Integrand size = 18, antiderivative size = 121 \[ \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx=\frac {20 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right ),2\right ) \sqrt {i \sinh (a+b x)}}{147 b^2 \sqrt {\sinh (a+b x)}}+\frac {20 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{147 b^2}-\frac {4 \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x)}{49 b^2}+\frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b} \]
-4/49*cosh(b*x+a)*sinh(b*x+a)^(5/2)/b^2+2/7*x*sinh(b*x+a)^(7/2)/b-20/147*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)*Ell ipticF(cos(1/2*I*a+1/4*Pi+1/2*I*b*x),2^(1/2))*(I*sinh(b*x+a))^(1/2)/b^2/si nh(b*x+a)^(1/2)+20/147*cosh(b*x+a)*sinh(b*x+a)^(1/2)/b^2
Time = 0.36 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.85 \[ \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx=\frac {63 b x-84 b x \cosh (2 (a+b x))+21 b x \cosh (4 (a+b x))-80 i \operatorname {EllipticF}\left (\frac {1}{4} (-2 i a+\pi -2 i b x),2\right ) \sqrt {i \sinh (a+b x)}+52 \sinh (2 (a+b x))-6 \sinh (4 (a+b x))}{588 b^2 \sqrt {\sinh (a+b x)}} \]
(63*b*x - 84*b*x*Cosh[2*(a + b*x)] + 21*b*x*Cosh[4*(a + b*x)] - (80*I)*Ell ipticF[((-2*I)*a + Pi - (2*I)*b*x)/4, 2]*Sqrt[I*Sinh[a + b*x]] + 52*Sinh[2 *(a + b*x)] - 6*Sinh[4*(a + b*x)])/(588*b^2*Sqrt[Sinh[a + b*x]])
Time = 0.49 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5895, 3042, 3115, 3042, 3115, 3042, 3121, 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.
\(\displaystyle \int x \sinh ^{\frac {5}{2}}(a+b x) \cosh (a+b x) \, dx\) |
\(\Big \downarrow \) 5895 |
\(\displaystyle \frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {2 \int \sinh ^{\frac {7}{2}}(a+b x)dx}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {2 \int (-i \sin (i a+i b x))^{7/2}dx}{7 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {2 \left (\frac {2 \sinh ^{\frac {5}{2}}(a+b x) \cosh (a+b x)}{7 b}-\frac {5}{7} \int \sinh ^{\frac {3}{2}}(a+b x)dx\right )}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {2 \left (\frac {2 \sinh ^{\frac {5}{2}}(a+b x) \cosh (a+b x)}{7 b}-\frac {5}{7} \int (-i \sin (i a+i b x))^{3/2}dx\right )}{7 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {2 \left (\frac {2 \sinh ^{\frac {5}{2}}(a+b x) \cosh (a+b x)}{7 b}-\frac {5}{7} \left (\frac {2 \sqrt {\sinh (a+b x)} \cosh (a+b x)}{3 b}-\frac {1}{3} \int \frac {1}{\sqrt {\sinh (a+b x)}}dx\right )\right )}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {2 \left (\frac {2 \sinh ^{\frac {5}{2}}(a+b x) \cosh (a+b x)}{7 b}-\frac {5}{7} \left (\frac {2 \sqrt {\sinh (a+b x)} \cosh (a+b x)}{3 b}-\frac {1}{3} \int \frac {1}{\sqrt {-i \sin (i a+i b x)}}dx\right )\right )}{7 b}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {2 \left (\frac {2 \sinh ^{\frac {5}{2}}(a+b x) \cosh (a+b x)}{7 b}-\frac {5}{7} \left (\frac {2 \sqrt {\sinh (a+b x)} \cosh (a+b x)}{3 b}-\frac {\sqrt {i \sinh (a+b x)} \int \frac {1}{\sqrt {i \sinh (a+b x)}}dx}{3 \sqrt {\sinh (a+b x)}}\right )\right )}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {2 \left (\frac {2 \sinh ^{\frac {5}{2}}(a+b x) \cosh (a+b x)}{7 b}-\frac {5}{7} \left (\frac {2 \sqrt {\sinh (a+b x)} \cosh (a+b x)}{3 b}-\frac {\sqrt {i \sinh (a+b x)} \int \frac {1}{\sqrt {\sin (i a+i b x)}}dx}{3 \sqrt {\sinh (a+b x)}}\right )\right )}{7 b}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 x \sinh ^{\frac {7}{2}}(a+b x)}{7 b}-\frac {2 \left (\frac {2 \sinh ^{\frac {5}{2}}(a+b x) \cosh (a+b x)}{7 b}-\frac {5}{7} \left (\frac {2 \sqrt {\sinh (a+b x)} \cosh (a+b x)}{3 b}+\frac {2 i \sqrt {i \sinh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right ),2\right )}{3 b \sqrt {\sinh (a+b x)}}\right )\right )}{7 b}\) |
(2*x*Sinh[a + b*x]^(7/2))/(7*b) - (2*((-5*((((2*I)/3)*EllipticF[(I*a - Pi/ 2 + I*b*x)/2, 2]*Sqrt[I*Sinh[a + b*x]])/(b*Sqrt[Sinh[a + b*x]]) + (2*Cosh[ a + b*x]*Sqrt[Sinh[a + b*x]])/(3*b)))/7 + (2*Cosh[a + b*x]*Sinh[a + b*x]^( 5/2))/(7*b)))/(7*b)
3.6.44.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.) ]^(p_.), x_Symbol] :> Simp[x^(m - n + 1)*(Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1))), x] - Simp[(m - n + 1)/(b*n*(p + 1)) Int[x^(m - n)*Sinh[a + b*x^n]^ (p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
\[\int x \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{\frac {5}{2}}d x\]
Exception generated. \[ \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
Timed out. \[ \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx=\text {Timed out} \]
\[ \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx=\int { x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{\frac {5}{2}} \,d x } \]
Exception generated. \[ \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Unable to divide, perhaps due to rounding error%%%{1,[0 ,1,1,0]%%%} / %%%{1,[0,0,0,2]%%%} Error: Bad Argument Value
Timed out. \[ \int x \cosh (a+b x) \sinh ^{\frac {5}{2}}(a+b x) \, dx=\int x\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^{5/2} \,d x \]