3.2.71 \(\int \frac {\sinh ^6(c+d x)}{a+b \sinh ^3(c+d x)} \, dx\) [171]

Optimal. Leaf size=328 \[ -\frac {a x}{b^2}-\frac {2 (-1)^{2/3} a^{4/3} \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} b^2 d}-\frac {2 (-1)^{2/3} a^{4/3} \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} b^2 d}-\frac {2 a^{4/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {a^{2/3}+b^{2/3}} b^2 d}-\frac {\cosh (c+d x)}{b d}+\frac {\cosh ^3(c+d x)}{3 b d} \]

[Out]

-a*x/b^2-cosh(d*x+c)/b/d+1/3*cosh(d*x+c)^3/b/d-2/3*(-1)^(2/3)*a^(4/3)*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/d/((-1)^(1/3)*a^(2/3)-b^(2/3))^(1/2)-2/3*
a^(4/3)*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/d/(a^(2/3)+b^(2/3))^(1/2)-2
/3*(-1)^(2/3)*a^(4/3)*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^2/d/((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3))^(1/2)

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Rubi [A]
time = 0.63, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3299, 2713, 3292, 2739, 632, 210} \begin {gather*} -\frac {2 (-1)^{2/3} a^{4/3} \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 b^2 d \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac {2 (-1)^{2/3} a^{4/3} \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 b^2 d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}-\frac {2 a^{4/3} \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 d \sqrt {a^{2/3}+b^{2/3}}}-\frac {a x}{b^2}+\frac {\cosh ^3(c+d x)}{3 b d}-\frac {\cosh (c+d x)}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^6/(a + b*Sinh[c + d*x]^3),x]

[Out]

-((a*x)/b^2) - (2*(-1)^(2/3)*a^(4/3)*ArcTan[((-1)^(1/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(c + d*x)/2]))/Sq
rt[(-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^2*d) - (2*(-1
)^(2/3)*a^(4/3)*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^2*d) - (2*a^(4/3)*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^2*d) - Cosh[c + d*x]/(b*d) + Cosh[c + d*x]^3/(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 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 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 ^6(c+d x)}{a+b \sinh ^3(c+d x)} \, dx &=-\int \left (\frac {a}{b^2}-\frac {\sinh ^3(c+d x)}{b}-\frac {a^2}{b^2 \left (a+b \sinh ^3(c+d x)\right )}\right ) \, dx\\ &=-\frac {a x}{b^2}+\frac {a^2 \int \frac {1}{a+b \sinh ^3(c+d x)} \, dx}{b^2}+\frac {\int \sinh ^3(c+d x) \, dx}{b}\\ &=-\frac {a x}{b^2}+\frac {a^2 \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^2}-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{b d}\\ &=-\frac {a x}{b^2}-\frac {\cosh (c+d x)}{b d}+\frac {\cosh ^3(c+d x)}{3 b d}+\frac {\left (\sqrt [6]{-1} a^{4/3}\right ) \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 b^2}+\frac {\left (\sqrt [6]{-1} a^{4/3}\right ) \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}+\sqrt [6]{-1} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 b^2}+\frac {\left (\sqrt [6]{-1} a^{4/3}\right ) \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}+(-1)^{5/6} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 b^2}\\ &=-\frac {a x}{b^2}-\frac {\cosh (c+d x)}{b d}+\frac {\cosh ^3(c+d x)}{3 b d}-\frac {\left (2 (-1)^{2/3} a^{4/3}\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 b^2 d}-\frac {\left (2 (-1)^{2/3} a^{4/3}\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 b^2 d}-\frac {\left (2 (-1)^{2/3} a^{4/3}\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 b^2 d}\\ &=-\frac {a x}{b^2}-\frac {\cosh (c+d x)}{b d}+\frac {\cosh ^3(c+d x)}{3 b d}+\frac {\left (4 (-1)^{2/3} a^{4/3}\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 b^2 d}+\frac {\left (4 (-1)^{2/3} a^{4/3}\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 b^2 d}+\frac {\left (4 (-1)^{2/3} a^{4/3}\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 b^2 d}\\ &=-\frac {a x}{b^2}+\frac {2 (-1)^{2/3} a^{4/3} \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 \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} b^2 d}-\frac {2 (-1)^{2/3} a^{4/3} \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 \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} b^2 d}-\frac {2 a^{4/3} \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 d}-\frac {\cosh (c+d x)}{b d}+\frac {\cosh ^3(c+d x)}{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 = 0.26, size = 168, normalized size = 0.51 \begin {gather*} \frac {-12 a c-12 a d x-9 b \cosh (c+d x)+b \cosh (3 (c+d x))+8 a^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 ]}{12 b^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^6/(a + b*Sinh[c + d*x]^3),x]

[Out]

