 - 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^
2 + 4*((a^4 - 2*a^2*b^2 + b^4)*cosh(d*x + c)^3 - (a^4 - 2*a^2*b^2 + b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a)
*log(-(2*a^2*cosh(d*x + c)^4 + 2*a^2*sinh(d*x + c)^4 + 4*a*b*cosh(d*x + c)^3 + 4*(2*a^2*cosh(d*x + c) + a*b)*s
inh(d*x + c)^3 + 4*a*b*cosh(d*x + c) + (4*a^2 + b^2)*cosh(d*x + c)^2 + (12*a^2*cosh(d*x + c)^2 + 12*a*b*cosh(d
*x + c) + 4*a^2 + b^2)*sinh(d*x + c)^2 + 2*a^2 + 2*(a*cosh(d*x + c)^4 + a*sinh(d*x + c)^4 + b*cosh(d*x + c)^3
+ (4*a*cosh(d*x + c) + b)*sinh(d*x + c)^3 + 2*a*cosh(d*x + c)^2 + (6*a*cosh(d*x + c)^2 + 3*b*cosh(d*x + c) + 2
*a)*sinh(d*x + c)^2 + b*cosh(d*x + c) + (4*a*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)^2 + 4*a*cosh(d*x + c) + b)*si
nh(d*x + c) + a)*sqrt(a)*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c)) + 2*(4*a^2*cosh(d*x + c)^3 + 6*a*b*cosh(d*x
+ c)^2 + 2*a*b + (4*a^2 + b^2)*cosh(d*x + c))*sinh(d*x + c))/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c)
+ sinh(d*x + c)^2)) - 8*((a^4 - a^2*b^2)*cosh(d*x + c)^4 + (a^4 - a^2*b^2)*sinh(d*x + c)^4 + a^4 - a^2*b^2 -
2*(a^3*b - a*b^3)*cosh(d*x + c)^3 - 2*(a^3*b - ...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{3}{\left (c + d x \right )}}{\sqrt {a + b \operatorname {sech}{\left (c + d x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)**3/(a+b*sech(d*x+c))**(1/2),x)
[Out]
Integral(coth(c + d*x)**3/sqrt(a + b*sech(c + d*x)), x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)^3/(a+b*sech(d*x+c))^(1/2),x, algorithm="giac")
[Out]
integrate(coth(d*x + c)^3/sqrt(b*sech(d*x + c) + a), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {coth}\left (c+d\,x\right )}^3}{\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(c + d*x)^3/(a + b/cosh(c + d*x))^(1/2),x)
[Out]
int(coth(c + d*x)^3/(a + b/cosh(c + d*x))^(1/2), x)
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