### 3.552 $$\int x \cosh (a+b x) \text{csch}^{\frac{9}{2}}(a+b x) \, dx$$

Optimal. Leaf size=121 $-\frac{4 \cosh (a+b x) \text{csch}^{\frac{5}{2}}(a+b x)}{35 b^2}+\frac{12 \cosh (a+b x) \sqrt{\text{csch}(a+b x)}}{35 b^2}+\frac{12 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{35 b^2 \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}}-\frac{2 x \text{csch}^{\frac{7}{2}}(a+b x)}{7 b}$

[Out]

(12*Cosh[a + b*x]*Sqrt[Csch[a + b*x]])/(35*b^2) - (4*Cosh[a + b*x]*Csch[a + b*x]^(5/2))/(35*b^2) - (2*x*Csch[a
+ b*x]^(7/2))/(7*b) + (((12*I)/35)*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 = 0.0661242, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {5445, 3768, 3771, 2639} $-\frac{4 \cosh (a+b x) \text{csch}^{\frac{5}{2}}(a+b x)}{35 b^2}+\frac{12 \cosh (a+b x) \sqrt{\text{csch}(a+b x)}}{35 b^2}+\frac{12 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{35 b^2 \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}}-\frac{2 x \text{csch}^{\frac{7}{2}}(a+b x)}{7 b}$

Antiderivative was successfully veriﬁed.

[In]

Int[x*Cosh[a + b*x]*Csch[a + b*x]^(9/2),x]

[Out]

(12*Cosh[a + b*x]*Sqrt[Csch[a + b*x]])/(35*b^2) - (4*Cosh[a + b*x]*Csch[a + b*x]^(5/2))/(35*b^2) - (2*x*Csch[a
+ b*x]^(7/2))/(7*b) + (((12*I)/35)*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{9}{2}}(a+b x) \, dx &=-\frac{2 x \text{csch}^{\frac{7}{2}}(a+b x)}{7 b}+\frac{2 \int \text{csch}^{\frac{7}{2}}(a+b x) \, dx}{7 b}\\ &=-\frac{4 \cosh (a+b x) \text{csch}^{\frac{5}{2}}(a+b x)}{35 b^2}-\frac{2 x \text{csch}^{\frac{7}{2}}(a+b x)}{7 b}-\frac{6 \int \text{csch}^{\frac{3}{2}}(a+b x) \, dx}{35 b}\\ &=\frac{12 \cosh (a+b x) \sqrt{\text{csch}(a+b x)}}{35 b^2}-\frac{4 \cosh (a+b x) \text{csch}^{\frac{5}{2}}(a+b x)}{35 b^2}-\frac{2 x \text{csch}^{\frac{7}{2}}(a+b x)}{7 b}-\frac{6 \int \frac{1}{\sqrt{\text{csch}(a+b x)}} \, dx}{35 b}\\ &=\frac{12 \cosh (a+b x) \sqrt{\text{csch}(a+b x)}}{35 b^2}-\frac{4 \cosh (a+b x) \text{csch}^{\frac{5}{2}}(a+b x)}{35 b^2}-\frac{2 x \text{csch}^{\frac{7}{2}}(a+b x)}{7 b}-\frac{6 \int \sqrt{i \sinh (a+b x)} \, dx}{35 b \sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}}\\ &=\frac{12 \cosh (a+b x) \sqrt{\text{csch}(a+b x)}}{35 b^2}-\frac{4 \cosh (a+b x) \text{csch}^{\frac{5}{2}}(a+b x)}{35 b^2}-\frac{2 x \text{csch}^{\frac{7}{2}}(a+b x)}{7 b}+\frac{12 i E\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right )}{35 b^2 \sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.514715, size = 83, normalized size = 0.69 $-\frac{2 \sqrt{\text{csch}(a+b x)} \left (-6 \cosh (a+b x)+(\sinh (2 (a+b x))+5 b x) \text{csch}^3(a+b x)+6 \sqrt{i \sinh (a+b x)} E\left (\left .\frac{1}{4} (-2 i a-2 i b x+\pi )\right |2\right )\right )}{35 b^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*Cosh[a + b*x]*Csch[a + b*x]^(9/2),x]

[Out]

(-2*Sqrt[Csch[a + b*x]]*(-6*Cosh[a + b*x] + 6*EllipticE[((-2*I)*a + Pi - (2*I)*b*x)/4, 2]*Sqrt[I*Sinh[a + b*x]
] + Csch[a + b*x]^3*(5*b*x + Sinh[2*(a + b*x)])))/(35*b^2)

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int x\cosh \left ( bx+a \right ) \left ({\rm csch} \left (bx+a\right ) \right ) ^{{\frac{9}{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)^(9/2),x)

[Out]

int(x*cosh(b*x+a)*csch(b*x+a)^(9/2),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{9}{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)^(9/2),x, algorithm="maxima")

[Out]

integrate(x*cosh(b*x + a)*csch(b*x + a)^(9/2), 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)^(9/2),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)**(9/2),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{9}{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)^(9/2),x, algorithm="giac")

[Out]

integrate(x*cosh(b*x + a)*csch(b*x + a)^(9/2), x)