(-12*a*c - 12*a*d*x - 9*b*Cosh[c + d*x] + b*Cosh[3*(c + d*x)] + 8*a^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) & ])/(12*b^2*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.30, size = 236, normalized size = 0.72

method	result	size
derivativedivides	\(\frac {\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}-\frac {a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 b^{2}}-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}}{d}\)	\(236\)
default	\(\frac {\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}-\frac {a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 b^{2}}-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}}{d}\)	\(236\)
risch	\(-\frac {a x}{b^{2}}+\frac {{\mathrm e}^{3 d x +3 c}}{24 b d}-\frac {3 \,{\mathrm e}^{d x +c}}{8 b d}-\frac {3 \,{\mathrm e}^{-d x -c}}{8 b d}+\frac {{\mathrm e}^{-3 d x -3 c}}{24 b d}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (729 a^{2} b^{12} d^{6}+729 b^{14} d^{6}\right ) \textit {\_Z}^{6}-243 a^{4} b^{8} d^{4} \textit {\_Z}^{4}+27 a^{6} d^{2} \textit {\_Z}^{2} b^{4}-a^{8}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (-\frac {486 b^{9} d^{5}}{a^{4}}-\frac {486 b^{11} d^{5}}{a^{6}}\right ) \textit {\_R}^{5}+\left (\frac {81 b^{7} d^{4}}{a^{3}}+\frac {81 b^{9} d^{4}}{a^{5}}\right ) \textit {\_R}^{4}+\left (\frac {135 b^{5} d^{3}}{a^{2}}-\frac {27 b^{7} d^{3}}{a^{4}}\right ) \textit {\_R}^{3}-\frac {27 b^{3} d^{2} \textit {\_R}^{2}}{a}-9 b d \textit {\_R} +\frac {2 a}{b}\right )\right )\)	\(252\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^6/(a+b*sinh(d*x+c)^3),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/3/b/(tanh(1/2*d*x+1/2*c)+1)^3-1/2/b/(tanh(1/2*d*x+1/2*c)+1)^2-1/2/b/(tanh(1/2*d*x+1/2*c)+1)-a/b^2*ln(ta
nh(1/2*d*x+1/2*c)+1)-1/3*a^2/b^2*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))-1/3/b/(tanh(1/2*d*x+1/2*c)-1)^3-1/2/b/(tanh(1/2*d*x+1/2*c)-1
)^2+1/2/b/(tanh(1/2*d*x+1/2*c)-1)+a/b^2*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)^6/(a+b*sinh(d*x+c)^3),x, algorithm="maxima")

[Out]

8*a^2*integrate(e^(3*d*x + 3*c)/(b^3*e^(6*d*x + 6*c) - 3*b^3*e^(4*d*x + 4*c) + 8*a*b^2*e^(3*d*x + 3*c) + 3*b^3
*e^(2*d*x + 2*c) - b^3), x) - 1/24*(24*a*d*x*e^(3*d*x + 3*c) - b*e^(6*d*x + 6*c) + 9*b*e^(4*d*x + 4*c) + 9*b*e
^(2*d*x + 2*c) - b)*e^(-3*d*x - 3*c)/(b^2*d)

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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. 28816 vs. \(2 (237) = 474\).
time = 5.59, size = 28816, normalized size = 87.85 \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)^6/(a+b*sinh(d*x+c)^3),x, algorithm="fricas")

[Out]

1/24*(b*cosh(d*x + c)^6 + 6*b*cosh(d*x + c)*sinh(d*x + c)^5 + b*sinh(d*x + c)^6 - 24*a*d*x*cosh(d*x + c)^3 - 9
*b*cosh(d*x + c)^4 + 3*(5*b*cosh(d*x + c)^2 - 3*b)*sinh(d*x + c)^4 + 4*(5*b*cosh(d*x + c)^3 - 6*a*d*x - 9*b*co
sh(d*x + c))*sinh(d*x + c)^3 + 12*sqrt(2/3)*sqrt(1/6)*(b^2*d*cosh(d*x + c)^3 + 3*b^2*d*cosh(d*x + c)^2*sinh(d*
x + c) + 3*b^2*d*cosh(d*x + c)*sinh(d*x + c)^2 + b^2*d*sinh(d*x + c)^3)*sqrt((6*a^4 - (a^2*b^4 + b^6)*(2*a^4/(
a^2*b^4*d^2 + b^6*d^2) - 2*(1/2)^(2/3)*(a^8/(a^2*b^4*d^2 + b^6*d^2)^2 - a^6/(a^2*b^8*d^4 + b^10*d^4))*(-I*sqrt
(3) + 1)/(2*a^12/(a^2*b^4*d^2 + b^6*d^2)^3 - 3*a^10/((a^2*b^8*d^4 + b^10*d^4)*(a^2*b^4*d^2 + b^6*d^2)) + a^8/(
a^2*b^12*d^6 + b^14*d^6) + a^8/((a^2 + b^2)^2*b^10*d^6))^(1/3) - (1/2)^(1/3)*(2*a^12/(a^2*b^4*d^2 + b^6*d^2)^3
- 3*a^10/((a^2*b^8*d^4 + b^10*d^4)*(a^2*b^4*d^2 + b^6*d^2)) + a^8/(a^2*b^12*d^6 + b^14*d^6) + a^8/((a^2 + b^2
)^2*b^10*d^6))^(1/3)*(I*sqrt(3) + 1))*d^2 - 3*sqrt(1/3)*(a^2*b^4 + b^6)*d^2*sqrt(-(4*a^8 + 16*a^6*b^2 + (a^4*b
^8 + 2*a^2 ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^{6}{\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)**6/(a+b*sinh(d*x+c)**3),x)

