### 3.558 $$\int \frac{x \cosh (a+b x)}{\text{csch}^{\frac{3}{2}}(a+b x)} \, dx$$

Optimal. Leaf size=98 $-\frac{4 \cosh (a+b x)}{25 b^2 \text{csch}^{\frac{3}{2}}(a+b x)}-\frac{12 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{25 b^2 \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}}+\frac{2 x}{5 b \text{csch}^{\frac{5}{2}}(a+b x)}$

[Out]

(2*x)/(5*b*Csch[a + b*x]^(5/2)) - (4*Cosh[a + b*x])/(25*b^2*Csch[a + b*x]^(3/2)) - (((12*I)/25)*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 = 0.0527631, 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, 3769, 3771, 2639} $-\frac{4 \cosh (a+b x)}{25 b^2 \text{csch}^{\frac{3}{2}}(a+b x)}-\frac{12 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{25 b^2 \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}}+\frac{2 x}{5 b \text{csch}^{\frac{5}{2}}(a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*x)/(5*b*Csch[a + b*x]^(5/2)) - (4*Cosh[a + b*x])/(25*b^2*Csch[a + b*x]^(3/2)) - (((12*I)/25)*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 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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 \frac{x \cosh (a+b x)}{\text{csch}^{\frac{3}{2}}(a+b x)} \, dx &=\frac{2 x}{5 b \text{csch}^{\frac{5}{2}}(a+b x)}-\frac{2 \int \frac{1}{\text{csch}^{\frac{5}{2}}(a+b x)} \, dx}{5 b}\\ &=\frac{2 x}{5 b \text{csch}^{\frac{5}{2}}(a+b x)}-\frac{4 \cosh (a+b x)}{25 b^2 \text{csch}^{\frac{3}{2}}(a+b x)}+\frac{6 \int \frac{1}{\sqrt{\text{csch}(a+b x)}} \, dx}{25 b}\\ &=\frac{2 x}{5 b \text{csch}^{\frac{5}{2}}(a+b x)}-\frac{4 \cosh (a+b x)}{25 b^2 \text{csch}^{\frac{3}{2}}(a+b x)}+\frac{6 \int \sqrt{i \sinh (a+b x)} \, dx}{25 b \sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}}\\ &=\frac{2 x}{5 b \text{csch}^{\frac{5}{2}}(a+b x)}-\frac{4 \cosh (a+b x)}{25 b^2 \text{csch}^{\frac{3}{2}}(a+b x)}-\frac{12 i E\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right )}{25 b^2 \sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}}\\ \end{align*}

Mathematica [C]  time = 1.91286, size = 111, normalized size = 1.13 $\frac{e^{-2 (a+b x)} \left (-\frac{48 e^{2 (a+b x)} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};e^{2 (a+b x)}\right )}{\sqrt{1-e^{2 (a+b x)}}}+(24-10 b x) e^{2 (a+b x)}+(5 b x-2) e^{4 (a+b x)}+5 b x+2\right )}{50 b^2 \sqrt{\text{csch}(a+b x)}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2 + 5*b*x + E^(2*(a + b*x))*(24 - 10*b*x) + E^(4*(a + b*x))*(-2 + 5*b*x) - (48*E^(2*(a + b*x))*Hypergeometric
2F1[-1/4, 1/2, 3/4, E^(2*(a + b*x))])/Sqrt[1 - E^(2*(a + b*x))])/(50*b^2*E^(2*(a + b*x))*Sqrt[Csch[a + b*x]])

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Maple [F]  time = 0.03, 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 \frac{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 \frac{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)