3.2.72 \(\int \frac {\sinh ^5(c+d x)}{a+b \sinh ^3(c+d x)} \, dx\) [172]

Optimal. Leaf size=295 \[ -\frac {x}{2 b}+\frac {2 a \text {ArcTan}\left (\frac {(-1)^{5/6} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}} b^{5/3} d}+\frac {2 a \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} b^{5/3} d}+\frac {2 a \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {a^{2/3}+b^{2/3}} b^{5/3} d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d} \]

[Out]

-1/2*x/b+1/2*cosh(d*x+c)*sinh(d*x+c)/b/d+2/3*a*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^(5/3)/d/((-1)^(1/3)*a^(2/3)-b^(2/3))^(1/2)+2/3*a*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^(5/3)/d/(-(-1)^(2/3
)*a^(2/3)-b^(2/3))^(1/2)+2/3*a*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*d*x+1/2*c))/(a^(2/3)+b^(2/3))^(1/2))/b^(5/3)/
d/(a^(2/3)+b^(2/3))^(1/2)

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Rubi [A]
time = 0.42, antiderivative size = 295, 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, 2715, 8, 2739, 632, 212, 210} \begin {gather*} \frac {2 a \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^{5/3} d \sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}+\frac {2 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^{5/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}+\frac {2 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^{5/3} d \sqrt {a^{2/3}+b^{2/3}}}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {x}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^5/(a + b*Sinh[c + d*x]^3),x]

[Out]

-1/2*x/b + (2*a*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^(5/3)*d) + (2*a*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^(5/3)*d) + (2*a*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^(5/3)*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 2715

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

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 ^5(c+d x)}{a+b \sinh ^3(c+d x)} \, dx &=-\left (i \int \left (\frac {i \sinh ^2(c+d x)}{b}-\frac {i a \sinh ^2(c+d x)}{b \left (a+b \sinh ^3(c+d x)\right )}\right ) \, dx\right )\\ &=\frac {\int \sinh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {\sinh ^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx}{b}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {\int 1 \, dx}{2 b}+\frac {a \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}{b}\\ &=-\frac {x}{2 b}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d}+\frac {(i a) \int \frac {1}{-i \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 b^{5/3}}+\frac {(i a) \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 b^{5/3}}+\frac {(i a) \int \frac {1}{(-1)^{5/6} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 b^{5/3}}\\ &=-\frac {x}{2 b}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d}+\frac {(2 a) \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^{5/3} d}+\frac {(2 a) \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^{5/3} d}+\frac {(2 a) \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^{5/3} d}\\ &=-\frac {x}{2 b}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {(4 a) \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^{5/3} d}-\frac {(4 a) \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^{5/3} d}-\frac {(4 a) \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^{5/3} d}\\ &=-\frac {x}{2 b}-\frac {2 a \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^{5/3} d}-\frac {2 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^{5/3} d}+\frac {2 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^{5/3} d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 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 = 0.23, size = 299, normalized size = 1.01 \begin {gather*} \frac {-6 (c+d x)-2 a \text {RootSum}\left [-b+3 b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {c+d x+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-2 c \text {$\#$1}^2-2 d x \text {$\#$1}^2-4 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+c \text {$\#$1}^4+d x \text {$\#$1}^4+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+4 a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]+3 \sinh (2 (c+d x))}{12 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^5/(a + b*Sinh[c + d*x]^3),x]

[Out]

(-6*(c + d*x) - 2*a*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) & ] + 3*Sinh[2*(c + d*x)])/(12*b*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.21, size = 190, normalized size = 0.64

method	result	size
derivativedivides	\(\frac {\frac {4 a \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 b}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b}}{d}\)	\(190\)
default	\(\frac {\frac {4 a \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 b}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b}}{d}\)	\(190\)
risch	\(-\frac {x}{2 b}+\frac {{\mathrm e}^{2 d x +2 c}}{8 b d}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 b d}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (729 a^{2} b^{10} d^{6}+729 b^{12} d^{6}\right ) \textit {\_Z}^{6}-243 a^{2} b^{8} d^{4} \textit {\_Z}^{4}+27 a^{4} d^{2} \textit {\_Z}^{2} b^{4}-a^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (\frac {243 d^{5} b^{8}}{a^{3}}+\frac {243 d^{5} b^{10}}{a^{5}}\right ) \textit {\_R}^{5}+\left (\frac {81 b^{7} d^{4}}{a^{3}}+\frac {81 b^{9} d^{4}}{a^{5}}\right ) \textit {\_R}^{4}-\frac {81 d^{3} b^{6} \textit {\_R}^{3}}{a^{3}}+\left (\frac {9 d^{2} b^{3}}{a}-\frac {18 d^{2} b^{5}}{a^{3}}\right ) \textit {\_R}^{2}+\frac {9 d \,b^{2} \textit {\_R}}{a}+\frac {b}{a}\right )\right )\)	\(224\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^5/(a+b*sinh(d*x+c)^3),x,method=_RETURNVERBOSE)

[Out]

1/d*(4/3*a/b*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))-1/2/b/(tanh(1/2*d*x+1/2*c)+1)^2+1/2/b/(tanh(1/2*d*x+1/2*c)+1)-1/2/b*ln(tanh(1/2*d*x+1/2*c)+
1)+1/2/b/(tanh(1/2*d*x+1/2*c)-1)^2+1/2/b/(tanh(1/2*d*x+1/2*c)-1)+1/2/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)^5/(a+b*sinh(d*x+c)^3),x, algorithm="maxima")

[Out]

