3.237 \(\int \frac {\sinh ^2(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\)

Optimal. Leaf size=125 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {b} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {b} d \sqrt {\sqrt {a}-\sqrt {b}}} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3217, 1130, 208} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {b} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {b} d \sqrt {\sqrt {a}-\sqrt {b}}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^2/(a - b*Sinh[c + d*x]^4),x]

[Out]

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

Rule 208

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

Rule 1130

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*(b/q + 1))/2, Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2*(b/q - 1))/2, 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 3217

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

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Mathematica [A]  time = 0.36, size = 127, normalized size = 1.02 \[ \frac {\frac {\tanh ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}+a}}\right )}{\sqrt {\sqrt {a} \sqrt {b}+a}}+\frac {\tan ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}-a}}\right )}{\sqrt {\sqrt {a} \sqrt {b}-a}}}{2 \sqrt {b} d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^2/(a - b*Sinh[c + d*x]^4),x]

[Out]

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

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fricas [B]  time = 1.02, size = 975, normalized size = 7.80 \[ -\frac {1}{4} \, \sqrt {\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + 1}{{\left (a b - b^{2}\right )} d^{2}}} \log \left (2 \, {\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\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^{2} b - a b^{2}\right )} d^{3} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - a d\right )} \sqrt {\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + 1}{{\left (a b - b^{2}\right )} d^{2}}} - 1\right ) + \frac {1}{4} \, \sqrt {\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + 1}{{\left (a b - b^{2}\right )} d^{2}}} \log \left (2 \, {\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\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^{2} b - a b^{2}\right )} d^{3} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - a d\right )} \sqrt {\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + 1}{{\left (a b - b^{2}\right )} d^{2}}} - 1\right ) + \frac {1}{4} \, \sqrt {-\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - 1}{{\left (a b - b^{2}\right )} d^{2}}} \log \left (-2 \, {\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\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^{2} b - a b^{2}\right )} d^{3} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + a d\right )} \sqrt {-\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - 1}{{\left (a b - b^{2}\right )} d^{2}}} - 1\right ) - \frac {1}{4} \, \sqrt {-\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - 1}{{\left (a b - b^{2}\right )} d^{2}}} \log \left (-2 \, {\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\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^{2} b - a b^{2}\right )} d^{3} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + a d\right )} \sqrt {-\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - 1}{{\left (a b - b^{2}\right )} d^{2}}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2/(a-b*sinh(d*x+c)^4),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.35, size = 1, normalized size = 0.01 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2/(a-b*sinh(d*x+c)^4),x, algorithm="giac")

[Out]

0

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maple [C]  time = 0.08, size = 94, normalized size = 0.75 \[ -\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-\textit {\_R}^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^2/(a-b*sinh(d*x+c)^4),x)

[Out]

-1/d*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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {\sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right )^{4} - a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2/(a-b*sinh(d*x+c)^4),x, algorithm="maxima")

[Out]

-integrate(sinh(d*x + c)^2/(b*sinh(d*x + c)^4 - a), x)

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mupad [B]  time = 12.90, size = 1859, normalized size = 14.87 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^2/(a - b*sinh(c + d*x)^4),x)

[Out]

