3.2.75 \(\int \frac {\sinh ^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx\) [175]

Optimal. Leaf size=262 \[ -\frac {2 \text {ArcTan}\left (\frac {(-1)^{5/6} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}} b^{2/3} d}-\frac {2 \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} b^{2/3} d}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {a^{2/3}+b^{2/3}} b^{2/3} d} \]

[Out]

-2/3*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))/
b^(2/3)/d/((-1)^(1/3)*a^(2/3)-b^(2/3))^(1/2)-2/3*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))/b^(2/3)/d/(-(-1)^(2/3)*a^(2/3)-b^(2/3))^(1/2)-2/3*arctanh((b^(1/3
)-a^(1/3)*tanh(1/2*d*x+1/2*c))/(a^(2/3)+b^(2/3))^(1/2))/b^(2/3)/d/(a^(2/3)+b^(2/3))^(1/2)

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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 = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3299, 2739, 632, 212, 210} \begin {gather*} -\frac {2 \text {ArcTan}\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 b^{2/3} d \sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}-\frac {2 \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 b^{2/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}-\frac {2 \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 b^{2/3} d \sqrt {a^{2/3}+b^{2/3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^2/(a + b*Sinh[c + d*x]^3),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sinh ^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx &=-\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\\ &=-\frac {i \int \frac {1}{-i \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 b^{2/3}}-\frac {i \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 b^{2/3}}-\frac {i \int \frac {1}{(-1)^{5/6} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 b^{2/3}}\\ &=-\frac {2 \text {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 b^{2/3} d}-\frac {2 \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 b^{2/3} d}-\frac {2 \text {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 b^{2/3} d}\\ &=\frac {4 \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 b^{2/3} d}+\frac {4 \text {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 b^{2/3} d}+\frac {4 \text {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 b^{2/3} d}\\ &=\frac {2 \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 \sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}} b^{2/3} d}+\frac {2 \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 \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} b^{2/3} d}-\frac {2 \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 \sqrt {a^{2/3}+b^{2/3}} b^{2/3} 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 = 0.11, size = 275, normalized size = 1.05 \begin {gather*} \frac {\text {RootSum}\left [-b+3 b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {c+d x+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-2 c \text {$\#$1}^2-2 d x \text {$\#$1}^2-4 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+c \text {$\#$1}^4+d x \text {$\#$1}^4+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+4 a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^2/(a + b*Sinh[c + d*x]^3),x]

[Out]

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*Log[-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*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.02, size = 78, normalized size = 0.30

method	result	size
derivativedivides	\(-\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d}\)	\(78\)
default	\(-\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d}\)	\(78\)
risch	\(\munderset {\textit {\_R} =\RootOf \left (-1+\left (729 a^{2} b^{4} d^{6}+729 b^{6} d^{6}\right ) \textit {\_Z}^{6}-243 b^{4} d^{4} \textit {\_Z}^{4}+27 b^{2} d^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (-243 b^{3} d^{5} a^{2}-243 d^{5} b^{5}\right ) \textit {\_R}^{5}+\left (81 a \,b^{3} d^{4}+\frac {81 b^{5} d^{4}}{a}\right ) \textit {\_R}^{4}+81 b^{3} d^{3} \textit {\_R}^{3}+\left (9 a \,d^{2} b -\frac {18 d^{2} b^{3}}{a}\right ) \textit {\_R}^{2}-9 b d \textit {\_R} +\frac {b}{a}\right )\)	\(156\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^2/(a+b*sinh(d*x+c)^3),x,method=_RETURNVERBOSE)

[Out]

-4/3/d*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))

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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)^2/(a+b*sinh(d*x+c)^3),x, algorithm="maxima")

[Out]

integrate(sinh(d*x + c)^2/(b*sinh(d*x + c)^3 + a), x)

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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. 24063 vs. \(2 (181) = 362\).
time = 1.67, size = 24063, normalized size = 91.84 \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)^2/(a+b*sinh(d*x+c)^3),x, algorithm="fricas")

[Out]

