\(\int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx\) [556]

Optimal result

Integrand size = 18, antiderivative size = 71 \[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\frac {2 x}{b \sqrt {\text {csch}(a+b x)}}+\frac {4 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{b^2 \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \]

[Out]

2*x/b/csch(b*x+a)^(1/2)-4*I*(sin(1/2*I*a+1/4*Pi+1/2*I*b*x)^2)^(1/2)/sin(1/2*I*a+1/4*Pi+1/2*I*b*x)*EllipticE(co
s(1/2*I*a+1/4*Pi+1/2*I*b*x),2^(1/2))/b^2/csch(b*x+a)^(1/2)/(I*sinh(b*x+a))^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5553, 3856, 2719} \[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\frac {2 x}{b \sqrt {\text {csch}(a+b x)}}+\frac {4 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b^2 \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}} \]

[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 2719

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

Rule 3856

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 5553

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.90 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.41 \[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\frac {\sqrt {2} e^{-a-b x} \sqrt {\frac {e^{a+b x}}{-1+e^{2 (a+b x)}}} \left (\left (-1+e^{2 (a+b x)}\right ) (-2+b x)-4 \sqrt {1-e^{2 (a+b x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},e^{2 (a+b x)}\right )\right )}{b^2} \]

[In]

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

[Out]

(Sqrt[2]*E^(-a - b*x)*Sqrt[E^(a + b*x)/(-1 + E^(2*(a + b*x)))]*((-1 + E^(2*(a + b*x)))*(-2 + b*x) - 4*Sqrt[1 -
E^(2*(a + b*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, E^(2*(a + b*x))]))/b^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(228\) vs. \(2(92)=184\).

Time = 0.28 (sec) , antiderivative size = 229, normalized size of antiderivative = 3.23

method	result	size
risch	\(\frac {\left (b x -2\right ) \left ({\mathrm e}^{2 b x +2 a}-1\right ) \sqrt {2}\, \sqrt {\frac {{\mathrm e}^{b x +a}}{{\mathrm e}^{2 b x +2 a}-1}}\, {\mathrm e}^{-b x -a}}{b^{2}}+\frac {2 \left (\frac {2 \,{\mathrm e}^{2 b x +2 a}-2}{\sqrt {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}-1\right )}}-\frac {\sqrt {{\mathrm e}^{b x +a}+1}\, \sqrt {-2 \,{\mathrm e}^{b x +a}+2}\, \sqrt {-{\mathrm e}^{b x +a}}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {{\mathrm e}^{b x +a}+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {{\mathrm e}^{b x +a}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}-{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {\frac {{\mathrm e}^{b x +a}}{{\mathrm e}^{2 b x +2 a}-1}}\, \sqrt {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}-1\right )}\, {\mathrm e}^{-b x -a}}{b^{2}}\)	\(229\)

[In]

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

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

Fricas [F(-2)]

Exception generated. \[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x*cosh(b*x+a)*csch(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

\[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\int x \cosh {\left (a + b x \right )} \sqrt {\operatorname {csch}{\left (a + b x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\int { x \cosh \left (b x + a\right ) \sqrt {\operatorname {csch}\left (b x + a\right )} \,d x } \]

[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)

Giac [F]

\[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\int { x \cosh \left (b x + a\right ) \sqrt {\operatorname {csch}\left (b x + a\right )} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx=\int x\,\mathrm {cosh}\left (a+b\,x\right )\,\sqrt {\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}} \,d x \]

[In]

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

[Out]

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