### 3.556 $$\int x \cosh (a+b x) \sqrt{\text{csch}(a+b x)} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0405722, 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, 2639} $\frac{2 x}{b \sqrt{\text{csch}(a+b x)}}+\frac{4 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{b^2 \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}}$

Antiderivative was successfully veriﬁed.

[In]

Int[x*Cosh[a + b*x]*Sqrt[Csch[a + b*x]],x]

[Out]

(2*x)/(b*Sqrt[Csch[a + b*x]]) + ((4*I)*EllipticE[(I*a - Pi/2 + I*b*x)/2, 2])/(b^2*Sqrt[Csch[a + b*x]]*Sqrt[I*S
inh[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 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) \sqrt{\text{csch}(a+b x)} \, dx &=\frac{2 x}{b \sqrt{\text{csch}(a+b x)}}-\frac{2 \int \frac{1}{\sqrt{\text{csch}(a+b x)}} \, dx}{b}\\ &=\frac{2 x}{b \sqrt{\text{csch}(a+b x)}}-\frac{2 \int \sqrt{i \sinh (a+b x)} \, dx}{b \sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}}\\ &=\frac{2 x}{b \sqrt{\text{csch}(a+b x)}}+\frac{4 i E\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right )}{b^2 \sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}}\\ \end{align*}

Mathematica [C]  time = 1.01549, size = 183, normalized size = 2.58 $\frac{e^{-a-b x} \sqrt{2-2 e^{2 (a+b x)}} \sqrt{\frac{e^{a+b x}}{e^{2 (a+b x)}-1}} \left (-18 \text{HypergeometricPFQ}\left (\left \{-\frac{1}{4},-\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{3}{4},\frac{3}{4}\right \},e^{2 (a+b x)}\right )-2 e^{2 (a+b x)} \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{3}{4},\frac{3}{4}\right \},\left \{\frac{7}{4},\frac{7}{4}\right \},e^{2 (a+b x)}\right )-3 b x \left (3 \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};e^{2 (a+b x)}\right )-e^{2 (a+b x)} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{2 (a+b x)}\right )\right )\right )}{9 b^2}$

Warning: Unable to verify antiderivative.

[In]

Integrate[x*Cosh[a + b*x]*Sqrt[Csch[a + b*x]],x]

[Out]

(E^(-a - b*x)*Sqrt[2 - 2*E^(2*(a + b*x))]*Sqrt[E^(a + b*x)/(-1 + E^(2*(a + b*x)))]*(-3*b*x*(3*Hypergeometric2F
1[-1/4, 1/2, 3/4, E^(2*(a + b*x))] - E^(2*(a + b*x))*Hypergeometric2F1[1/2, 3/4, 7/4, E^(2*(a + b*x))]) - 18*H
ypergeometricPFQ[{-1/4, -1/4, 1/2}, {3/4, 3/4}, E^(2*(a + b*x))] - 2*E^(2*(a + b*x))*HypergeometricPFQ[{1/2, 3
/4, 3/4}, {7/4, 7/4}, E^(2*(a + b*x))]))/(9*b^2)

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Maple [B]  time = 0.062, size = 229, normalized size = 3.2 \begin{align*}{\frac{ \left ( bx-2 \right ) \left ( \left ({{\rm e}^{bx+a}} \right ) ^{2}-1 \right ) \sqrt{2}}{{b}^{2}{{\rm e}^{bx+a}}}\sqrt{{\frac{{{\rm e}^{bx+a}}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}-1}}}}+2\,{\frac{\sqrt{2}\sqrt{ \left ( \left ({{\rm e}^{bx+a}} \right ) ^{2}-1 \right ){{\rm e}^{bx+a}}}}{{b}^{2}{{\rm e}^{bx+a}}} \left ( 2\,{\frac{ \left ({{\rm e}^{bx+a}} \right ) ^{2}-1}{\sqrt{ \left ( \left ({{\rm e}^{bx+a}} \right ) ^{2}-1 \right ){{\rm e}^{bx+a}}}}}-{\frac{\sqrt{1+{{\rm e}^{bx+a}}}\sqrt{2-2\,{{\rm e}^{bx+a}}}\sqrt{-{{\rm e}^{bx+a}}} \left ( -2\,{\it EllipticE} \left ( \sqrt{1+{{\rm e}^{bx+a}}},1/2\,\sqrt{2} \right ) +{\it EllipticF} \left ( \sqrt{1+{{\rm e}^{bx+a}}},1/2\,\sqrt{2} \right ) \right ) }{\sqrt{ \left ({{\rm e}^{bx+a}} \right ) ^{3}-{{\rm e}^{bx+a}}}}} \right ) \sqrt{{\frac{{{\rm e}^{bx+a}}}{ \left ({{\rm e}^{bx+a}} \right ) ^{2}-1}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*cosh(b*x+a)*csch(b*x+a)^(1/2),x)

[Out]

(b*x-2)*(exp(b*x+a)^2-1)/b^2*2^(1/2)*(exp(b*x+a)/(exp(b*x+a)^2-1))^(1/2)/exp(b*x+a)+2/b^2*(2*(exp(b*x+a)^2-1)/
((exp(b*x+a)^2-1)*exp(b*x+a))^(1/2)-(1+exp(b*x+a))^(1/2)*(2-2*exp(b*x+a))^(1/2)*(-exp(b*x+a))^(1/2)/(exp(b*x+a
)^3-exp(b*x+a))^(1/2)*(-2*EllipticE((1+exp(b*x+a))^(1/2),1/2*2^(1/2))+EllipticF((1+exp(b*x+a))^(1/2),1/2*2^(1/
2))))*2^(1/2)*(exp(b*x+a)/(exp(b*x+a)^2-1))^(1/2)*((exp(b*x+a)^2-1)*exp(b*x+a))^(1/2)/exp(b*x+a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh \left (b x + a\right ) \sqrt{\operatorname{csch}\left (b x + a\right )}\,{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)^(1/2),x, algorithm="maxima")

[Out]

integrate(x*cosh(b*x + a)*sqrt(csch(b*x + a)), 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)^(1/2),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)**(1/2),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 ) \sqrt{\operatorname{csch}\left (b x + a\right )}\,{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)^(1/2),x, algorithm="giac")

[Out]

integrate(x*cosh(b*x + a)*sqrt(csch(b*x + a)), x)