3.3.36 \(\int \frac {\sinh ^4(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\) [236]

Optimal. Leaf size=127 \[ -\frac {x}{b}+\frac {\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b d}+\frac {\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b d} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 127, 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, 1180, 214} \begin {gather*} \frac {\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 b d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 b d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {x}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 1180

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

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

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Mathematica [A]
time = 0.33, size = 143, normalized size = 1.13 \begin {gather*} \frac {-2 (c+d x)-\frac {\sqrt {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 {\sqrt {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}}}}{2 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(c + d*x) - (Sqrt[a]*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqr
t[a]*Sqrt[b]] + (Sqrt[a]*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt
[a]*Sqrt[b]])/(2*b*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.52, size = 139, normalized size = 1.09

method	result	size
risch	\(-\frac {x}{b}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (256 a \,b^{4} d^{4}-256 b^{5} d^{4}\right ) \textit {\_Z}^{4}-32 a \,b^{2} d^{2} \textit {\_Z}^{2}+a \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (-128 a \,b^{2} d^{3}+128 b^{3} d^{3}\right ) \textit {\_R}^{3}+\left (32 a \,d^{2} b -32 b^{2} d^{2}\right ) \textit {\_R}^{2}+\left (8 a d +8 b d \right ) \textit {\_R} -\frac {2 a}{b}-1\right )\right )\)	\(121\)
derivativedivides	\(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b}-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+3 \textit {\_R}^{2}-1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}\right )}{4 b}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b}}{d}\)	\(139\)
default	\(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b}-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+3 \textit {\_R}^{2}-1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}\right )}{4 b}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b}}{d}\)	\(139\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-1/b*ln(tanh(1/2*d*x+1/2*c)+1)-1/4*a/b*sum((_R^6-3*_R^4+3*_R^2-1)/(_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/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)^4/(a-b*sinh(d*x+c)^4),x, algorithm="maxima")

[Out]

-16*a*integrate(e^(4*d*x + 4*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) - x/b

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1009 vs. \(2 (91) = 182\).
time = 0.42, size = 1009, normalized size = 7.94 \begin {gather*} -\frac {b \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (2 \, {\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + \cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 2 \, {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - b d\right )} \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} - 1\right ) - b \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (2 \, {\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + \cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 2 \, {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - b d\right )} \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} - 1\right ) - b \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (-2 \, {\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + \cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 2 \, {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + b d\right )} \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} - 1\right ) + b \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (-2 \, {\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + \cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 2 \, {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + b d\right )} \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} - 1\right ) + 4 \, x}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4/(a-b*sinh(d*x+c)^4),x, algorithm="fricas")

[Out]

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

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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)**4/(a-b*sinh(d*x+c)**4),x)

[Out]

Timed out

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Giac [A]
time = 0.58, size = 13, normalized size = 0.10 \begin {gather*} -\frac {d x + c}{b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4/(a-b*sinh(d*x+c)^4),x, algorithm="giac")

[Out]

-(d*x + c)/(b*d)

