3.3.35 \(\int \frac {\sinh ^6(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\) [235]

Optimal. Leaf size=175 \[ \frac {x}{2 b}-\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{3/2} d}+\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{3/2} d}-\frac {1}{4 b d (1-\tanh (c+d x))}+\frac {1}{4 b d (1+\tanh (c+d x))} \]

[Out]

1/2*x/b-1/2*a^(3/4)*arctanh((a^(1/2)-b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))/b^(3/2)/d/(a^(1/2)-b^(1/2))^(1/2)+1/2
*a^(3/4)*arctanh((a^(1/2)+b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))/b^(3/2)/d/(a^(1/2)+b^(1/2))^(1/2)-1/4/b/d/(1-tan
h(d*x+c))+1/4/b/d/(1+tanh(d*x+c))

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Rubi [A]
time = 0.18, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3296, 1301, 213, 1144, 214} \begin {gather*} -\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 b^{3/2} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 b^{3/2} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {1}{4 b d (1-\tanh (c+d x))}+\frac {1}{4 b d (\tanh (c+d x)+1)}+\frac {x}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^6/(a - b*Sinh[c + d*x]^4),x]

[Out]

x/(2*b) - (a^(3/4)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(3/2
)*d) + (a^(3/4)*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(3/2)*d
) - 1/(4*b*d*(1 - Tanh[c + d*x])) + 1/(4*b*d*(1 + Tanh[c + d*x]))

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 1144

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2/2)*(b/q + 1), Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2/2)*(b/q - 1), Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]

Rule 1301

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]

Rule 3296

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*
x^2)^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\sinh ^6(c+d x)}{a-b \sinh ^4(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2 \left (a-2 a x^2+(a-b) x^4\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{4 b (-1+x)^2}-\frac {1}{4 b (1+x)^2}-\frac {1}{2 b \left (-1+x^2\right )}+\frac {a x^2}{b \left (a-2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {1}{4 b d (1-\tanh (c+d x))}+\frac {1}{4 b d (1+\tanh (c+d x))}-\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{2 b d}+\frac {a \text {Subst}\left (\int \frac {x^2}{a-2 a x^2+(a-b) x^4} \, dx,x,\tanh (c+d x)\right )}{b d}\\ &=\frac {x}{2 b}-\frac {1}{4 b d (1-\tanh (c+d x))}+\frac {1}{4 b d (1+\tanh (c+d x))}+\frac {\left (a \left (\sqrt {a}+\sqrt {b}\right )\right ) \text {Subst}\left (\int \frac {1}{-a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{2 b^{3/2} d}+\frac {\left (a \left (1-\frac {\sqrt {a}}{\sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {1}{-a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{2 b d}\\ &=\frac {x}{2 b}-\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{3/2} d}+\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{3/2} d}-\frac {1}{4 b d (1-\tanh (c+d x))}+\frac {1}{4 b d (1+\tanh (c+d x))}\\ \end {align*}

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Mathematica [A]
time = 0.66, size = 158, normalized size = 0.90 \begin {gather*} \frac {2 \sqrt {b} (c+d x)+\frac {2 a \text {ArcTan}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {2 a \tanh ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}-\sqrt {b} \sinh (2 (c+d x))}{4 b^{3/2} d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^6/(a - b*Sinh[c + d*x]^4),x]

[Out]

(2*Sqrt[b]*(c + d*x) + (2*a*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a +
Sqrt[a]*Sqrt[b]] + (2*a*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[
a]*Sqrt[b]] - Sqrt[b]*Sinh[2*(c + d*x)])/(4*b^(3/2)*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.69, size = 206, normalized size = 1.18

method	result	size
risch	\(\frac {x}{2 b}-\frac {{\mathrm e}^{2 d x +2 c}}{8 b d}+\frac {{\mathrm e}^{-2 d x -2 c}}{8 b d}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (256 a \,b^{6} d^{4}-256 b^{7} d^{4}\right ) \textit {\_Z}^{4}-32 a^{2} b^{3} d^{2} \textit {\_Z}^{2}+a^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (-\frac {128 b^{4} d^{3}}{a}+\frac {128 b^{5} d^{3}}{a^{2}}\right ) \textit {\_R}^{3}+\left (32 b^{2} d^{2}-\frac {32 d^{2} b^{3}}{a}\right ) \textit {\_R}^{2}+16 b d \textit {\_R} -\frac {2 a}{b}-1\right )\right )\)	\(162\)
derivativedivides	\(\frac {\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b}-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-\textit {\_R}^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}\right )}{b}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b}}{d}\)	\(206\)
default	\(\frac {\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b}-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-\textit {\_R}^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}\right )}{b}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b}}{d}\)	\(206\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^6/(a-b*sinh(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