log((((((262144*a^2*d^2*(102*a*b - 128*a^2 - 22*b^2 - 272*a^2*exp(2*c + 2*d*x) + 19*b^2*exp(2*c + 2*d*x) + 189
*a*b*exp(2*c + 2*d*x)))/(b^6*(a - b)) - (131072*a^2*d^3*((a*b + (a*b^3)^(1/2))/(a*b^2*d^2*(a - b)))^(1/2)*(119
*a*b^2 - 136*a^2*b + b^3 - 1024*a^3*exp(2*c + 2*d*x) + 9*b^3*exp(2*c + 2*d*x) - 809*a*b^2*exp(2*c + 2*d*x) + 1
808*a^2*b*exp(2*c + 2*d*x)))/(b^6*(a - b)))*((a*b + (a*b^3)^(1/2))/(a*b^2*d^2*(a - b)))^(1/2))/4 + (32768*a*d*
(120*a^2*b - 129*a*b^2 + b^3 - 1024*a^3*exp(2*c + 2*d*x) - b^3*exp(2*c + 2*d*x) + 201*a*b^2*exp(2*c + 2*d*x) +
816*a^2*b*exp(2*c + 2*d*x)))/(b^7*(a - b)))*((a*b + (a*b^3)^(1/2))/(a*b^2*d^2*(a - b)))^(1/2))/4 - (16384*a*(
106*a*b - 128*a^2 - 2*b^2 + 240*a^2*exp(2*c + 2*d*x) + 3*b^2*exp(2*c + 2*d*x) - 275*a*b*exp(2*c + 2*d*x)))/(b^
7*(a - b)))*(-(a*b + (a*b^3)^(1/2))/(16*(a*b^3*d^2 - a^2*b^2*d^2)))^(1/2) - log((((((262144*a^2*d^2*(102*a*b -
128*a^2 - 22*b^2 - 272*a^2*exp(2*c + 2*d*x) + 19*b^2*exp(2*c + 2*d*x) + 189*a*b*exp(2*c + 2*d*x)))/(b^6*(a -
b)) + (131072*a^2*d^3*((a*b + (a*b^3)^(1/2))/(a*b^2*d^2*(a - b)))^(1/2)*(119*a*b^2 - 136*a^2*b + b^3 - 1024*a^
3*exp(2*c + 2*d*x) + 9*b^3*exp(2*c + 2*d*x) - 809*a*b^2*exp(2*c + 2*d*x) + 1808*a^2*b*exp(2*c + 2*d*x)))/(b^6*
(a - b)))*((a*b + (a*b^3)^(1/2))/(a*b^2*d^2*(a - b)))^(1/2))/4 - (32768*a*d*(120*a^2*b - 129*a*b^2 + b^3 - 102
4*a^3*exp(2*c + 2*d*x) - b^3*exp(2*c + 2*d*x) + 201*a*b^2*exp(2*c + 2*d*x) + 816*a^2*b*exp(2*c + 2*d*x)))/(b^7
*(a - b)))*((a*b + (a*b^3)^(1/2))/(a*b^2*d^2*(a - b)))^(1/2))/4 - (16384*a*(106*a*b - 128*a^2 - 2*b^2 + 240*a^
2*exp(2*c + 2*d*x) + 3*b^2*exp(2*c + 2*d*x) - 275*a*b*exp(2*c + 2*d*x)))/(b^7*(a - b)))*(-(a*b + (a*b^3)^(1/2)
)/(16*(a*b^3*d^2 - a^2*b^2*d^2)))^(1/2) + log((((((262144*a^2*d^2*(102*a*b - 128*a^2 - 22*b^2 - 272*a^2*exp(2*
c + 2*d*x) + 19*b^2*exp(2*c + 2*d*x) + 189*a*b*exp(2*c + 2*d*x)))/(b^6*(a - b)) - (131072*a^2*d^3*((a*b - (a*b
^3)^(1/2))/(a*b^2*d^2*(a - b)))^(1/2)*(119*a*b^2 - 136*a^2*b + b^3 - 1024*a^3*exp(2*c + 2*d*x) + 9*b^3*exp(2*c
+ 2*d*x) - 809*a*b^2*exp(2*c + 2*d*x) + 1808*a^2*b*exp(2*c + 2*d*x)))/(b^6*(a - b)))*((a*b - (a*b^3)^(1/2))/(
a*b^2*d^2*(a - b)))^(1/2))/4 + (32768*a*d*(120*a^2*b - 129*a*b^2 + b^3 - 1024*a^3*exp(2*c + 2*d*x) - b^3*exp(2
*c + 2*d*x) + 201*a*b^2*exp(2*c + 2*d*x) + 816*a^2*b*exp(2*c + 2*d*x)))/(b^7*(a - b)))*((a*b - (a*b^3)^(1/2))/
(a*b^2*d^2*(a - b)))^(1/2))/4 - (16384*a*(106*a*b - 128*a^2 - 2*b^2 + 240*a^2*exp(2*c + 2*d*x) + 3*b^2*exp(2*c
+ 2*d*x) - 275*a*b*exp(2*c + 2*d*x)))/(b^7*(a - b)))*(-(a*b - (a*b^3)^(1/2))/(16*(a*b^3*d^2 - a^2*b^2*d^2)))^
(1/2) - log((((((262144*a^2*d^2*(102*a*b - 128*a^2 - 22*b^2 - 272*a^2*exp(2*c + 2*d*x) + 19*b^2*exp(2*c + 2*d*
x) + 189*a*b*exp(2*c + 2*d*x)))/(b^6*(a - b)) + (131072*a^2*d^3*((a*b - (a*b^3)^(1/2))/(a*b^2*d^2*(a - b)))^(1
/2)*(119*a*b^2 - 136*a^2*b + b^3 - 1024*a^3*exp(2*c + 2*d*x) + 9*b^3*exp(2*c + 2*d*x) - 809*a*b^2*exp(2*c + 2*
d*x) + 1808*a^2*b*exp(2*c + 2*d*x)))/(b^6*(a - b)))*((a*b - (a*b^3)^(1/2))/(a*b^2*d^2*(a - b)))^(1/2))/4 - (32
768*a*d*(120*a^2*b - 129*a*b^2 + b^3 - 1024*a^3*exp(2*c + 2*d*x) - b^3*exp(2*c + 2*d*x) + 201*a*b^2*exp(2*c +
2*d*x) + 816*a^2*b*exp(2*c + 2*d*x)))/(b^7*(a - b)))*((a*b - (a*b^3)^(1/2))/(a*b^2*d^2*(a - b)))^(1/2))/4 - (1
6384*a*(106*a*b - 128*a^2 - 2*b^2 + 240*a^2*exp(2*c + 2*d*x) + 3*b^2*exp(2*c + 2*d*x) - 275*a*b*exp(2*c + 2*d*
x)))/(b^7*(a - b)))*(-(a*b - (a*b^3)^(1/2))/(16*(a*b^3*d^2 - a^2*b^2*d^2)))^(1/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**2/(a-b*sinh(d*x+c)**4),x)

[Out]

Timed out

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