\(\int x \text {sech}^{\frac {9}{2}}(a+b x) \sinh (a+b x) \, dx\) [294]

Optimal result

Integrand size = 18, antiderivative size = 107 \[ \int x \text {sech}^{\frac {9}{2}}(a+b x) \sinh (a+b x) \, dx=\frac {12 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{35 b^2}-\frac {2 x \text {sech}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {12 \sqrt {\text {sech}(a+b x)} \sinh (a+b x)}{35 b^2}+\frac {4 \text {sech}^{\frac {5}{2}}(a+b x) \sinh (a+b x)}{35 b^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.64 \[ \int x \text {sech}^{\frac {9}{2}}(a+b x) \sinh (a+b x) \, dx=\frac {\text {sech}^{\frac {7}{2}}(a+b x) \left (-20 b x+24 i \cosh ^{\frac {7}{2}}(a+b x) E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )+10 \sinh (2 (a+b x))+3 \sinh (4 (a+b x))\right )}{70 b^2} \] Input:

Integrate[x*Sech[a + b*x]^(9/2)*Sinh[a + b*x],x]
 

Output:

(Sech[a + b*x]^(7/2)*(-20*b*x + (24*I)*Cosh[a + b*x]^(7/2)*EllipticE[(I/2) 
*(a + b*x), 2] + 10*Sinh[2*(a + b*x)] + 3*Sinh[4*(a + b*x)]))/(70*b^2)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5967, 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.

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

\(\Big \downarrow \) 5967

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4255

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4255

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4258

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

Input:

Int[x*Sech[a + b*x]^(9/2)*Sinh[a + b*x],x]
 

Output:

(-2*x*Sech[a + b*x]^(7/2))/(7*b) + (2*((2*Sech[a + b*x]^(5/2)*Sinh[a + b*x 
])/(5*b) + (3*(((2*I)*Sqrt[Cosh[a + b*x]]*EllipticE[(I/2)*(a + b*x), 2]*Sq 
rt[Sech[a + b*x]])/b + (2*Sqrt[Sech[a + b*x]]*Sinh[a + b*x])/b))/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 5967

Int[(x_)^(m_.)*Sech[(a_.) + (b_.)*(x_)^(n_.)]^(p_)*Sinh[(a_.) + (b_.)*(x_)^ 
(n_.)], x_Symbol] :> Simp[(-x^(m - n + 1))*(Sech[a + b*x^n]^(p - 1)/(b*n*(p 
 - 1))), x] + Simp[(m - n + 1)/(b*n*(p - 1))   Int[x^(m - n)*Sech[a + b*x^n 
]^(p - 1), x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && 
 NeQ[p, 1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F(-1)]

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

integrate(x*sech(b*x+a)**(9/2)*sinh(b*x+a),x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate(x*sech(b*x + a)^(9/2)*sinh(b*x + a), x)
 

Giac [F]

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

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

Output:

integrate(x*sech(b*x + a)^(9/2)*sinh(b*x + a), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(sech(a + b*x))*sech(a + b*x)**4*sinh(a + b*x)*x,x)