\(\int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx\) [558]

Optimal result

Integrand size = 18, antiderivative size = 98 \[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\frac {2 x}{5 b \text {csch}^{\frac {5}{2}}(a+b x)}-\frac {4 \cosh (a+b x)}{25 b^2 \text {csch}^{\frac {3}{2}}(a+b x)}-\frac {12 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{25 b^2 \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \]

[Out]

2/5*x/b/csch(b*x+a)^(5/2)-4/25*cosh(b*x+a)/b^2/csch(b*x+a)^(3/2)+12/25*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(cos(1/2*I*a+1/4*Pi+1/2*I*b*x),2^(1/2))/b^2/csch(b*x+a)^(1/2)/(I*sin
h(b*x+a))^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5553, 3854, 3856, 2719} \[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=-\frac {4 \cosh (a+b x)}{25 b^2 \text {csch}^{\frac {3}{2}}(a+b x)}-\frac {12 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{25 b^2 \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}+\frac {2 x}{5 b \text {csch}^{\frac {5}{2}}(a+b x)} \]

[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 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 3854

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 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}{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] (verified)

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

Time = 3.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.13 \[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\frac {e^{-2 (a+b x)} \left (2+5 b x+e^{2 (a+b x)} (24-10 b x)+e^{4 (a+b x)} (-2+5 b x)-\frac {48 e^{2 (a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},e^{2 (a+b x)}\right )}{\sqrt {1-e^{2 (a+b x)}}}\right )}{50 b^2 \sqrt {\text {csch}(a+b x)}} \]

[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]])

Maple [F]

\[\int \frac {x \cosh \left (b x +a \right )}{\operatorname {csch}\left (b x +a \right )^{\frac {3}{2}}}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {x \cosh {\left (a + b x \right )}}{\operatorname {csch}^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {x \cosh \left (b x + a\right )}{\operatorname {csch}\left (b x + a\right )^{\frac {3}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {x \cosh \left (b x + a\right )}{\operatorname {csch}\left (b x + a\right )^{\frac {3}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x \cosh (a+b x)}{\text {csch}^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {x\,\mathrm {cosh}\left (a+b\,x\right )}{{\left (\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int((x*cosh(a + b*x))/(1/sinh(a + b*x))^(3/2),x)

[Out]

int((x*cosh(a + b*x))/(1/sinh(a + b*x))^(3/2), x)