-1/2*sqrt(2/3)*sqrt(1/6)*sqrt(-((2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(1/(a^2*b^2*d^4 + b^4*d^4) - 1/(a^2*d^2 + b^2*
d^2)^2)/(1/(a^2*b^4*d^6 + b^6*d^6) - 3/((a^2*b^2*d^4 + b^4*d^4)*(a^2*d^2 + b^2*d^2)) + 2/(a^2*d^2 + b^2*d^2)^3
+ a^2/((a^2 + b^2)^2*b^4*d^6))^(1/3) - (1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(a^2*b^4*d^6 + b^6*d^6) - 3/((a^2*b^2*d
^4 + b^4*d^4)*(a^2*d^2 + b^2*d^2)) + 2/(a^2*d^2 + b^2*d^2)^3 + a^2/((a^2 + b^2)^2*b^4*d^6))^(1/3) + 2/(a^2*d^2
+ b^2*d^2))*(a^2 + b^2)*d^2 + 3*sqrt(1/3)*(a^2 + b^2)*d^2*sqrt(-((a^4*b^2 + 2*a^2*b^4 + b^6)*(2*(1/2)^(2/3)*(
-I*sqrt(3) + 1)*(1/(a^2*b^2*d^4 + b^4*d^4) - 1/(a^2*d^2 + b^2*d^2)^2)/(1/(a^2*b^4*d^6 + b^6*d^6) - 3/((a^2*b^2
*d^4 + b^4*d^4)*(a^2*d^2 + b^2*d^2)) + 2/(a^2*d^2 + b^2*d^2)^3 + a^2/((a^2 + b^2)^2*b^4*d^6))^(1/3) - (1/2)^(1
/3)*(I*sqrt(3) + 1)*(1/(a^2*b^4*d^6 + b^6*d^6) - 3/((a^2*b^2*d^4 + b^4*d^4)*(a^2*d^2 + b^2*d^2)) + 2/(a^2*d^2
+ b^2*d^2)^3 + a^2/((a^2 + b^2)^2*b^4*d^6))^(1/3) + 2/(a^2*d^2 + b^2*d^2))^2*d^4 - 4*(a^2*b^2 + b^4)*(2*(1/2)^
(2/3)*(-I* ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^{2}{\left (c + d x \right )}}{a + b \sinh ^{3}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**2/(a+b*sinh(d*x+c)**3),x)

[Out]

Integral(sinh(c + d*x)**2/(a + b*sinh(c + d*x)**3), 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(sinh(d*x+c)^2/(a+b*sinh(d*x+c)^3),x, algorithm="giac")

[Out]

integrate(sinh(d*x + c)^2/(b*sinh(d*x + c)^3 + a), x)

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Mupad [B]
time = 11.13, size = 932, normalized size = 3.56 \begin {gather*} \sum _{k=1}^6\ln \left (\mathrm {root}\left (729\,a^2\,b^4\,d^6\,z^6+729\,b^6\,d^6\,z^6-243\,b^4\,d^4\,z^4+27\,b^2\,d^2\,z^2-1,z,k\right )\,\left (\mathrm {root}\left (729\,a^2\,b^4\,d^6\,z^6+729\,b^6\,d^6\,z^6-243\,b^4\,d^4\,z^4+27\,b^2\,d^2\,z^2-1,z,k\right )\,\left (\mathrm {root}\left (729\,a^2\,b^4\,d^6\,z^6+729\,b^6\,d^6\,z^6-243\,b^4\,d^4\,z^4+27\,b^2\,d^2\,z^2-1,z,k\right )\,\left (\mathrm {root}\left (729\,a^2\,b^4\,d^6\,z^6+729\,b^6\,d^6\,z^6-243\,b^4\,d^4\,z^4+27\,b^2\,d^2\,z^2-1,z,k\right )\,\left (\frac {663552\,\left (8\,a^5\,d^4+4\,a^3\,b^2\,d^4-5\,a^4\,b\,d^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^4\,d^6\,z^6+729\,b^6\,d^6\,z^6-243\,b^4\,d^4\,z^4+27\,b^2\,d^2\,z^2-1,z,k\right )}\right )}{b^6}-\frac {\mathrm {root}\left (729\,a^2\,b^4\,d^6\,z^6+729\,b^6\,d^6\,z^6-243\,b^4\,d^4\,z^4+27\,b^2\,d^2\,z^2-1,z,k\right )\,\left (4\,a^5\,d^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^4\,d^6\,z^6+729\,b^6\,d^6\,z^6-243\,b^4\,d^4\,z^4+27\,b^2\,d^2\,z^2-1,z,k\right )}-a^4\,b\,d^5+5\,a^3\,b^2\,d^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^4\,d^6\,z^6+729\,b^6\,d^6\,z^6-243\,b^4\,d^4\,z^4+27\,b^2\,d^2\,z^2-1,z,k\right )}\right )\,1990656}{b^5}\right )-\frac {442368\,\left (4\,a^4\,b\,d^3+8\,a^5\,d^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^4\,d^6\,z^6+729\,b^6\,d^6\,z^6-243\,b^4\,d^4\,z^4+27\,b^2\,d^2\,z^2-1,z,k\right )}-5\,a^3\,b^2\,d^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^4\,d^6\,z^6+729\,b^6\,d^6\,z^6-243\,b^4\,d^4\,z^4+27\,b^2\,d^2\,z^2-1,z,k\right )}\right )}{b^7}\right )-\frac {294912\,a^3\,d^2\,\left (2\,b-5\,a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^4\,d^6\,z^6+729\,b^6\,d^6\,z^6-243\,b^4\,d^4\,z^4+27\,b^2\,d^2\,z^2-1,z,k\right )}\right )}{b^7}\right )+\frac {24576\,a^3\,d\,\left (8\,a-5\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^4\,d^6\,z^6+729\,b^6\,d^6\,z^6-243\,b^4\,d^4\,z^4+27\,b^2\,d^2\,z^2-1,z,k\right )}\right )}{b^8}\right )+\frac {32768\,a^3\,\left (b-4\,a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^4\,d^6\,z^6+729\,b^6\,d^6\,z^6-243\,b^4\,d^4\,z^4+27\,b^2\,d^2\,z^2-1,z,k\right )}\right )}{b^9}\right )\,\mathrm {root}\left (729\,a^2\,b^4\,d^6\,z^6+729\,b^6\,d^6\,z^6-243\,b^4\,d^4\,z^4+27\,b^2\,d^2\,z^2-1,z,k\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^2/(a + b*sinh(c + d*x)^3),x)