-1/8*(4*d*x*e^(2*d*x + 2*c) - e^(4*d*x + 4*c) + 1)*e^(-2*d*x - 2*c)/(b*d) - 1/32*integrate(64*(a*e^(5*d*x + 5*
c) - 2*a*e^(3*d*x + 3*c) + a*e^(d*x + 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)

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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. 28427 vs. \(2 (210) = 420\).
time = 3.34, size = 28427, normalized size = 96.36 \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)^5/(a+b*sinh(d*x+c)^3),x, algorithm="fricas")

[Out]

-1/24*(12*d*x*cosh(d*x + c)^2 - 3*cosh(d*x + c)^4 - 12*cosh(d*x + c)*sinh(d*x + c)^3 - 3*sinh(d*x + c)^4 - 12*
sqrt(2/3)*sqrt(1/6)*(b*d*cosh(d*x + c)^2 + 2*b*d*cosh(d*x + c)*sinh(d*x + c) + b*d*sinh(d*x + c)^2)*sqrt(-((a^
2*b^2 + b^4)*(2*(1/2)^(2/3)*(a^4/(a^2*b^6*d^4 + b^8*d^4) - a^4/(a^2*b^2*d^2 + b^4*d^2)^2)*(-I*sqrt(3) + 1)/(a^
6/(a^2*b^10*d^6 + b^12*d^6) - 3*a^6/((a^2*b^6*d^4 + b^8*d^4)*(a^2*b^2*d^2 + b^4*d^2)) + 2*a^6/(a^2*b^2*d^2 + b
^4*d^2)^3 + a^8/((a^2 + b^2)^2*b^10*d^6))^(1/3) - (1/2)^(1/3)*(a^6/(a^2*b^10*d^6 + b^12*d^6) - 3*a^6/((a^2*b^6
*d^4 + b^8*d^4)*(a^2*b^2*d^2 + b^4*d^2)) + 2*a^6/(a^2*b^2*d^2 + b^4*d^2)^3 + a^8/((a^2 + b^2)^2*b^10*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^6 + 2*
a^2*b^8 + b^10)*(2*(1/2)^(2/3)*(a^4/(a^2*b^6*d^4 + b^8*d^4) - a^4/(a^2*b^2*d^2 + b^4*d^2)^2)*(-I*sqrt(3) + 1)/
(a^6/(a^2*b^10*d^6 + b^12*d^6) - 3*a^6/((a^2*b^6*d^4 + b^8*d^4)*(a^2*b^2*d^2 + b^4*d^2)) + 2*a^6/(a^2*b^2*d^2
+ b^4*d^2) ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^{5}{\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)**5/(a+b*sinh(d*x+c)**3),x)

[Out]

Integral(sinh(c + d*x)**5/(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)^5/(a+b*sinh(d*x+c)^3),x, algorithm="giac")

[Out]

integrate(sinh(d*x + c)^5/(b*sinh(d*x + c)^3 + a), x)

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Mupad [B]
time = 11.48, size = 1114, normalized size = 3.78 \begin {gather*} \left (\sum _{k=1}^6\ln \left (-\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )\,\left (\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )\,\left (\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )\,\left (\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )\,\left (\frac {\left (8\,a^6\,d^4+4\,a^4\,b^2\,d^4-a^5\,b\,d^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,5\right )\,663552}{b^7}+\frac {\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )\,\left (-a^4\,b\,d^5+a^5\,d^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,4+a^3\,b^2\,d^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,5\right )\,1990656}{b^5}\right )+\frac {\left (4\,a^6\,b\,d^3+a^7\,d^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,8-a^5\,b^2\,d^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,5\right )\,442368}{b^9}\right )-\frac {a^6\,d^2\,\left (2\,b-a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,5\right )\,294912}{b^{10}}\right )-\frac {a^7\,d\,\left (8\,a-b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,5\right )\,24576}{b^{12}}\right )-\frac {a^8\,\left (b-a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,4\right )\,32768}{b^{14}}\right )\,\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )\right )-\frac {x}{2\,b}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^5/(a + b*sinh(c + d*x)^3),x)

[Out]

symsum(log(- root(729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z,
k)*(root(729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k)*(roo
t(729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k)*(root(729*a^
2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k)*((663552*(8*a^6*d^4
+ 4*a^4*b^2*d^4 - 5*a^5*b*d^4*exp(d*x)*exp(root(729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4
+ 27*a^4*b^4*d^2*z^2 - a^6, z, k))))/b^7 + (1990656*root(729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8
*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k)*(4*a^5*d^5*exp(d*x)*exp(root(729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^
6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k)) - a^4*b*d^5 + 5*a^3*b^2*d^5*exp(d*x)*exp(root(729*a
^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k))))/b^5) + (442368*(
4*a^6*b*d^3 + 8*a^7*d^3*exp(d*x)*exp(root(729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a
^4*b^4*d^2*z^2 - a^6, z, k)) - 5*a^5*b^2*d^3*exp(d*x)*exp(root(729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a
^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k))))/b^9) - (294912*a^6*d^2*(2*b - 5*a*exp(d*x)*exp(root(729*a^
2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k))))/b^10) - (24576*a^
7*d*(8*a - 5*b*exp(d*x)*exp(root(729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^
2*z^2 - a^6, z, k))))/b^12) - (32768*a^8*(b - 4*a*exp(d*x)*exp(root(729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 -
243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k))))/b^14)*root(729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 -
243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k), k, 1, 6) - x/(2*b) - exp(- 2*c - 2*d*x)/(8*b*d) + exp(2
*c + 2*d*x)/(8*b*d)

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