3.178 \(\int \frac {\text {csch}(c+d x)}{a+b \sinh ^3(c+d x)} \, dx\)

Optimal. Leaf size=286 \[ \frac {2 \sqrt [3]{b} \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a d \sqrt {a^{2/3}+b^{2/3}}}+\frac {2 \sqrt [3]{b} \tan ^{-1}\left (\frac {(-1)^{5/6} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right )}{3 a d \sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}+\frac {2 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d} \]

[Out]

-arctanh(cosh(d*x+c))/a/d+2/3*b^(1/3)*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*d*x+1/2*c))/((-
1)^(1/3)*a^(2/3)-b^(2/3))^(1/2))/a/d/((-1)^(1/3)*a^(2/3)-b^(2/3))^(1/2)+2/3*b^(1/3)*arctan((-1)^(5/6)*((-1)^(1
/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*d*x+1/2*c))/(-(-1)^(2/3)*a^(2/3)-b^(2/3))^(1/2))/a/d/(-(-1)^(2/3)*a^(2/3)-b^(2/
3))^(1/2)+2/3*b^(1/3)*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*d*x+1/2*c))/(a^(2/3)+b^(2/3))^(1/2))/a/d/(a^(2/3)+b^(2
/3))^(1/2)

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Rubi [A]  time = 0.43, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3220, 3770, 2660, 618, 206, 204} \[ \frac {2 \sqrt [3]{b} \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a d \sqrt {a^{2/3}+b^{2/3}}}+\frac {2 \sqrt [3]{b} \tan ^{-1}\left (\frac {(-1)^{5/6} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right )}{3 a d \sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}+\frac {2 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^(1/3)*ArcTan[((-1)^(5/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(c + d*x)/2]))/Sqrt[-((-1)^(2/3)*a^(2/3)) -
b^(2/3)]])/(3*a*Sqrt[-((-1)^(2/3)*a^(2/3)) - b^(2/3)]*d) + (2*b^(1/3)*ArcTan[((-1)^(1/6)*((-1)^(5/6)*b^(1/3)
+ I*a^(1/3)*Tanh[(c + d*x)/2]))/Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*a*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*d
) - ArcTanh[Cosh[c + d*x]]/(a*d) + (2*b^(1/3)*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(c + d*x)/2])/Sqrt[a^(2/3) + b^(
2/3)]])/(3*a*Sqrt[a^(2/3) + b^(2/3)]*d)

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 206

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 3220

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {\text {csch}(c+d x)}{a+b \sinh ^3(c+d x)} \, dx &=i \int \left (-\frac {i \text {csch}(c+d x)}{a}+\frac {i b \sinh ^2(c+d x)}{a \left (a+b \sinh ^3(c+d x)\right )}\right ) \, dx\\ &=\frac {\int \text {csch}(c+d x) \, dx}{a}-\frac {b \int \frac {\sinh ^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx}{a}\\ &=-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {b \int \left (\frac {i}{3 b^{2/3} \left (-i \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)\right )}+\frac {i}{3 b^{2/3} \left (\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)\right )}+\frac {i}{3 b^{2/3} \left ((-1)^{5/6} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)\right )}\right ) \, dx}{a}\\ &=-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {\left (i \sqrt [3]{b}\right ) \int \frac {1}{-i \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a}+\frac {\left (i \sqrt [3]{b}\right ) \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a}+\frac {\left (i \sqrt [3]{b}\right ) \int \frac {1}{(-1)^{5/6} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a}\\ &=-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {\left (2 \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-i \sqrt [3]{a}-2 \sqrt [3]{b} x-i \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a d}+\frac {\left (2 \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-2 \sqrt [3]{b} x+\sqrt [6]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a d}+\frac {\left (2 \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1)^{5/6} \sqrt [3]{a}-2 \sqrt [3]{b} x+(-1)^{5/6} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a d}\\ &=-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}-\frac {\left (4 \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (\sqrt [3]{-1} a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{b}+2 \sqrt [6]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a d}-\frac {\left (4 \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (a^{2/3}+b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{b}-2 i \sqrt [3]{a} \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a d}-\frac {\left (4 \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left ((-1)^{2/3} a^{2/3}+b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{b}+2 (-1)^{5/6} \sqrt [3]{a} \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a d}\\ &=-\frac {2 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right )}{3 a \sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}} d}-\frac {2 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {2 \sqrt [3]{b} \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a \sqrt {a^{2/3}+b^{2/3}} d}\\ \end {align*}

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Mathematica [C]  time = 0.26, size = 295, normalized size = 1.03 \[ \frac {6 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-b \text {RootSum}\left [\text {$\#$1}^6 b-3 \text {$\#$1}^4 b+8 \text {$\#$1}^3 a+3 \text {$\#$1}^2 b-b\& ,\frac {2 \text {$\#$1}^4 \log \left (-\text {$\#$1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )+\text {$\#$1}^4 c+\text {$\#$1}^4 d x-4 \text {$\#$1}^2 \log \left (-\text {$\#$1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )-2 \text {$\#$1}^2 c-2 \text {$\#$1}^2 d x+2 \log \left (-\text {$\#$1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )+c+d x}{\text {$\#$1}^5 b-2 \text {$\#$1}^3 b+4 \text {$\#$1}^2 a+\text {$\#$1} b}\& \right ]}{6 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Log[Tanh[(c + d*x)/2]] - b*RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (c + d*x + 2*Log[-Cosh[
(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 2*c*#1^2 - 2*d*x*#1^2 - 4*Lo
g[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + c*#1^4 + d*x*#1
^4 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4)/(b*#1 +
4*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/(6*a*d)

