3.2.74 \(\int \frac {\sinh ^3(c+d x)}{a+b \sinh ^3(c+d x)} \, dx\) [174]

Optimal. Leaf size=294 \[ \frac {x}{b}+\frac {2 (-1)^{2/3} \sqrt [3]{a} \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} b d}+\frac {2 (-1)^{2/3} \sqrt [3]{a} \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} b d}+\frac {2 \sqrt [3]{a} \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {a^{2/3}+b^{2/3}} b d} \]

[Out]

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

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Rubi [A]
time = 0.40, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3299, 3292, 2739, 632, 210} \begin {gather*} \frac {2 (-1)^{2/3} \sqrt [3]{a} \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 b d \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac {2 (-1)^{2/3} \sqrt [3]{a} \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 d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}+\frac {2 \sqrt [3]{a} \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 d \sqrt {a^{2/3}+b^{2/3}}}+\frac {x}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sinh ^3(c+d x)}{a+b \sinh ^3(c+d x)} \, dx &=i \int \left (-\frac {i}{b}+\frac {i a}{b \left (a+b \sinh ^3(c+d x)\right )}\right ) \, dx\\ &=\frac {x}{b}-\frac {a \int \frac {1}{a+b \sinh ^3(c+d x)} \, dx}{b}\\ &=\frac {x}{b}-\frac {a \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}{b}\\ &=\frac {x}{b}-\frac {\left (\sqrt [6]{-1} \sqrt [3]{a}\right ) \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 b}-\frac {\left (\sqrt [6]{-1} \sqrt [3]{a}\right ) \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}+\sqrt [6]{-1} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 b}-\frac {\left (\sqrt [6]{-1} \sqrt [3]{a}\right ) \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}+(-1)^{5/6} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 b}\\ &=\frac {x}{b}+\frac {\left (2 (-1)^{2/3} \sqrt [3]{a}\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 b d}+\frac {\left (2 (-1)^{2/3} \sqrt [3]{a}\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 b d}+\frac {\left (2 (-1)^{2/3} \sqrt [3]{a}\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 b d}\\ &=\frac {x}{b}-\frac {\left (4 (-1)^{2/3} \sqrt [3]{a}\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 b d}-\frac {\left (4 (-1)^{2/3} \sqrt [3]{a}\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 b d}-\frac {\left (4 (-1)^{2/3} \sqrt [3]{a}\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 b d}\\ &=\frac {x}{b}-\frac {2 (-1)^{2/3} \sqrt [3]{a} \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 d}+\frac {2 (-1)^{2/3} \sqrt [3]{a} \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 {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} b d}+\frac {2 \sqrt [3]{a} \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 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 = 0.15, size = 145, normalized size = 0.49 \begin {gather*} \frac {3 c+3 d x-2 a \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 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*c + 3*d*x - 2*a*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*b*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.10, size = 124, normalized size = 0.42

method	result	size
derivativedivides	\(\frac {\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 b}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b}}{d}\)	\(124\)
default	\(\frac {\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 b}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b}}{d}\)	\(124\)
risch	\(\frac {x}{b}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (729 a^{2} b^{6} d^{6}+729 b^{8} d^{6}\right ) \textit {\_Z}^{6}-243 a^{2} b^{4} d^{4} \textit {\_Z}^{4}+27 a^{2} b^{2} d^{2} \textit {\_Z}^{2}-a^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (486 a \,b^{4} d^{5}+\frac {486 b^{6} d^{5}}{a}\right ) \textit {\_R}^{5}+\left (81 a \,b^{3} d^{4}+\frac {81 b^{5} d^{4}}{a}\right ) \textit {\_R}^{4}+\left (-135 a \,b^{2} d^{3}+\frac {27 b^{4} d^{3}}{a}\right ) \textit {\_R}^{3}-27 a b \,d^{2} \textit {\_R}^{2}+9 a d \textit {\_R} +\frac {2 a}{b}\right )\right )\)	\(175\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(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/b*ln(tanh(1/2*d*x+1/2*c)-1)+1/b*ln(tanh(1/2*d*x+1/2*c)+1))