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Mupad [B]
time = 11.05, size = 1861, normalized size = 14.65 \begin {gather*} \ln \left (\frac {\left (\frac {\left (\frac {524288\,a^3\,d^2\,\left (31\,a\,b^2-128\,a^2\,b+128\,a^3-b^3+256\,a^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-240\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^8\,\left (a-b\right )}+\frac {1048576\,a^3\,d^3\,\sqrt {\frac {a\,b^2-\sqrt {a\,b^5}}{b^4\,d^2\,\left (a-b\right )}}\,\left (45\,a\,b^2-104\,a^2\,b+64\,a^3-3\,b^3+4\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}-50\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+48\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^7\,\left (a-b\right )}\right )\,\sqrt {\frac {a\,b^2-\sqrt {a\,b^5}}{b^4\,d^2\,\left (a-b\right )}}}{4}+\frac {262144\,a^4\,d\,\left (72\,a\,b-64\,a^2-9\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+31\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-288\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^9\,\left (a-b\right )}\right )\,\sqrt {\frac {a\,b^2-\sqrt {a\,b^5}}{b^4\,d^2\,\left (a-b\right )}}}{4}+\frac {32768\,a^4\,\left (128\,a\,b-128\,a^2-15\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+29\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-304\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^{10}\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b^2-\sqrt {a\,b^5}}{16\,\left (b^5\,d^2-a\,b^4\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {524288\,a^3\,d^2\,\left (31\,a\,b^2-128\,a^2\,b+128\,a^3-b^3+256\,a^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-240\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^8\,\left (a-b\right )}-\frac {1048576\,a^3\,d^3\,\sqrt {\frac {a\,b^2-\sqrt {a\,b^5}}{b^4\,d^2\,\left (a-b\right )}}\,\left (45\,a\,b^2-104\,a^2\,b+64\,a^3-3\,b^3+4\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}-50\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+48\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^7\,\left (a-b\right )}\right )\,\sqrt {\frac {a\,b^2-\sqrt {a\,b^5}}{b^4\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^4\,d\,\left (72\,a\,b-64\,a^2-9\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+31\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-288\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^9\,\left (a-b\right )}\right )\,\sqrt {\frac {a\,b^2-\sqrt {a\,b^5}}{b^4\,d^2\,\left (a-b\right )}}}{4}+\frac {32768\,a^4\,\left (128\,a\,b-128\,a^2-15\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+29\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-304\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^{10}\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b^2-\sqrt {a\,b^5}}{16\,\left (b^5\,d^2-a\,b^4\,d^2\right )}}-\frac {x}{b}-\ln \left (\frac {\left (\frac {\left (\frac {524288\,a^3\,d^2\,\left (31\,a\,b^2-128\,a^2\,b+128\,a^3-b^3+256\,a^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-240\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^8\,\left (a-b\right )}-\frac {1048576\,a^3\,d^3\,\sqrt {\frac {a\,b^2+\sqrt {a\,b^5}}{b^4\,d^2\,\left (a-b\right )}}\,\left (45\,a\,b^2-104\,a^2\,b+64\,a^3-3\,b^3+4\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}-50\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+48\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^7\,\left (a-b\right )}\right )\,\sqrt {\frac {a\,b^2+\sqrt {a\,b^5}}{b^4\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^4\,d\,\left (72\,a\,b-64\,a^2-9\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+31\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-288\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^9\,\left (a-b\right )}\right )\,\sqrt {\frac {a\,b^2+\sqrt {a\,b^5}}{b^4\,d^2\,\left (a-b\right )}}}{4}+\frac {32768\,a^4\,\left (128\,a\,b-128\,a^2-15\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+29\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-304\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^{10}\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b^2+\sqrt {a\,b^5}}{16\,\left (b^5\,d^2-a\,b^4\,d^2\right )}}+\ln \left (\frac {\left (\frac {\left (\frac {524288\,a^3\,d^2\,\left (31\,a\,b^2-128\,a^2\,b+128\,a^3-b^3+256\,a^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-240\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^8\,\left (a-b\right )}+\frac {1048576\,a^3\,d^3\,\sqrt {\frac {a\,b^2+\sqrt {a\,b^5}}{b^4\,d^2\,\left (a-b\right )}}\,\left (45\,a\,b^2-104\,a^2\,b+64\,a^3-3\,b^3+4\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}-50\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+48\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^7\,\left (a-b\right )}\right )\,\sqrt {\frac {a\,b^2+\sqrt {a\,b^5}}{b^4\,d^2\,\left (a-b\right )}}}{4}+\frac {262144\,a^4\,d\,\left (72\,a\,b-64\,a^2-9\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+31\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-288\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^9\,\left (a-b\right )}\right )\,\sqrt {\frac {a\,b^2+\sqrt {a\,b^5}}{b^4\,d^2\,\left (a-b\right )}}}{4}+\frac {32768\,a^4\,\left (128\,a\,b-128\,a^2-15\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+29\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-304\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^{10}\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b^2+\sqrt {a\,b^5}}{16\,\left (b^5\,d^2-a\,b^4\,d^2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((((((524288*a^3*d^2*(31*a*b^2 - 128*a^2*b + 128*a^3 - b^3 + 256*a^3*exp(2*c + 2*d*x) + b^3*exp(2*c + 2*d*x
) + 21*a*b^2*exp(2*c + 2*d*x) - 240*a^2*b*exp(2*c + 2*d*x)))/(b^8*(a - b)) + (1048576*a^3*d^3*((a*b^2 - (a*b^5
)^(1/2))/(b^4*d^2*(a - b)))^(1/2)*(45*a*b^2 - 104*a^2*b + 64*a^3 - 3*b^3 + 4*b^3*exp(2*c + 2*d*x) - 50*a*b^2*e
xp(2*c + 2*d*x) + 48*a^2*b*exp(2*c + 2*d*x)))/(b^7*(a - b)))*((a*b^2 - (a*b^5)^(1/2))/(b^4*d^2*(a - b)))^(1/2)
)/4 + (262144*a^4*d*(72*a*b - 64*a^2 - 9*b^2 + 256*a^2*exp(2*c + 2*d*x) + 31*b^2*exp(2*c + 2*d*x) - 288*a*b*ex
p(2*c + 2*d*x)))/(b^9*(a - b)))*((a*b^2 - (a*b^5)^(1/2))/(b^4*d^2*(a - b)))^(1/2))/4 + (32768*a^4*(128*a*b - 1
28*a^2 - 15*b^2 + 256*a^2*exp(2*c + 2*d*x) + 29*b^2*exp(2*c + 2*d*x) - 304*a*b*exp(2*c + 2*d*x)))/(b^10*(a - b
)))*(-(a*b^2 - (a*b^5)^(1/2))/(16*(b^5*d^2 - a*b^4*d^2)))^(1/2) - log((((((524288*a^3*d^2*(31*a*b^2 - 128*a^2*
b + 128*a^3 - b^3 + 256*a^3*exp(2*c + 2*d*x) + b^3*exp(2*c + 2*d*x) + 21*a*b^2*exp(2*c + 2*d*x) - 240*a^2*b*ex
p(2*c + 2*d*x)))/(b^8*(a - b)) - (1048576*a^3*d^3*((a*b^2 - (a*b^5)^(1/2))/(b^4*d^2*(a - b)))^(1/2)*(45*a*b^2
- 104*a^2*b + 64*a^3 - 3*b^3 + 4*b^3*exp(2*c + 2*d*x) - 50*a*b^2*exp(2*c + 2*d*x) + 48*a^2*b*exp(2*c + 2*d*x))
)/(b^7*(a - b)))*((a*b^2 - (a*b^5)^(1/2))/(b^4*d^2*(a - b)))^(1/2))/4 - (262144*a^4*d*(72*a*b - 64*a^2 - 9*b^2
+ 256*a^2*exp(2*c + 2*d*x) + 31*b^2*exp(2*c + 2*d*x) - 288*a*b*exp(2*c + 2*d*x)))/(b^9*(a - b)))*((a*b^2 - (a
*b^5)^(1/2))/(b^4*d^2*(a - b)))^(1/2))/4 + (32768*a^4*(128*a*b - 128*a^2 - 15*b^2 + 256*a^2*exp(2*c + 2*d*x) +
29*b^2*exp(2*c + 2*d*x) - 304*a*b*exp(2*c + 2*d*x)))/(b^10*(a - b)))*(-(a*b^2 - (a*b^5)^(1/2))/(16*(b^5*d^2 -
a*b^4*d^2)))^(1/2) - x/b - log((((((524288*a^3*d^2*(31*a*b^2 - 128*a^2*b + 128*a^3 - b^3 + 256*a^3*exp(2*c +
2*d*x) + b^3*exp(2*c + 2*d*x) + 21*a*b^2*exp(2*c + 2*d*x) - 240*a^2*b*exp(2*c + 2*d*x)))/(b^8*(a - b)) - (1048
576*a^3*d^3*((a*b^2 + (a*b^5)^(1/2))/(b^4*d^2*(a - b)))^(1/2)*(45*a*b^2 - 104*a^2*b + 64*a^3 - 3*b^3 + 4*b^3*e
xp(2*c + 2*d*x) - 50*a*b^2*exp(2*c + 2*d*x) + 48*a^2*b*exp(2*c + 2*d*x)))/(b^7*(a - b)))*((a*b^2 + (a*b^5)^(1/
2))/(b^4*d^2*(a - b)))^(1/2))/4 - (262144*a^4*d*(72*a*b - 64*a^2 - 9*b^2 + 256*a^2*exp(2*c + 2*d*x) + 31*b^2*e
xp(2*c + 2*d*x) - 288*a*b*exp(2*c + 2*d*x)))/(b^9*(a - b)))*((a*b^2 + (a*b^5)^(1/2))/(b^4*d^2*(a - b)))^(1/2))
/4 + (32768*a^4*(128*a*b - 128*a^2 - 15*b^2 + 256*a^2*exp(2*c + 2*d*x) + 29*b^2*exp(2*c + 2*d*x) - 304*a*b*exp
(2*c + 2*d*x)))/(b^10*(a - b)))*(-(a*b^2 + (a*b^5)^(1/2))/(16*(b^5*d^2 - a*b^4*d^2)))^(1/2) + log((((((524288*
a^3*d^2*(31*a*b^2 - 128*a^2*b + 128*a^3 - b^3 + 256*a^3*exp(2*c + 2*d*x) + b^3*exp(2*c + 2*d*x) + 21*a*b^2*exp
(2*c + 2*d*x) - 240*a^2*b*exp(2*c + 2*d*x)))/(b^8*(a - b)) + (1048576*a^3*d^3*((a*b^2 + (a*b^5)^(1/2))/(b^4*d^
2*(a - b)))^(1/2)*(45*a*b^2 - 104*a^2*b + 64*a^3 - 3*b^3 + 4*b^3*exp(2*c + 2*d*x) - 50*a*b^2*exp(2*c + 2*d*x)
+ 48*a^2*b*exp(2*c + 2*d*x)))/(b^7*(a - b)))*((a*b^2 + (a*b^5)^(1/2))/(b^4*d^2*(a - b)))^(1/2))/4 + (262144*a^
4*d*(72*a*b - 64*a^2 - 9*b^2 + 256*a^2*exp(2*c + 2*d*x) + 31*b^2*exp(2*c + 2*d*x) - 288*a*b*exp(2*c + 2*d*x)))
/(b^9*(a - b)))*((a*b^2 + (a*b^5)^(1/2))/(b^4*d^2*(a - b)))^(1/2))/4 + (32768*a^4*(128*a*b - 128*a^2 - 15*b^2
+ 256*a^2*exp(2*c + 2*d*x) + 29*b^2*exp(2*c + 2*d*x) - 304*a*b*exp(2*c + 2*d*x)))/(b^10*(a - b)))*(-(a*b^2 + (
a*b^5)^(1/2))/(16*(b^5*d^2 - a*b^4*d^2)))^(1/2)

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