3.554 \(\int x \cosh (a+b x) \text{csch}^{\frac{5}{2}}(a+b x) \, dx\)

Optimal. Leaf size=98 \[ -\frac{4 \cosh (a+b x) \sqrt{\text{csch}(a+b x)}}{3 b^2}-\frac{4 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{3 b^2 \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}}-\frac{2 x \text{csch}^{\frac{3}{2}}(a+b x)}{3 b} \]

[Out]

(-4*Cosh[a + b*x]*Sqrt[Csch[a + b*x]])/(3*b^2) - (2*x*Csch[a + b*x]^(3/2))/(3*b) - (((4*I)/3)*EllipticE[(I*a -
Pi/2 + I*b*x)/2, 2])/(b^2*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])

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Rubi [A]  time = 0.0507373, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5445, 3768, 3771, 2639} \[ -\frac{4 \cosh (a+b x) \sqrt{\text{csch}(a+b x)}}{3 b^2}-\frac{4 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{3 b^2 \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}}-\frac{2 x \text{csch}^{\frac{3}{2}}(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[x*Cosh[a + b*x]*Csch[a + b*x]^(5/2),x]

[Out]

(-4*Cosh[a + b*x]*Sqrt[Csch[a + b*x]])/(3*b^2) - (2*x*Csch[a + b*x]^(3/2))/(3*b) - (((4*I)/3)*EllipticE[(I*a -
Pi/2 + I*b*x)/2, 2])/(b^2*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])

Rule 5445

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*Csch[(a_.) + (b_.)*(x_)^(n_.)]^(p_)*(x_)^(m_.), x_Symbol] :> -Simp[(x^(m -
n + 1)*Csch[a + b*x^n]^(p - 1))/(b*n*(p - 1)), x] + Dist[(m - n + 1)/(b*n*(p - 1)), Int[x^(m - n)*Csch[a + b*x
^n]^(p - 1), x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int x \cosh (a+b x) \text{csch}^{\frac{5}{2}}(a+b x) \, dx &=-\frac{2 x \text{csch}^{\frac{3}{2}}(a+b x)}{3 b}+\frac{2 \int \text{csch}^{\frac{3}{2}}(a+b x) \, dx}{3 b}\\ &=-\frac{4 \cosh (a+b x) \sqrt{\text{csch}(a+b x)}}{3 b^2}-\frac{2 x \text{csch}^{\frac{3}{2}}(a+b x)}{3 b}+\frac{2 \int \frac{1}{\sqrt{\text{csch}(a+b x)}} \, dx}{3 b}\\ &=-\frac{4 \cosh (a+b x) \sqrt{\text{csch}(a+b x)}}{3 b^2}-\frac{2 x \text{csch}^{\frac{3}{2}}(a+b x)}{3 b}+\frac{2 \int \sqrt{i \sinh (a+b x)} \, dx}{3 b \sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}}\\ &=-\frac{4 \cosh (a+b x) \sqrt{\text{csch}(a+b x)}}{3 b^2}-\frac{2 x \text{csch}^{\frac{3}{2}}(a+b x)}{3 b}-\frac{4 i E\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right )}{3 b^2 \sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.207189, size = 70, normalized size = 0.71 \[ -\frac{2 \sqrt{\text{csch}(a+b x)} \left (2 \cosh (a+b x)+b x \text{csch}(a+b x)-2 \sqrt{i \sinh (a+b x)} E\left (\left .\frac{1}{4} (-2 i a-2 i b x+\pi )\right |2\right )\right )}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cosh[a + b*x]*Csch[a + b*x]^(5/2),x]

[Out]

(-2*Sqrt[Csch[a + b*x]]*(2*Cosh[a + b*x] + b*x*Csch[a + b*x] - 2*EllipticE[((-2*I)*a + Pi - (2*I)*b*x)/4, 2]*S
qrt[I*Sinh[a + b*x]]))/(3*b^2)

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \begin{align*} \int x\cosh \left ( bx+a \right ) \left ({\rm csch} \left (bx+a\right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cosh(b*x+a)*csch(b*x+a)^(5/2),x)

[Out]

int(x*cosh(b*x+a)*csch(b*x+a)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh \left (b x + a\right ) \operatorname{csch}\left (b x + a\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(b*x+a)*csch(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

integrate(x*cosh(b*x + a)*csch(b*x + a)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(b*x+a)*csch(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(b*x+a)*csch(b*x+a)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh \left (b x + a\right ) \operatorname{csch}\left (b x + a\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(b*x+a)*csch(b*x+a)^(5/2),x, algorithm="giac")

[Out]

integrate(x*cosh(b*x + a)*csch(b*x + a)^(5/2), x)