3.2.76 \(\int \frac {\sinh (c+d x)}{a+b \sinh ^3(c+d x)} \, dx\) [176]

Optimal. Leaf size=290 \[ \frac {2 \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 \sqrt [3]{a} \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} \sqrt [3]{b} d}-\frac {2 \sqrt [3]{-1} \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 \sqrt [3]{a} \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} \sqrt [3]{b} d}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 \sqrt [3]{a} \sqrt {a^{2/3}+b^{2/3}} \sqrt [3]{b} d} \]

[Out]

-2/3*(-1)^(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^(1/3)/b^(1/3)/d/((-1)^(1/3)*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))/a^(1/3)/b^(1/3)/d/(a^(2/3)+b^(2/3))^(1/2)+2/3*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^(1/3)/b^(1/3)/d/((-1)^(1/3)
*a^(2/3)-(-1)^(2/3)*b^(2/3))^(1/2)

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Rubi [A]
time = 0.32, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3299, 2739, 632, 210} \begin {gather*} \frac {2 \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 \sqrt [3]{a} \sqrt [3]{b} d \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac {2 \sqrt [3]{-1} \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 [3]{a} \sqrt [3]{b} 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 \sqrt [3]{a} \sqrt [3]{b} d \sqrt {a^{2/3}+b^{2/3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*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^(1/3)*Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^(1/3)*d) - (2*(-1)^(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^(1/3)*Sqrt[
(-1)^(1/3)*a^(2/3) - b^(2/3)]*b^(1/3)*d) + (2*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(c + d*x)/2])/Sqrt[a^(2/3) + b^(
2/3)]])/(3*a^(1/3)*Sqrt[a^(2/3) + b^(2/3)]*b^(1/3)*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 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 (c+d x)}{a+b \sinh ^3(c+d x)} \, dx &=-\left (i \int \left (\frac {\sqrt [3]{-1}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)\right )}-\frac {(-1)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [6]{-1} \sqrt [3]{a}+\sqrt [6]{-1} \sqrt [3]{b} \sinh (c+d x)\right )}-\frac {1}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [6]{-1} \sqrt [3]{a}+(-1)^{5/6} \sqrt [3]{b} \sinh (c+d x)\right )}\right ) \, dx\right )\\ &=\frac {i \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}+(-1)^{5/6} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\sqrt [6]{-1} \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}+\sqrt [6]{-1} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {(-1)^{5/6} \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\\ &=\frac {2 \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 \sqrt [3]{a} \sqrt [3]{b} d}-\frac {\left (2 \sqrt [3]{-1}\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 \sqrt [3]{a} \sqrt [3]{b} d}+\frac {\left (2 (-1)^{2/3}\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 \sqrt [3]{a} \sqrt [3]{b} d}\\ &=-\frac {4 \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 \sqrt [3]{a} \sqrt [3]{b} d}+\frac {\left (4 \sqrt [3]{-1}\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 \sqrt [3]{a} \sqrt [3]{b} d}-\frac {\left (4 (-1)^{2/3}\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 \sqrt [3]{a} \sqrt [3]{b} d}\\ &=\frac {2 \sqrt [3]{-1} \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 [3]{a} \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} \sqrt [3]{b} d}+\frac {2 \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 \sqrt [3]{a} \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} \sqrt [3]{b} 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 [3]{a} \sqrt {a^{2/3}+b^{2/3}} \sqrt [3]{b} 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 = 0.14, size = 199, normalized size = 0.69 \begin {gather*} \frac {\text {RootSum}\left [-b+3 b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {-c-d x-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+c \text {$\#$1}^2+d x \text {$\#$1}^2+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2}{b+4 a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]/(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] + c*#1^2 + d*x*#1^2 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c +
d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2)/(b + 4*a*#1 - 2*b*#1^2 + b*#1^4) & ]/(3*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.19, size = 82, normalized size = 0.28

