3.2.80 \(\int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^3(c+d x)} \, dx\) [180]

Optimal. Leaf size=322 \[ \frac {2 (-1)^{2/3} b \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} d}+\frac {2 (-1)^{2/3} b \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{5/3} \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} d}+\frac {\tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac {2 b \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a^{5/3} \sqrt {a^{2/3}+b^{2/3}} d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \]

[Out]

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

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Rubi [A]
time = 0.43, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3299, 3853, 3855, 3292, 2739, 632, 210} \begin {gather*} \frac {2 (-1)^{2/3} b \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac {2 (-1)^{2/3} b \text {ArcTan}\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^{5/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}+\frac {2 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^{5/3} d \sqrt {a^{2/3}+b^{2/3}}}+\frac {\tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 632

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 2739

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 3292

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Int[ExpandTrig[(a + b*(c*sin[e + f*
x])^n)^p, x], x] /; FreeQ[{a, b, c, e, f, n}, x] && (IGtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rule 3299

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 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.36, size = 178, normalized size = 0.55 \begin {gather*} -\frac {16 b \text {RootSum}\left [-b+3 b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {c \text {$\#$1}+d x \text {$\#$1}+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}}{b+4 a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]+3 \left (\text {csch}^2\left (\frac {1}{2} (c+d x)\right )+4 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+\text {sech}^2\left (\frac {1}{2} (c+d x)\right )\right )}{24 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*(16*b*RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (c*#1 + d*x*#1 + 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)/(b + 4*a*#1 - 2*b*#1^2 + b*#1^4) & ] +
3*(Csch[(c + d*x)/2]^2 + 4*Log[Tanh[(c + d*x)/2]] + Sech[(c + d*x)/2]^2))/(a*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.87, size = 138, normalized size = 0.43

method	result	size
derivativedivides	\(\frac {\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}+\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{d}\)	\(138\)
default	\(\frac {\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}+\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{d}\)	\(138\)
risch	\(-\frac {{\mathrm e}^{d x +c} \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+8 \left (\munderset {\textit {\_R} =\RootOf \left (\left (191102976 a^{12} d^{6}[phone] a^{10} b^{2} d^{6}\right ) \textit {\_Z}^{6}-995328 a^{8} b^{2} d^{4} \textit {\_Z}^{4}+1728 a^{4} d^{2} \textit {\_Z}^{2} b^{4}-b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (\frac {15925248 d^{5} a^{11}}{b^{6}}+\frac {15925248 d^{5} a^{9}}{b^{4}}\right ) \textit {\_R}^{5}+\left (\frac {331776 d^{4} a^{9}}{b^{5}}+\frac {331776 d^{4} a^{7}}{b^{3}}\right ) \textit {\_R}^{4}+\left (-\frac {69120 d^{3} a^{7}}{b^{4}}+\frac {13824 d^{3} a^{5}}{b^{2}}\right ) \textit {\_R}^{3}-\frac {1728 d^{2} a^{5} \textit {\_R}^{2}}{b^{3}}+\frac {72 d \,a^{3} \textit {\_R}}{b^{2}}+\frac {2 a}{b}\right )\right )+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d a}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d a}\)	\(259\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/a+1/3/a*b*sum((_R^4-2*_R^2+1)/(_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))-1/8/a/tanh(1/2*d*x+1/2*c)^2-1/2/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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 29179 vs. \(2 (229) = 458\).
time = 8.96, size = 29179, normalized size = 90.62 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*sqrt(2/3)*sqrt(1/6)*(a*d*cosh(d*x + c)^4 + 4*a*d*cosh(d*x + c)*sinh(d*x + c)^3 + a*d*sinh(d*x + c)^4 -
2*a*d*cosh(d*x + c)^2 + 2*(3*a*d*cosh(d*x + c)^2 - a*d)*sinh(d*x + c)^2 + a*d + 4*(a*d*cosh(d*x + c)^3 - a*d*
cosh(d*x + c))*sinh(d*x + c))*sqrt(-((a^4 + a^2*b^2)*(2*(1/2)^(2/3)*(b^4/(a^8*d^4 + a^6*b^2*d^4) - b^4/(a^4*d^
2 + a^2*b^2*d^2)^2)*(-I*sqrt(3) + 1)/(b^6/(a^12*d^6 + a^10*b^2*d^6) - 3*b^6/((a^8*d^4 + a^6*b^2*d^4)*(a^4*d^2
+ a^2*b^2*d^2)) + 2*b^6/(a^4*d^2 + a^2*b^2*d^2)^3 + b^8/((a^2 + b^2)^2*a^10*d^6))^(1/3) - (1/2)^(1/3)*(b^6/(a^
12*d^6 + a^10*b^2*d^6) - 3*b^6/((a^8*d^4 + a^6*b^2*d^4)*(a^4*d^2 + a^2*b^2*d^2)) + 2*b^6/(a^4*d^2 + a^2*b^2*d^
2)^3 + b^8/((a^2 + b^2)^2*a^10*d^6))^(1/3)*(I*sqrt(3) + 1) + 2*b^2/(a^4*d^2 + a^2*b^2*d^2))*d^2 + 3*sqrt(1/3)*
(a^4 + a^2*b^2)*d^2*sqrt(-((a^10 + 2*a^8*b^2 + a^6*b^4)*(2*(1/2)^(2/3)*(b^4/(a^8*d^4 + a^6*b^2*d^4) - b^4/(a^4
*d^2 + a^2*b^2*d^2)^2)*(-I*sqrt(3) + 1)/(b^6/(a^12*d^6 + a^10*b^2*d^6) - 3*b^6/((a^8*d^4 + a^6*b^2*d^4)*(a^4*d
^2 + a^2*b ...

