3.2.77 \(\int \frac {1}{a+b \sinh ^3(c+d x)} \, dx\) [177]

Optimal. Leaf size=280 \[ -\frac {2 (-1)^{2/3} \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{2/3} \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} d}-\frac {2 (-1)^{2/3} \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{2/3} \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} d}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}+b^{2/3}} d} \]

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3292, 2739, 632, 210} \begin {gather*} -\frac {2 (-1)^{2/3} \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^{2/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac {2 (-1)^{2/3} \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^{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 a^{2/3} d \sqrt {a^{2/3}+b^{2/3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[c + d*x]^3)^(-1),x]

[Out]

(-2*(-1)^(2/3)*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^(2/3)*Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) - (2*(-1)^(2/3)*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^(2/3)*S
qrt[(-1)^(1/3)*a^(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*a^(2/3)*Sqrt[a^(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 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]))

Rubi steps

\begin {align*} \int \frac {1}{a+b \sinh ^3(c+d x)} \, dx &=\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\\ &=\frac {\sqrt [6]{-1} \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^{2/3}}+\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 a^{2/3}}+\frac {\sqrt [6]{-1} \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}+(-1)^{5/6} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^{2/3}}\\ &=-\frac {\left (2 (-1)^{2/3}\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^{2/3} d}-\frac {\left (2 (-1)^{2/3}\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^{2/3} 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 a^{2/3} d}\\ &=\frac {\left (4 (-1)^{2/3}\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^{2/3} 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 a^{2/3} d}+\frac {\left (4 (-1)^{2/3}\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^{2/3} d}\\ &=\frac {2 (-1)^{2/3} \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^{2/3} \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} d}-\frac {2 (-1)^{2/3} \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^{2/3} \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{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 a^{2/3} \sqrt {a^{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 = 131, normalized size = 0.47 \begin {gather*} \frac {2 \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 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sinh[c + d*x]^3)^(-1),x]

[Out]

(2*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*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.96, size = 87, normalized size = 0.31

method	result	size
derivativedivides	\(\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (-\textit {\_R}^{4}+2 \textit {\_R}^{2}-1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}}{3 d}\)	\(87\)
default	\(\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (-\textit {\_R}^{4}+2 \textit {\_R}^{2}-1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}}{3 d}\)	\(87\)
risch	\(\munderset {\textit {\_R} =\RootOf \left (-1+\left (729 a^{6} d^{6}+729 a^{4} b^{2} d^{6}\right ) \textit {\_Z}^{6}-243 a^{4} d^{4} \textit {\_Z}^{4}+27 a^{2} d^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (-\frac {486 d^{5} a^{6}}{b}-486 b \,d^{5} a^{4}\right ) \textit {\_R}^{5}+\left (\frac {81 d^{4} a^{5}}{b}+81 b \,d^{4} a^{3}\right ) \textit {\_R}^{4}+\left (\frac {135 d^{3} a^{4}}{b}-27 a^{2} b \,d^{3}\right ) \textit {\_R}^{3}-\frac {27 d^{2} a^{3} \textit {\_R}^{2}}{b}-\frac {9 d \,a^{2} \textit {\_R}}{b}+\frac {2 a}{b}\right )\)	\(168\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/d*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))

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

[Out]

integrate(1/(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. 24084 vs. \(2 (191) = 382\).
time = 1.22, size = 24084, normalized size = 86.01 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{a + b \sinh ^{3}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(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(1/(a+b*sinh(d*x+c)^3),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 9.49, size = 1261, normalized size = 4.50 \begin {gather*} \sum _{k=1}^6\ln \left (\frac {\left (-4\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )+d\,x}+\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )\,b\,d+{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )}^2\,a\,b\,d^2\,12-\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )\,a\,d\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )+d\,x}\,20+{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )}^2\,a^2\,d^2\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )+d\,x}\,24+{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )}^3\,a^3\,d^3\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )+d\,x}\,216+{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )}^4\,a^4\,d^4\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )+d\,x}\,108-{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )}^5\,a^5\,d^5\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )+d\,x}\,324+{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )}^3\,a^2\,b\,d^3\,54+{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )}^4\,a^3\,b\,d^4\,108+{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )}^5\,a^4\,b\,d^5\,81-{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )}^4\,a^2\,b^2\,d^4\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )+d\,x}\,27-{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )}^5\,a^3\,b^2\,d^5\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right )+d\,x}\,405\right )\,24576}{b^5}\right )\,\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^6\,d^6\,z^6-243\,a^4\,d^4\,z^4+27\,a^2\,d^2\,z^2-1,z,k\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*sinh(c + d*x)^3),x)

[Out]

symsum(log((24576*(root(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k)*b*
d - 4*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k) + d*x) + 12
*root(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k)^2*a*b*d^2 - 20*root(
729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k)*a*d*exp(root(729*a^4*b^2*d
^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k) + d*x) + 24*root(729*a^4*b^2*d^6*z^6 +
729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k)^2*a^2*d^2*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^6
*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k) + d*x) + 216*root(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6
- 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k)^3*a^3*d^3*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*
a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k) + d*x) + 108*root(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*
z^4 + 27*a^2*d^2*z^2 - 1, z, k)^4*a^4*d^4*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 2
7*a^2*d^2*z^2 - 1, z, k) + d*x) - 324*root(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^
2*z^2 - 1, z, k)^5*a^5*d^5*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 -
1, z, k) + d*x) + 54*root(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k)
^3*a^2*b*d^3 + 108*root(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k)^4*
a^3*b*d^4 + 81*root(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k)^5*a^4*
b*d^5 - 27*root(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k)^4*a^2*b^2*
d^4*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k) + d*x) - 405*
root(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k)^5*a^3*b^2*d^5*exp(roo
t(729*a^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k) + d*x)))/b^5)*root(729*a
^4*b^2*d^6*z^6 + 729*a^6*d^6*z^6 - 243*a^4*d^4*z^4 + 27*a^2*d^2*z^2 - 1, z, k), k, 1, 6)

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