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

[Out]

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

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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. 27931 vs. \(2 (205) = 410\).
time = 1.24, size = 27931, normalized size = 95.00 \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)^3/(a+b*sinh(d*x+c)^3),x, algorithm="fricas")

[Out]

-1/2*(sqrt(2/3)*sqrt(1/6)*b*sqrt(((a^2*b^2 + b^4)*(2*(1/2)^(2/3)*(a^4/(a^2*b^2*d^2 + b^4*d^2)^2 - a^2/(a^2*b^4
*d^4 + b^6*d^4))*(-I*sqrt(3) + 1)/(2*a^6/(a^2*b^2*d^2 + b^4*d^2)^3 - 3*a^4/((a^2*b^4*d^4 + b^6*d^4)*(a^2*b^2*d
^2 + b^4*d^2)) + a^2/(a^2*b^6*d^6 + b^8*d^6) + a^2/((a^2 + b^2)^2*b^4*d^6))^(1/3) + (1/2)^(1/3)*(2*a^6/(a^2*b^
2*d^2 + b^4*d^2)^3 - 3*a^4/((a^2*b^4*d^4 + b^6*d^4)*(a^2*b^2*d^2 + b^4*d^2)) + a^2/(a^2*b^6*d^6 + b^8*d^6) + a
^2/((a^2 + b^2)^2*b^4*d^6))^(1/3)*(I*sqrt(3) + 1) - 2*a^2/(a^2*b^2*d^2 + b^4*d^2))*d^2 + 3*sqrt(1/3)*(a^2*b^2
+ b^4)*d^2*sqrt(-((a^4*b^4 + 2*a^2*b^6 + b^8)*(2*(1/2)^(2/3)*(a^4/(a^2*b^2*d^2 + b^4*d^2)^2 - a^2/(a^2*b^4*d^4
+ b^6*d^4))*(-I*sqrt(3) + 1)/(2*a^6/(a^2*b^2*d^2 + b^4*d^2)^3 - 3*a^4/((a^2*b^4*d^4 + b^6*d^4)*(a^2*b^2*d^2 +
b^4*d^2)) + a^2/(a^2*b^6*d^6 + b^8*d^6) + a^2/((a^2 + b^2)^2*b^4*d^6))^(1/3) + (1/2)^(1/3)*(2*a^6/(a^2*b^2*d^
2 + b^4*d^2)^3 - 3*a^4/((a^2*b^4*d^4 + b^6*d^4)*(a^2*b^2*d^2 + b^4*d^2)) + a^2/(a^2*b^6*d^6 + b^8*d^6) + a^2/(
(a^2 + b^2 ...

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

[Out]

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

[Out]