[Out]

symsum(log(root(729*a^2*b^4*d^6*z^6 + 729*b^6*d^6*z^6 - 243*b^4*d^4*z^4 + 27*b^2*d^2*z^2 - 1, z, k)*(root(729*
a^2*b^4*d^6*z^6 + 729*b^6*d^6*z^6 - 243*b^4*d^4*z^4 + 27*b^2*d^2*z^2 - 1, z, k)*(root(729*a^2*b^4*d^6*z^6 + 72
9*b^6*d^6*z^6 - 243*b^4*d^4*z^4 + 27*b^2*d^2*z^2 - 1, z, k)*(root(729*a^2*b^4*d^6*z^6 + 729*b^6*d^6*z^6 - 243*
b^4*d^4*z^4 + 27*b^2*d^2*z^2 - 1, z, k)*((663552*(8*a^5*d^4 + 4*a^3*b^2*d^4 - 5*a^4*b*d^4*exp(d*x)*exp(root(72
9*a^2*b^4*d^6*z^6 + 729*b^6*d^6*z^6 - 243*b^4*d^4*z^4 + 27*b^2*d^2*z^2 - 1, z, k))))/b^6 - (1990656*root(729*a
^2*b^4*d^6*z^6 + 729*b^6*d^6*z^6 - 243*b^4*d^4*z^4 + 27*b^2*d^2*z^2 - 1, z, k)*(4*a^5*d^5*exp(d*x)*exp(root(72
9*a^2*b^4*d^6*z^6 + 729*b^6*d^6*z^6 - 243*b^4*d^4*z^4 + 27*b^2*d^2*z^2 - 1, z, k)) - a^4*b*d^5 + 5*a^3*b^2*d^5
*exp(d*x)*exp(root(729*a^2*b^4*d^6*z^6 + 729*b^6*d^6*z^6 - 243*b^4*d^4*z^4 + 27*b^2*d^2*z^2 - 1, z, k))))/b^5)
- (442368*(4*a^4*b*d^3 + 8*a^5*d^3*exp(d*x)*exp(root(729*a^2*b^4*d^6*z^6 + 729*b^6*d^6*z^6 - 243*b^4*d^4*z^4
+ 27*b^2*d^2*z^2 - 1, z, k)) - 5*a^3*b^2*d^3*exp(d*x)*exp(root(729*a^2*b^4*d^6*z^6 + 729*b^6*d^6*z^6 - 243*b^4
*d^4*z^4 + 27*b^2*d^2*z^2 - 1, z, k))))/b^7) - (294912*a^3*d^2*(2*b - 5*a*exp(d*x)*exp(root(729*a^2*b^4*d^6*z^
6 + 729*b^6*d^6*z^6 - 243*b^4*d^4*z^4 + 27*b^2*d^2*z^2 - 1, z, k))))/b^7) + (24576*a^3*d*(8*a - 5*b*exp(d*x)*e
xp(root(729*a^2*b^4*d^6*z^6 + 729*b^6*d^6*z^6 - 243*b^4*d^4*z^4 + 27*b^2*d^2*z^2 - 1, z, k))))/b^8) + (32768*a
^3*(b - 4*a*exp(d*x)*exp(root(729*a^2*b^4*d^6*z^6 + 729*b^6*d^6*z^6 - 243*b^4*d^4*z^4 + 27*b^2*d^2*z^2 - 1, z,
k))))/b^9)*root(729*a^2*b^4*d^6*z^6 + 729*b^6*d^6*z^6 - 243*b^4*d^4*z^4 + 27*b^2*d^2*z^2 - 1, z, k), k, 1, 6)

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