1/d*(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)-a/b*sum((_R
^4-_R^2)/(_R^7*a-3*_R^5*a+3*_R^3*a-8*_R^3*b-_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(a*_Z^8-4*a*_Z^6+(6*a-16
*b)*_Z^4-4*a*_Z^2+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))

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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)^4),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) - 1/64*integrate(256*(a*e^(6*d*x + 6*
c) - 2*a*e^(4*d*x + 4*c) + a*e^(2*d*x + 2*c))/(b^2*e^(8*d*x + 8*c) - 4*b^2*e^(6*d*x + 6*c) - 4*b^2*e^(2*d*x +
2*c) + b^2 - 2*(8*a*b*e^(4*c) - 3*b^2*e^(4*c))*e^(4*d*x)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1441 vs. \(2 (127) = 254\).
time = 0.44, size = 1441, normalized size = 8.23 \begin {gather*} \frac {4 \, d x \cosh \left (d x + c\right )^{2} - \cosh \left (d x + c\right )^{4} - 4 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} - \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, d x - 3 \, \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} - 2 \, {\left (b d \cosh \left (d x + c\right )^{2} + 2 \, b d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b d \sinh \left (d x + c\right )^{2}\right )} \sqrt {\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}} \log \left (a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} b^{2} - a b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a^{2} + 2 \, {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a^{2} b d\right )} \sqrt {\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}}\right ) + 2 \, {\left (b d \cosh \left (d x + c\right )^{2} + 2 \, b d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b d \sinh \left (d x + c\right )^{2}\right )} \sqrt {\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}} \log \left (a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} b^{2} - a b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a^{2} - 2 \, {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a^{2} b d\right )} \sqrt {\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}}\right ) + 2 \, {\left (b d \cosh \left (d x + c\right )^{2} + 2 \, b d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b d \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}} \log \left (a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )^{2} - 2 \, {\left (a^{2} b^{2} - a b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a^{2} + 2 \, {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a^{2} b d\right )} \sqrt {-\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}}\right ) - 2 \, {\left (b d \cosh \left (d x + c\right )^{2} + 2 \, b d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b d \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}} \log \left (a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )^{2} - 2 \, {\left (a^{2} b^{2} - a b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a^{2} - 2 \, {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a^{2} b d\right )} \sqrt {-\frac {{\left (a b^{3} - b^{4}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a^{2}}{{\left (a b^{3} - b^{4}\right )} d^{2}}}\right ) + 4 \, {\left (2 \, d x \cosh \left (d x + c\right ) - \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right ) + 1}{8 \, {\left (b d \cosh \left (d x + c\right )^{2} + 2 \, b d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b d \sinh \left (d x + c\right )^{2}\right )}} \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)^4),x, algorithm="fricas")

[Out]

1/8*(4*d*x*cosh(d*x + c)^2 - cosh(d*x + c)^4 - 4*cosh(d*x + c)*sinh(d*x + c)^3 - sinh(d*x + c)^4 + 2*(2*d*x -
3*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 2*(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*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))*log(a^2
*cosh(d*x + c)^2 + 2*a^2*cosh(d*x + c)*sinh(d*x + c) + a^2*sinh(d*x + c)^2 + 2*(a^2*b^2 - a*b^3)*d^2*sqrt(a^3/
((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2 + 2*((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2*
b*d)*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))) + 2*(b*d*c
osh(d*x + c)^2 + 2*b*d*cosh(d*x + c)*sinh(d*x + c) + b*d*sinh(d*x + c)^2)*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^3/((a
^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))*log(a^2*cosh(d*x + c)^2 + 2*a^2*cosh(d*x + c)*sinh(d
*x + c) + a^2*sinh(d*x + c)^2 + 2*(a^2*b^2 - a*b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2 - 2*((
a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2*b*d)*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b
^5 - 2*a*b^6 + b^7)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))) + 2*(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*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2)/((a*b^3
- b^4)*d^2))*log(a^2*cosh(d*x + c)^2 + 2*a^2*cosh(d*x + c)*sinh(d*x + c) + a^2*sinh(d*x + c)^2 - 2*(a^2*b^2 -
a*b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2 + 2*((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^
6 + b^7)*d^4)) + a^2*b*d)*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2)/((a*b^3 -
b^4)*d^2))) - 2*(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*b^3
- b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))*log(a^2*cosh(d*x + c)^2 + 2*a
^2*cosh(d*x + c)*sinh(d*x + c) + a^2*sinh(d*x + c)^2 - 2*(a^2*b^2 - a*b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 +
b^7)*d^4)) - a^2 - 2*((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2*b*d)*sqrt(-((a*b^3 - b
^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))) + 4*(2*d*x*cosh(d*x + c) - cosh
(d*x + c)^3)*sinh(d*x + c) + 1)/(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
)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**4),x)

