3.78 \(\int \frac{\sinh ^2(x)}{a+b \text{csch}(x)} \, dx\)

Optimal. Leaf size=80 \[ -\frac{x \left (a^2-2 b^2\right )}{2 a^3}+\frac{2 b^3 \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2}}-\frac{b \cosh (x)}{a^2}+\frac{\sinh (x) \cosh (x)}{2 a} \]

[Out]

-((a^2 - 2*b^2)*x)/(2*a^3) + (2*b^3*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^3*Sqrt[a^2 + b^2]) - (b*Cos
h[x])/a^2 + (Cosh[x]*Sinh[x])/(2*a)

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Rubi [A]  time = 0.288274, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {3853, 4104, 3919, 3831, 2660, 618, 206} \[ -\frac{x \left (a^2-2 b^2\right )}{2 a^3}+\frac{2 b^3 \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2}}-\frac{b \cosh (x)}{a^2}+\frac{\sinh (x) \cosh (x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^2/(a + b*Csch[x]),x]

[Out]

-((a^2 - 2*b^2)*x)/(2*a^3) + (2*b^3*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^3*Sqrt[a^2 + b^2]) - (b*Cos
h[x])/a^2 + (Cosh[x]*Sinh[x])/(2*a)

Rule 3853

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(Cot[e + f*
x]*(d*Csc[e + f*x])^n)/(a*f*n), x] - Dist[1/(a*d*n), Int[((d*Csc[e + f*x])^(n + 1)*Simp[b*n - a*(n + 1)*Csc[e
+ f*x] - b*(n + 1)*Csc[e + f*x]^2, x])/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 -
b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]

Rule 4104

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m +
1)*(d*Csc[e + f*x])^n)/(a*f*n), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[
a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ
[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2660

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 618

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 206

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

Rubi steps

\begin{align*} \int \frac{\sinh ^2(x)}{a+b \text{csch}(x)} \, dx &=\frac{\cosh (x) \sinh (x)}{2 a}-\frac{i \int \frac{\left (-2 i b-i a \text{csch}(x)-i b \text{csch}^2(x)\right ) \sinh (x)}{a+b \text{csch}(x)} \, dx}{2 a}\\ &=-\frac{b \cosh (x)}{a^2}+\frac{\cosh (x) \sinh (x)}{2 a}+\frac{\int \frac{-a^2+2 b^2-a b \text{csch}(x)}{a+b \text{csch}(x)} \, dx}{2 a^2}\\ &=-\frac{\left (a^2-2 b^2\right ) x}{2 a^3}-\frac{b \cosh (x)}{a^2}+\frac{\cosh (x) \sinh (x)}{2 a}-\frac{b^3 \int \frac{\text{csch}(x)}{a+b \text{csch}(x)} \, dx}{a^3}\\ &=-\frac{\left (a^2-2 b^2\right ) x}{2 a^3}-\frac{b \cosh (x)}{a^2}+\frac{\cosh (x) \sinh (x)}{2 a}-\frac{b^2 \int \frac{1}{1+\frac{a \sinh (x)}{b}} \, dx}{a^3}\\ &=-\frac{\left (a^2-2 b^2\right ) x}{2 a^3}-\frac{b \cosh (x)}{a^2}+\frac{\cosh (x) \sinh (x)}{2 a}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{2 a x}{b}-x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{a^3}\\ &=-\frac{\left (a^2-2 b^2\right ) x}{2 a^3}-\frac{b \cosh (x)}{a^2}+\frac{\cosh (x) \sinh (x)}{2 a}+\frac{\left (4 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (1+\frac{a^2}{b^2}\right )-x^2} \, dx,x,\frac{2 a}{b}-2 \tanh \left (\frac{x}{2}\right )\right )}{a^3}\\ &=-\frac{\left (a^2-2 b^2\right ) x}{2 a^3}+\frac{2 b^3 \tanh ^{-1}\left (\frac{b \left (\frac{a}{b}-\tanh \left (\frac{x}{2}\right )\right )}{\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2}}-\frac{b \cosh (x)}{a^2}+\frac{\cosh (x) \sinh (x)}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.135727, size = 82, normalized size = 1.02 \[ \frac{-\frac{8 b^3 \tan ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}-2 a^2 x+a^2 \sinh (2 x)-4 a b \cosh (x)+4 b^2 x}{4 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]^2/(a + b*Csch[x]),x]