method	result	size
derivativedivides	\(\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R} \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d}\)	\(82\)
default	\(\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R} \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d}\)	\(82\)
risch	\(\munderset {\textit {\_R} =\RootOf \left (-1+\left (729 a^{4} b^{2} d^{6}+729 a^{2} b^{4} d^{6}\right ) \textit {\_Z}^{6}+243 a^{2} b^{2} d^{4} \textit {\_Z}^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (\frac {243 d^{5} b^{2} a^{5}}{a^{2}-b^{2}}+\frac {243 d^{5} b^{4} a^{3}}{a^{2}-b^{2}}\right ) \textit {\_R}^{5}+\left (\frac {81 d^{4} b \,a^{5}}{a^{2}-b^{2}}+\frac {81 d^{4} b^{3} a^{3}}{a^{2}-b^{2}}\right ) \textit {\_R}^{4}+\left (\frac {54 d^{3} b^{2} a^{3}}{a^{2}-b^{2}}-\frac {27 d^{3} b^{4} a}{a^{2}-b^{2}}\right ) \textit {\_R}^{3}+\left (\frac {18 d^{2} b \,a^{3}}{a^{2}-b^{2}}-\frac {9 d^{2} b^{3} a}{a^{2}-b^{2}}\right ) \textit {\_R}^{2}+\left (\frac {3 d \,a^{3}}{a^{2}-b^{2}}-\frac {6 d \,b^{2} a}{a^{2}-b^{2}}\right ) \textit {\_R} -\frac {a b}{a^{2}-b^{2}}\right )\)	\(299\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(sinh(d*x + c)/(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. 18312 vs. \(2 (197) = 394\).
time = 1.26, size = 18312, normalized size = 63.14 \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)/(a+b*sinh(d*x+c)^3),x, algorithm="fricas")

[Out]

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

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

[Out]

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

[Out]

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

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Mupad [B]
time = 21.84, size = 857, normalized size = 2.96 \begin {gather*} \sum _{k=1}^6\ln \left (\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )\,\left (\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )\,\left (\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )\,\left (\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )\,\left (\frac {\left (4\,a^4\,b\,d^4+a^5\,d^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,16+a^3\,b^2\,d^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,11\right )\,663552}{b^6}-\frac {\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )\,\left (-a^4\,b\,d^5+a^5\,d^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,4+a^3\,b^2\,d^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,5\right )\,1990656}{b^5}\right )+\frac {\left (8\,a^4\,d^3+a^2\,b^2\,d^3-a^3\,b\,d^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,25\right )\,221184}{b^6}\right )-\frac {a^2\,d^2\,\left (b-a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,7\right )\,294912}{b^6}\right )+\frac {a^2\,d\,\left (b-a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,2\right )\,196608}{b^7}\right )-\frac {a\,\left (8\,a-b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\right )\,8192}{b^7}\right )\,\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4 - 1, z, k)*(root(729*a^4*b^2*d
^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4 - 1, z, k)*(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6
+ 243*a^2*b^2*d^4*z^4 - 1, z, k)*(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4 - 1, z,
k)*((663552*(4*a^4*b*d^4 + 16*a^5*d^4*exp(d*x)*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b
^2*d^4*z^4 - 1, z, k)) + 11*a^3*b^2*d^4*exp(d*x)*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*
b^2*d^4*z^4 - 1, z, k))))/b^6 - (1990656*root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4
- 1, z, k)*(4*a^5*d^5*exp(d*x)*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4 - 1, z
, k)) - a^4*b*d^5 + 5*a^3*b^2*d^5*exp(d*x)*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^
4*z^4 - 1, z, k))))/b^5) + (221184*(8*a^4*d^3 + a^2*b^2*d^3 - 25*a^3*b*d^3*exp(d*x)*exp(root(729*a^4*b^2*d^6*z
^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4 - 1, z, k))))/b^6) - (294912*a^2*d^2*(b - 7*a*exp(d*x)*exp(root
(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4 - 1, z, k))))/b^6) + (196608*a^2*d*(b - 2*a*e
xp(d*x)*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4 - 1, z, k))))/b^7) - (8192*a*
(8*a - b*exp(d*x)*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4 - 1, z, k))))/b^7)*
root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4 - 1, z, k), k, 1, 6)

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