Integrand size = 18, antiderivative size = 107 \[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}+\frac {20 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{147 b^2}-\frac {4 \sinh (a+b x)}{49 b^2 \text {sech}^{\frac {5}{2}}(a+b x)}-\frac {20 \sinh (a+b x)}{147 b^2 \sqrt {\text {sech}(a+b x)}} \]
2/7*x/b/sech(b*x+a)^(7/2)-4/49*sinh(b*x+a)/b^2/sech(b*x+a)^(5/2)-20/147*si nh(b*x+a)/b^2/sech(b*x+a)^(1/2)+20/147*I*(cosh(1/2*a+1/2*b*x)^2)^(1/2)/cos h(1/2*a+1/2*b*x)*EllipticF(I*sinh(1/2*a+1/2*b*x),2^(1/2))*cosh(b*x+a)^(1/2 )*sech(b*x+a)^(1/2)/b^2
Time = 0.42 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.87 \[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\frac {\sqrt {\text {sech}(a+b x)} \left (63 b x+84 b x \cosh (2 (a+b x))+21 b x \cosh (4 (a+b x))+80 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )-52 \sinh (2 (a+b x))-6 \sinh (4 (a+b x))\right )}{588 b^2} \]
(Sqrt[Sech[a + b*x]]*(63*b*x + 84*b*x*Cosh[2*(a + b*x)] + 21*b*x*Cosh[4*(a + b*x)] + (80*I)*Sqrt[Cosh[a + b*x]]*EllipticF[(I/2)*(a + b*x), 2] - 52*S inh[2*(a + b*x)] - 6*Sinh[4*(a + b*x)]))/(588*b^2)
Time = 0.51 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5967, 3042, 4256, 3042, 4256, 3042, 4258, 3042, 3120}
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.
\(\displaystyle \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx\) |
\(\Big \downarrow \) 5967 |
\(\displaystyle \frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}-\frac {2 \int \frac {1}{\text {sech}^{\frac {7}{2}}(a+b x)}dx}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}-\frac {2 \int \frac {1}{\csc \left (i a+i b x+\frac {\pi }{2}\right )^{7/2}}dx}{7 b}\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}-\frac {2 \left (\frac {5}{7} \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)}dx+\frac {2 \sinh (a+b x)}{7 b \text {sech}^{\frac {5}{2}}(a+b x)}\right )}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}-\frac {2 \left (\frac {2 \sinh (a+b x)}{7 b \text {sech}^{\frac {5}{2}}(a+b x)}+\frac {5}{7} \int \frac {1}{\csc \left (i a+i b x+\frac {\pi }{2}\right )^{3/2}}dx\right )}{7 b}\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}-\frac {2 \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {\text {sech}(a+b x)}dx+\frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}\right )+\frac {2 \sinh (a+b x)}{7 b \text {sech}^{\frac {5}{2}}(a+b x)}\right )}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}-\frac {2 \left (\frac {2 \sinh (a+b x)}{7 b \text {sech}^{\frac {5}{2}}(a+b x)}+\frac {5}{7} \left (\frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}+\frac {1}{3} \int \sqrt {\csc \left (i a+i b x+\frac {\pi }{2}\right )}dx\right )\right )}{7 b}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}-\frac {2 \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \int \frac {1}{\sqrt {\cosh (a+b x)}}dx+\frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}\right )+\frac {2 \sinh (a+b x)}{7 b \text {sech}^{\frac {5}{2}}(a+b x)}\right )}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}-\frac {2 \left (\frac {2 \sinh (a+b x)}{7 b \text {sech}^{\frac {5}{2}}(a+b x)}+\frac {5}{7} \left (\frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}+\frac {1}{3} \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \int \frac {1}{\sqrt {\sin \left (i a+i b x+\frac {\pi }{2}\right )}}dx\right )\right )}{7 b}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 x}{7 b \text {sech}^{\frac {7}{2}}(a+b x)}-\frac {2 \left (\frac {2 \sinh (a+b x)}{7 b \text {sech}^{\frac {5}{2}}(a+b x)}+\frac {5}{7} \left (\frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}-\frac {2 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{3 b}\right )\right )}{7 b}\) |
(2*x)/(7*b*Sech[a + b*x]^(7/2)) - (2*((2*Sinh[a + b*x])/(7*b*Sech[a + b*x] ^(5/2)) + (5*((((-2*I)/3)*Sqrt[Cosh[a + b*x]]*EllipticF[(I/2)*(a + b*x), 2 ]*Sqrt[Sech[a + b*x]])/b + (2*Sinh[a + b*x])/(3*b*Sqrt[Sech[a + b*x]])))/7 ))/(7*b)
3.6.43.3.1 Defintions of rubi rules used
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
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] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* n]
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]
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]
\[\int \frac {x \sinh \left (b x +a \right )}{\operatorname {sech}\left (b x +a \right )^{\frac {5}{2}}}d x\]
Exception generated. \[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {x \sinh {\left (a + b x \right )}}{\operatorname {sech}^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
\[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \sinh \left (b x + a\right )}{\operatorname {sech}\left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \sinh \left (b x + a\right )}{\operatorname {sech}\left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x \sinh (a+b x)}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {x\,\mathrm {sinh}\left (a+b\,x\right )}{{\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{5/2}} \,d x \]