[Out]

(-2*a^2*x + 4*b^2*x - (8*b^3*ArcTan[(a - b*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 4*a*b*Cosh[x] + a^
2*Sinh[2*x])/(4*a^3)

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Maple [B]  time = 0.034, size = 174, normalized size = 2.2 \begin{align*} -{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{b}{{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{2\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{{b}^{2}}{{a}^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-2\,{\frac{{b}^{3}}{{a}^{3}\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,\tanh \left ( x/2 \right ) b-2\,a}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }+{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{b}{{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{1}{2\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }-{\frac{{b}^{2}}{{a}^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^2/(a+b*csch(x)),x)

[Out]

-1/2/a/(tanh(1/2*x)+1)^2+1/2/a/(tanh(1/2*x)+1)-1/a^2/(tanh(1/2*x)+1)*b-1/2/a*ln(tanh(1/2*x)+1)+1/a^3*ln(tanh(1
/2*x)+1)*b^2-2*b^3/a^3/(a^2+b^2)^(1/2)*arctanh(1/2*(2*tanh(1/2*x)*b-2*a)/(a^2+b^2)^(1/2))+1/2/a/(tanh(1/2*x)-1
)^2+1/2/a/(tanh(1/2*x)-1)+1/a^2/(tanh(1/2*x)-1)*b+1/2/a*ln(tanh(1/2*x)-1)-1/a^3*ln(tanh(1/2*x)-1)*b^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^2/(a+b*csch(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.9658, size = 1142, normalized size = 14.28 \begin{align*} \frac{{\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right )^{4} +{\left (a^{4} + a^{2} b^{2}\right )} \sinh \left (x\right )^{4} - a^{4} - a^{2} b^{2} - 4 \,{\left (a^{4} - a^{2} b^{2} - 2 \, b^{4}\right )} x \cosh \left (x\right )^{2} - 4 \,{\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )^{3} - 4 \,{\left (a^{3} b + a b^{3} -{\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \,{\left (3 \,{\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right )^{2} - 2 \,{\left (a^{4} - a^{2} b^{2} - 2 \, b^{4}\right )} x - 6 \,{\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 8 \,{\left (b^{3} \cosh \left (x\right )^{2} + 2 \, b^{3} \cosh \left (x\right ) \sinh \left (x\right ) + b^{3} \sinh \left (x\right )^{2}\right )} \sqrt{a^{2} + b^{2}} \log \left (\frac{a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, b^{2} + 2 \,{\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt{a^{2} + b^{2}}{\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \,{\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) - a}\right ) - 4 \,{\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right ) - 4 \,{\left (a^{3} b + a b^{3} -{\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right )^{3} + 2 \,{\left (a^{4} - a^{2} b^{2} - 2 \, b^{4}\right )} x \cosh \left (x\right ) + 3 \,{\left (a^{3} b + a b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )}{8 \,{\left ({\left (a^{5} + a^{3} b^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a^{5} + a^{3} b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a^{5} + a^{3} b^{2}\right )} \sinh \left (x\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^2/(a+b*csch(x)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{2}{\left (x \right )}}{a + b \operatorname{csch}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)**2/(a+b*csch(x)),x)

[Out]

Integral(sinh(x)**2/(a + b*csch(x)), x)

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Giac [A]  time = 1.15516, size = 155, normalized size = 1.94 \begin{align*} -\frac{b^{3} \log \left (\frac{{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{\sqrt{a^{2} + b^{2}} a^{3}} + \frac{a e^{\left (2 \, x\right )} - 4 \, b e^{x}}{8 \, a^{2}} - \frac{{\left (a^{2} - 2 \, b^{2}\right )} x}{2 \, a^{3}} - \frac{{\left (4 \, a b e^{x} + a^{2}\right )} e^{\left (-2 \, x\right )}}{8 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^2/(a+b*csch(x)),x, algorithm="giac")

[Out]

-b^3*log(abs(2*a*e^x + 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*e^x + 2*b + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^3)
+ 1/8*(a*e^(2*x) - 4*b*e^x)/a^2 - 1/2*(a^2 - 2*b^2)*x/a^3 - 1/8*(4*a*b*e^x + a^2)*e^(-2*x)/a^3