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fricas [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(csch(d*x+c)/(a+b*sinh(d*x+c)^3),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}\left (d x + c\right )}{b \sinh \left (d x + c\right )^{3} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csch(d*x + c)/(b*sinh(d*x + c)^3 + a), x)

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maple [C]  time = 0.15, size = 100, normalized size = 0.35 \[ \frac {4 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3/d/a*b*sum(_R^2/(_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))+1/d/a*ln(tanh(1/2*d*x+1/2*c))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{a d} + \frac {\log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{a d} - 2 \, \int \frac {b e^{\left (5 \, d x + 5 \, c\right )} - 2 \, b e^{\left (3 \, d x + 3 \, c\right )} + b e^{\left (d x + c\right )}}{a b e^{\left (6 \, d x + 6 \, c\right )} - 3 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} + 3 \, a b e^{\left (2 \, d x + 2 \, c\right )} - a b}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log((e^(d*x + c) + 1)*e^(-c))/(a*d) + log((e^(d*x + c) - 1)*e^(-c))/(a*d) - 2*integrate((b*e^(5*d*x + 5*c) -
2*b*e^(3*d*x + 3*c) + b*e^(d*x + c))/(a*b*e^(6*d*x + 6*c) - 3*a*b*e^(4*d*x + 4*c) + 8*a^2*e^(3*d*x + 3*c) + 3*
a*b*e^(2*d*x + 2*c) - a*b), x)

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mupad [B]  time = 55.94, size = 2970, normalized size = 10.38 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)*(a + b*sinh(c + d*x)^3)),x)

[Out]

symsum(log(-(2147483648*a*b*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*
d^2*z^2 - b^2, z, k) + d*x) - 1073741824*b^2 - 86973087744*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^
4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^4*a^6*d^4 + 86973087744*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6
*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^6*a^8*d^6 + 134217728*root(729*a^6*b^2*d^6*z^6 +
729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)*b^3*d + 3221225472*root(729*a^6*b^2*d^
6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)*a^2*b*d + 18589155328*root(729
*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^2*a^2*b^2*d^2 - 281
8572288*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a
^2*b^3*d^3 - 88181047296*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2
- b^2, z, k)^4*a^4*b^2*d^4 + 18119393280*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 2
7*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^4*b^3*d^5 + 70665633792*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^
4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^6*a^6*b^2*d^6 - 32614907904*root(729*a^6*b^2*d^6*z^6 + 729*a^8
*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^7*a^6*b^3*d^7 - 57982058496*root(729*a^6*b^2*
d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^5*d^3*exp(root(729*a^6*b
^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) - 333396836352*roo
t(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^7*d^5*exp(
root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) + 39
1378894848*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^
7*a^9*d^7*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z,
k) + d*x) - 17716740096*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2
- b^2, z, k)^3*a^4*b*d^3 + 30802968576*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a
^2*b^2*d^2*z^2 - b^2, z, k)^5*a^6*b*d^5 - 40768634880*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2
*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^7*a^8*b*d^7 + 268435456*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6
- 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^2*a*b^3*d^2*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6
*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) - 16642998272*root(729*a^6*b^2*d^6*z^6 + 7
29*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^2*a^3*b*d^2*exp(root(729*a^6*b^2*d^6*z^
6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) + 36238786560*root(729*a^6*
b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^4*a^5*b*d^4*exp(root(729
*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) + 2717908992
*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^6*a^7*b*d^
6*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x
) - 5637144576*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z,
k)*a*b^2*d*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z
, k) + d*x) + 100763959296*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z
^2 - b^2, z, k)^3*a^3*b^2*d^3*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^
2*d^2*z^2 - b^2, z, k) + d*x) - 4831838208*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 +
27*a^2*b^2*d^2*z^2 - b^2, z, k)^4*a^3*b^3*d^4*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4
*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) - 494659436544*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*
a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^5*b^2*d^5*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6
- 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) + 21743271936*root(729*a^6*b^2*d^6*z^6 + 729*a
^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^6*a^5*b^3*d^6*exp(root(729*a^6*b^2*d^6*z^6
+ 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) + 399532621824*root(729*a^6*b
^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^7*a^7*b^2*d^7*exp(root(72
9*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x))/b^9)*root(
729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k), k, 1, 6) + log(
exp(d*x + 1/(a*d)) - 1)/(a*d) - log(exp(d*x - 1/(a*d)) + 1)/(a*d)

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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(csch(d*x+c)/(a+b*sinh(d*x+c)**3),x)

[Out]

Timed out

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