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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)^4),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Not invertible Error: Bad Ar
gument Valu

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Mupad [B]
time = 11.24, size = 2191, normalized size = 12.52 \begin {gather*} \text {Too large to display} \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)^4),x)

[Out]

log((((((4194304*a^6*d^2*(512*a^4 - 1184*a^3*b - 253*a*b^3 - b^4 + 930*a^2*b^2 + b^4*exp(2*c + 2*d*x) + 627*a*
b^3*exp(2*c + 2*d*x) + 768*a^3*b*exp(2*c + 2*d*x) - 1392*a^2*b^2*exp(2*c + 2*d*x)))/(b^12*(a - b)^2) - (167772
16*a^6*d^3*(((a^3*b^7)^(1/2) + a^2*b^3)/(b^6*d^2*(a - b)))^(1/2)*(40*a*b^2 - 35*b^3 + 512*a^3*exp(2*c + 2*d*x)
+ 64*b^3*exp(2*c + 2*d*x) + 326*a*b^2*exp(2*c + 2*d*x) - 896*a^2*b*exp(2*c + 2*d*x)))/(b^11*(a - b)))*(((a^3*
b^7)^(1/2) + a^2*b^3)/(b^6*d^2*(a - b)))^(1/2))/4 - (2097152*a^7*d*(256*a^2*b - 256*a*b^2 - 5*b^3 - 1024*a^3*e
xp(2*c + 2*d*x) + 6*b^3*exp(2*c + 2*d*x) + 756*a*b^2*exp(2*c + 2*d*x) + 256*a^2*b*exp(2*c + 2*d*x)))/(b^14*(a
- b)))*(((a^3*b^7)^(1/2) + a^2*b^3)/(b^6*d^2*(a - b)))^(1/2))/4 - (524288*a^8*(185*a*b^2 - 464*a^2*b + 256*a^3
+ 24*b^3 - 1024*a^3*exp(2*c + 2*d*x) - 35*b^3*exp(2*c + 2*d*x) - 988*a*b^2*exp(2*c + 2*d*x) + 2048*a^2*b*exp(
2*c + 2*d*x)))/(b^15*(a - b)^2))*(-((a^3*b^7)^(1/2) + a^2*b^3)/(16*(b^7*d^2 - a*b^6*d^2)))^(1/2) - log((((((41
94304*a^6*d^2*(512*a^4 - 1184*a^3*b - 253*a*b^3 - b^4 + 930*a^2*b^2 + b^4*exp(2*c + 2*d*x) + 627*a*b^3*exp(2*c
+ 2*d*x) + 768*a^3*b*exp(2*c + 2*d*x) - 1392*a^2*b^2*exp(2*c + 2*d*x)))/(b^12*(a - b)^2) + (16777216*a^6*d^3*
(((a^3*b^7)^(1/2) + a^2*b^3)/(b^6*d^2*(a - b)))^(1/2)*(40*a*b^2 - 35*b^3 + 512*a^3*exp(2*c + 2*d*x) + 64*b^3*e
xp(2*c + 2*d*x) + 326*a*b^2*exp(2*c + 2*d*x) - 896*a^2*b*exp(2*c + 2*d*x)))/(b^11*(a - b)))*(((a^3*b^7)^(1/2)
+ a^2*b^3)/(b^6*d^2*(a - b)))^(1/2))/4 + (2097152*a^7*d*(256*a^2*b - 256*a*b^2 - 5*b^3 - 1024*a^3*exp(2*c + 2*
d*x) + 6*b^3*exp(2*c + 2*d*x) + 756*a*b^2*exp(2*c + 2*d*x) + 256*a^2*b*exp(2*c + 2*d*x)))/(b^14*(a - b)))*(((a
^3*b^7)^(1/2) + a^2*b^3)/(b^6*d^2*(a - b)))^(1/2))/4 - (524288*a^8*(185*a*b^2 - 464*a^2*b + 256*a^3 + 24*b^3 -
1024*a^3*exp(2*c + 2*d*x) - 35*b^3*exp(2*c + 2*d*x) - 988*a*b^2*exp(2*c + 2*d*x) + 2048*a^2*b*exp(2*c + 2*d*x