[Out]

Integral(sinh(c + d*x)**6/(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)^6/(a+b*sinh(d*x+c)^3),x, algorithm="giac")

[Out]

integrate(sinh(d*x + c)^6/(b*sinh(d*x + c)^3 + a), x)

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Mupad [B]
time = 10.80, size = 1579, normalized size = 4.81 \begin {gather*} \left (\sum _{k=1}^6\ln \left (\frac {-a^{10}\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}\,98304+{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}^2\,a^7\,b^5\,d^2\,294912+{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}^3\,a^6\,b^7\,d^3\,1327104+{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}^4\,a^5\,b^9\,d^4\,2654208+{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}^5\,a^4\,b^{11}\,d^5\,1990656+\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )\,a^8\,b^3\,d\,24576+{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}^2\,a^8\,b^4\,d^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}\,589824+{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}^3\,a^7\,b^6\,d^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}\,5308416-{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}^4\,a^4\,b^{10}\,d^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}\,663552+{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}^4\,a^6\,b^8\,d^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}\,2654208-{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}^5\,a^3\,b^{12}\,d^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}\,9953280-{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}^5\,a^5\,b^{10}\,d^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}\,7962624-\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )\,a^9\,b^2\,d\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )}\,491520}{b^{15}}\right )\,\mathrm {root}\left (729\,a^2\,b^{12}\,d^6\,z^6+729\,b^{14}\,d^6\,z^6-243\,a^4\,b^8\,d^4\,z^4+27\,a^6\,b^4\,d^2\,z^2-a^8,z,k\right )\right )-\frac {3\,{\mathrm {e}}^{c+d\,x}}{8\,b\,d}-\frac {3\,{\mathrm {e}}^{-c-d\,x}}{8\,b\,d}+\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,b\,d}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,b\,d}-\frac {a\,x}{b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^6/(a + b*sinh(c + d*x)^3),x)

[Out]

symsum(log((294912*root(729*a^2*b^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a
^8, z, k)^2*a^7*b^5*d^2 - 98304*a^10*exp(d*x)*exp(root(729*a^2*b^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b^8*d
^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8, z, k)) + 1327104*root(729*a^2*b^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b^8
*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8, z, k)^3*a^6*b^7*d^3 + 2654208*root(729*a^2*b^12*d^6*z^6 + 729*b^14*d^6*z^
6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8, z, k)^4*a^5*b^9*d^4 + 1990656*root(729*a^2*b^12*d^6*z^6 +
729*b^14*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8, z, k)^5*a^4*b^11*d^5 + 24576*root(729*a^2*b
^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8, z, k)*a^8*b^3*d + 589824*root
(729*a^2*b^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8, z, k)^2*a^8*b^4*d^2
*exp(d*x)*exp(root(729*a^2*b^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8, z
, k)) + 5308416*root(729*a^2*b^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8,
z, k)^3*a^7*b^6*d^3*exp(d*x)*exp(root(729*a^2*b^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*
b^4*d^2*z^2 - a^8, z, k)) - 663552*root(729*a^2*b^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6
*b^4*d^2*z^2 - a^8, z, k)^4*a^4*b^10*d^4*exp(d*x)*exp(root(729*a^2*b^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b
^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8, z, k)) + 2654208*root(729*a^2*b^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4
*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8, z, k)^4*a^6*b^8*d^4*exp(d*x)*exp(root(729*a^2*b^12*d^6*z^6 + 729*b^14
*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8, z, k)) - 9953280*root(729*a^2*b^12*d^6*z^6 + 729*b^
14*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8, z, k)^5*a^3*b^12*d^5*exp(d*x)*exp(root(729*a^2*b^
12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8, z, k)) - 7962624*root(729*a^2*
b^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8, z, k)^5*a^5*b^10*d^5*exp(d*x
)*exp(root(729*a^2*b^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8, z, k)) -
491520*root(729*a^2*b^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 - a^8, z, k)*a^
9*b^2*d*exp(d*x)*exp(root(729*a^2*b^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 -
a^8, z, k)))/b^15)*root(729*a^2*b^12*d^6*z^6 + 729*b^14*d^6*z^6 - 243*a^4*b^8*d^4*z^4 + 27*a^6*b^4*d^2*z^2 -
a^8, z, k), k, 1, 6) - (3*exp(c + d*x))/(8*b*d) - (3*exp(- c - d*x))/(8*b*d) + exp(- 3*c - 3*d*x)/(24*b*d) + e
xp(3*c + 3*d*x)/(24*b*d) - (a*x)/b^2

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