\(\int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx\) [310]

Optimal result

Integrand size = 18, antiderivative size = 121 \[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=\frac {12 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{35 b^2}-\frac {4 \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x)}{35 b^2}-\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {12 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{35 b^2 \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \] Output:

12/35*cosh(b*x+a)*csch(b*x+a)^(1/2)/b^2-4/35*cosh(b*x+a)*csch(b*x+a)^(5/2) 
/b^2-2/7*x*csch(b*x+a)^(7/2)/b-12/35*I*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*sinh(b*x+a))^(1/2)
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.69 \[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=-\frac {2 \sqrt {\text {csch}(a+b x)} \left (-6 \cosh (a+b x)+6 E\left (\left .\frac {1}{4} (-2 i a+\pi -2 i b x)\right |2\right ) \sqrt {i \sinh (a+b x)}+\text {csch}^3(a+b x) (5 b x+\sinh (2 (a+b x)))\right )}{35 b^2} \] Input:

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

Output:

(-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)
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5968, 3042, 4255, 3042, 4255, 3042, 4258, 3042, 3119}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5968

\(\displaystyle \frac {2 \int \text {csch}^{\frac {7}{2}}(a+b x)dx}{7 b}-\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {2 \int (i \csc (i a+i b x))^{7/2}dx}{7 b}\)

\(\Big \downarrow \) 4255

\(\displaystyle \frac {2 \left (-\frac {3}{5} \int \text {csch}^{\frac {3}{2}}(a+b x)dx-\frac {2 \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x)}{5 b}\right )}{7 b}-\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {2 \left (-\frac {2 \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x)}{5 b}-\frac {3}{5} \int (i \csc (i a+i b x))^{3/2}dx\right )}{7 b}\)

\(\Big \downarrow \) 4255

\(\displaystyle \frac {2 \left (-\frac {3}{5} \left (\int \frac {1}{\sqrt {\text {csch}(a+b x)}}dx-\frac {2 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{b}\right )-\frac {2 \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x)}{5 b}\right )}{7 b}-\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {2 \left (-\frac {2 \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x)}{5 b}-\frac {3}{5} \left (-\frac {2 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{b}+\int \frac {1}{\sqrt {i \csc (i a+i b x)}}dx\right )\right )}{7 b}\)

\(\Big \downarrow \) 4258

\(\displaystyle -\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {2 \left (-\frac {2 \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x)}{5 b}-\frac {3}{5} \left (-\frac {2 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{b}+\frac {\int \sqrt {i \sinh (a+b x)}dx}{\sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}\right )\right )}{7 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {2 \left (-\frac {2 \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x)}{5 b}-\frac {3}{5} \left (-\frac {2 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{b}+\frac {\int \sqrt {\sin (i a+i b x)}dx}{\sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}\right )\right )}{7 b}\)

\(\Big \downarrow \) 3119

\(\displaystyle -\frac {2 x \text {csch}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {2 \left (-\frac {2 \cosh (a+b x) \text {csch}^{\frac {5}{2}}(a+b x)}{5 b}-\frac {3}{5} \left (-\frac {2 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{b}-\frac {2 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}\right )\right )}{7 b}\)

Input:

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

Output:

(-2*x*Csch[a + b*x]^(7/2))/(7*b) + (2*((-2*Cosh[a + b*x]*Csch[a + b*x]^(5/ 
2))/(5*b) - (3*((-2*Cosh[a + b*x]*Sqrt[Csch[a + b*x]])/b - ((2*I)*Elliptic 
E[(I*a - Pi/2 + I*b*x)/2, 2])/(b*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]] 
)))/5))/(7*b)
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

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 4255

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] + Simp[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 4258

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(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 5968

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] + Simp[(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]
 

Maple [F]

\[\int x \cosh \left (b x +a \right ) \operatorname {csch}\left (b x +a \right )^{\frac {9}{2}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F(-1)]

Timed out. \[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=\int { x \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{\frac {9}{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=\int { x \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{\frac {9}{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=\int x\,\mathrm {cosh}\left (a+b\,x\right )\,{\left (\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}\right )}^{9/2} \,d x \] Input:

int(x*cosh(a + b*x)*(1/sinh(a + b*x))^(9/2),x)
 

Output:

int(x*cosh(a + b*x)*(1/sinh(a + b*x))^(9/2), x)
 

Reduce [F]

\[ \int x \cosh (a+b x) \text {csch}^{\frac {9}{2}}(a+b x) \, dx=\int \sqrt {\mathrm {csch}\left (b x +a \right )}\, \cosh \left (b x +a \right ) \mathrm {csch}\left (b x +a \right )^{4} x d x \] Input:

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

Output:

int(sqrt(csch(a + b*x))*cosh(a + b*x)*csch(a + b*x)**4*x,x)