### 3.555 $$\int x \cosh (a+b x) \text{csch}^{\frac{3}{2}}(a+b x) \, dx$$

Optimal. Leaf size=71 $-\frac{2 x \sqrt{\text{csch}(a+b x)}}{b}-\frac{4 i \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right ),2\right )}{b^2}$

[Out]

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

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Rubi [A]  time = 0.0416293, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {5445, 3771, 2641} $-\frac{2 x \sqrt{\text{csch}(a+b x)}}{b}-\frac{4 i \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)} F\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{b^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[x*Cosh[a + b*x]*Csch[a + b*x]^(3/2),x]

[Out]

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

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 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 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(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{3}{2}}(a+b x) \, dx &=-\frac{2 x \sqrt{\text{csch}(a+b x)}}{b}+\frac{2 \int \sqrt{\text{csch}(a+b x)} \, dx}{b}\\ &=-\frac{2 x \sqrt{\text{csch}(a+b x)}}{b}+\frac{\left (2 \sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}\right ) \int \frac{1}{\sqrt{i \sinh (a+b x)}} \, dx}{b}\\ &=-\frac{2 x \sqrt{\text{csch}(a+b x)}}{b}-\frac{4 i \sqrt{\text{csch}(a+b x)} F\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right ) \sqrt{i \sinh (a+b x)}}{b^2}\\ \end{align*}

Mathematica [A]  time = 0.145454, size = 56, normalized size = 0.79 $-\frac{2 \sqrt{\text{csch}(a+b x)} \left (b x-2 i \sqrt{i \sinh (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 i a-2 i b x+\pi ),2\right )\right )}{b^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*Cosh[a + b*x]*Csch[a + b*x]^(3/2),x]

[Out]

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

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int x\cosh \left ( bx+a \right ) \left ({\rm csch} \left (bx+a\right ) \right ) ^{{\frac{3}{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)^(3/2),x)

[Out]

int(x*cosh(b*x+a)*csch(b*x+a)^(3/2),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{3}{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)^(3/2),x, algorithm="maxima")

[Out]

integrate(x*cosh(b*x + a)*csch(b*x + a)^(3/2), 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)^(3/2),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)**(3/2),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{3}{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)^(3/2),x, algorithm="giac")

[Out]

integrate(x*cosh(b*x + a)*csch(b*x + a)^(3/2), x)