)))/(b^15*(a - b)^2))*(-((a^3*b^7)^(1/2) + a^2*b^3)/(16*(b^7*d^2 - a*b^6*d^2)))^(1/2) + log((((((4194304*a^6*d
^2*(512*a^4 - 1184*a^3*b - 253*a*b^3 - b^4 + 930*a^2*b^2 + b^4*exp(2*c + 2*d*x) + 627*a*b^3*exp(2*c + 2*d*x) +
768*a^3*b*exp(2*c + 2*d*x) - 1392*a^2*b^2*exp(2*c + 2*d*x)))/(b^12*(a - b)^2) - (16777216*a^6*d^3*(-((a^3*b^7
)^(1/2) - a^2*b^3)/(b^6*d^2*(a - b)))^(1/2)*(40*a*b^2 - 35*b^3 + 512*a^3*exp(2*c + 2*d*x) + 64*b^3*exp(2*c + 2
*d*x) + 326*a*b^2*exp(2*c + 2*d*x) - 896*a^2*b*exp(2*c + 2*d*x)))/(b^11*(a - b)))*(-((a^3*b^7)^(1/2) - a^2*b^3
)/(b^6*d^2*(a - b)))^(1/2))/4 - (2097152*a^7*d*(256*a^2*b - 256*a*b^2 - 5*b^3 - 1024*a^3*exp(2*c + 2*d*x) + 6*
b^3*exp(2*c + 2*d*x) + 756*a*b^2*exp(2*c + 2*d*x) + 256*a^2*b*exp(2*c + 2*d*x)))/(b^14*(a - b)))*(-((a^3*b^7)^
(1/2) - a^2*b^3)/(b^6*d^2*(a - b)))^(1/2))/4 - (524288*a^8*(185*a*b^2 - 464*a^2*b + 256*a^3 + 24*b^3 - 1024*a^
3*exp(2*c + 2*d*x) - 35*b^3*exp(2*c + 2*d*x) - 988*a*b^2*exp(2*c + 2*d*x) + 2048*a^2*b*exp(2*c + 2*d*x)))/(b^1
5*(a - b)^2))*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*(b^7*d^2 - a*b^6*d^2)))^(1/2) - log((((((4194304*a^6*d^2*(512*a
^4 - 1184*a^3*b - 253*a*b^3 - b^4 + 930*a^2*b^2 + b^4*exp(2*c + 2*d*x) + 627*a*b^3*exp(2*c + 2*d*x) + 768*a^3*
b*exp(2*c + 2*d*x) - 1392*a^2*b^2*exp(2*c + 2*d*x)))/(b^12*(a - b)^2) + (16777216*a^6*d^3*(-((a^3*b^7)^(1/2) -
a^2*b^3)/(b^6*d^2*(a - b)))^(1/2)*(40*a*b^2 - 35*b^3 + 512*a^3*exp(2*c + 2*d*x) + 64*b^3*exp(2*c + 2*d*x) + 3
26*a*b^2*exp(2*c + 2*d*x) - 896*a^2*b*exp(2*c + 2*d*x)))/(b^11*(a - b)))*(-((a^3*b^7)^(1/2) - a^2*b^3)/(b^6*d^
2*(a - b)))^(1/2))/4 + (2097152*a^7*d*(256*a^2*b - 256*a*b^2 - 5*b^3 - 1024*a^3*exp(2*c + 2*d*x) + 6*b^3*exp(2
*c + 2*d*x) + 756*a*b^2*exp(2*c + 2*d*x) + 256*a^2*b*exp(2*c + 2*d*x)))/(b^14*(a - b)))*(-((a^3*b^7)^(1/2) - a
^2*b^3)/(b^6*d^2*(a - b)))^(1/2))/4 - (524288*a^8*(185*a*b^2 - 464*a^2*b + 256*a^3 + 24*b^3 - 1024*a^3*exp(2*c
+ 2*d*x) - 35*b^3*exp(2*c + 2*d*x) - 988*a*b^2*exp(2*c + 2*d*x) + 2048*a^2*b*exp(2*c + 2*d*x)))/(b^15*(a - b)
^2))*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*(b^7*d^2 - a*b^6*d^2)))^(1/2) + x/(2*b) + exp(- 2*c - 2*d*x)/(8*b*d) - e
xp(2*c + 2*d*x)/(8*b*d)

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