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

[Out]

Timed out

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

[Out]

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

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Mupad [B]
time = 89.25, size = 2500, normalized size = 7.76 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(-(16777216*b^7*exp(d*x)*exp(root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27
*a^4*b^4*d^2*z^2 - b^6, z, k)) - 50331648*a*b^6 + 33554432*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*
a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)*a*b^7*d + 671088640*a^2*b^5*exp(d*x)*exp(root(729*a^10*b^2*d
^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)) + 201326592*root(729*a^10*b
^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^2*a^3*b^6*d^2 - 15099494
40*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^2*a^5*
b^4*d^2 - 2717908992*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 -
b^6, z, k)^3*a^5*b^5*d^3 + 2717908992*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27
*a^4*b^4*d^2*z^2 - b^6, z, k)^3*a^7*b^3*d^3 + 6039797760*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^
8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^4*a^7*b^4*d^4 - 4076863488*root(729*a^10*b^2*d^6*z^6 + 729*a^1
2*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^4*a^9*b^2*d^4 - 679477248*root(729*a^10*b^2*
d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^5*a^9*b^3*d^5 + 16307453952
*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^6*a^11*b
^2*d^6 - 32614907904*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 -
b^6, z, k)^7*a^11*b^3*d^7 + 452984832*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27
*a^4*b^4*d^2*z^2 - b^6, z, k)*a^3*b^5*d + 4076863488*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^
2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^5*a^11*b*d^5 - 40768634880*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^
6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^7*a^13*b*d^7 - 97844723712*root(729*a^10*b^2*d^6
*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^5*a^12*d^5*exp(d*x)*exp(root(7
29*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)) + 391378894848
*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^7*a^14*d
^7*exp(d*x)*exp(root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6,
z, k)) + 10871635968*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2
- b^6, z, k)^4*a^10*b*d^4*exp(d*x)*exp(root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27
*a^4*b^4*d^2*z^2 - b^6, z, k)) - 55717134336*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^
4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^6*a^12*b*d^6*exp(d*x)*exp(root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 2
43*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)) - 3061841920*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^
6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^2*a^4*b^5*d^2*exp(d*x)*exp(root(729*a^10*b^2*d^6*z^6
+ 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)) - 7247757312*root(729*a^10*b^2*d^
6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^2*a^6*b^3*d^2*exp(d*x)*exp(ro
ot(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)) + 96888422
40*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^3*a^6*
b^4*d^3*exp(d*x)*exp(root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 -
b^6, z, k)) + 36238786560*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2
*z^2 - b^6, z, k)^3*a^8*b^2*d^3*exp(d*x)*exp(root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^
4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)) - 301989888*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^
4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^4*a^6*b^5*d^4*exp(d*x)*exp(root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^
6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)) + 48695869440*root(729*a^10*b^2*d^6*z^6 + 729*a^12*
d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^4*a^8*b^3*d^4*exp(d*x)*exp(root(729*a^10*b^2*d
^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)) + 6341787648*root(729*a^10*
b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)^5*a^8*b^4*d^5*exp(d*x)*
exp(root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k)) - 24
4838301696*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - b^6, z, k
)^5*a^10*b^2*d^5*exp(d*x)*exp(root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^4*b^4*
d^2*z^2 - b^6, z, k)) - 74742497280*root(729*a^10*b^2*d^6*z^6 + 729*a^12*d^6*z^6 - 243*a^8*b^2*d^4*z^4 + 27*a^
4*b^4*d^2*z^2 - b^6, z, k)^6*a^10*b^3*d^6*exp(d...

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