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

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Mupad [B]
time = 10.76, size = 1498, normalized size = 5.10 \begin {gather*} \left (\sum _{k=1}^6\ln \left (-\frac {a^3\,\left (-4\,a^2\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )+d\,x}-\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )\,a\,b^2\,d-{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )}^4\,b^6\,d^4\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )+d\,x}\,27+{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )}^5\,b^7\,d^5\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )+d\,x}\,405+{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )}^2\,a\,b^3\,d^2\,12-{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )}^3\,a\,b^4\,d^3\,54+{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )}^4\,a\,b^5\,d^4\,108-{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )}^5\,a\,b^6\,d^5\,81+\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )\,a^2\,b\,d\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )+d\,x}\,20+{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )}^2\,a^2\,b^2\,d^2\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )+d\,x}\,24-{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )}^3\,a^2\,b^3\,d^3\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )+d\,x}\,216+{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )}^4\,a^2\,b^4\,d^4\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )+d\,x}\,108+{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )}^5\,a^2\,b^5\,d^5\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )+d\,x}\,324\right )\,24576}{b^{10}}\right )\,\mathrm {root}\left (729\,a^2\,b^6\,d^6\,z^6+729\,b^8\,d^6\,z^6-243\,a^2\,b^4\,d^4\,z^4+27\,a^2\,b^2\,d^2\,z^2-a^2,z,k\right )\right )+\frac {x}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(-(24576*a^3*(405*root(729*a^2*b^6*d^6*z^6 + 729*b^8*d^6*z^6 - 243*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d^2*
z^2 - a^2, z, k)^5*b^7*d^5*exp(root(729*a^2*b^6*d^6*z^6 + 729*b^8*d^6*z^6 - 243*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d
^2*z^2 - a^2, z, k) + d*x) - root(729*a^2*b^6*d^6*z^6 + 729*b^8*d^6*z^6 - 243*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d^2
*z^2 - a^2, z, k)*a*b^2*d - 27*root(729*a^2*b^6*d^6*z^6 + 729*b^8*d^6*z^6 - 243*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d
^2*z^2 - a^2, z, k)^4*b^6*d^4*exp(root(729*a^2*b^6*d^6*z^6 + 729*b^8*d^6*z^6 - 243*a^2*b^4*d^4*z^4 + 27*a^2*b^
2*d^2*z^2 - a^2, z, k) + d*x) - 4*a^2*exp(root(729*a^2*b^6*d^6*z^6 + 729*b^8*d^6*z^6 - 243*a^2*b^4*d^4*z^4 + 2
7*a^2*b^2*d^2*z^2 - a^2, z, k) + d*x) + 12*root(729*a^2*b^6*d^6*z^6 + 729*b^8*d^6*z^6 - 243*a^2*b^4*d^4*z^4 +
27*a^2*b^2*d^2*z^2 - a^2, z, k)^2*a*b^3*d^2 - 54*root(729*a^2*b^6*d^6*z^6 + 729*b^8*d^6*z^6 - 243*a^2*b^4*d^4*
z^4 + 27*a^2*b^2*d^2*z^2 - a^2, z, k)^3*a*b^4*d^3 + 108*root(729*a^2*b^6*d^6*z^6 + 729*b^8*d^6*z^6 - 243*a^2*b
^4*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - a^2, z, k)^4*a*b^5*d^4 - 81*root(729*a^2*b^6*d^6*z^6 + 729*b^8*d^6*z^6 - 243
*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - a^2, z, k)^5*a*b^6*d^5 + 20*root(729*a^2*b^6*d^6*z^6 + 729*b^8*d^6*z^6
- 243*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - a^2, z, k)*a^2*b*d*exp(root(729*a^2*b^6*d^6*z^6 + 729*b^8*d^6*z^
6 - 243*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - a^2, z, k) + d*x) + 24*root(729*a^2*b^6*d^6*z^6 + 729*b^8*d^6*z
^6 - 243*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - a^2, z, k)^2*a^2*b^2*d^2*exp(root(729*a^2*b^6*d^6*z^6 + 729*b^
8*d^6*z^6 - 243*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - a^2, z, k) + d*x) - 216*root(729*a^2*b^6*d^6*z^6 + 729*
b^8*d^6*z^6 - 243*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - a^2, z, k)^3*a^2*b^3*d^3*exp(root(729*a^2*b^6*d^6*z^6
+ 729*b^8*d^6*z^6 - 243*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - a^2, z, k) + d*x) + 108*root(729*a^2*b^6*d^6*z
^6 + 729*b^8*d^6*z^6 - 243*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - a^2, z, k)^4*a^2*b^4*d^4*exp(root(729*a^2*b^
6*d^6*z^6 + 729*b^8*d^6*z^6 - 243*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - a^2, z, k) + d*x) + 324*root(729*a^2*
b^6*d^6*z^6 + 729*b^8*d^6*z^6 - 243*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - a^2, z, k)^5*a^2*b^5*d^5*exp(root(7
29*a^2*b^6*d^6*z^6 + 729*b^8*d^6*z^6 - 243*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - a^2, z, k) + d*x)))/b^10)*ro
ot(729*a^2*b^6*d^6*z^6 + 729*b^8*d^6*z^6 - 243*a^2*b^4*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - a^2, z, k), k, 1, 6) + x
/b

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