Integrand size = 18, antiderivative size = 98 \[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=-\frac {12 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{25 b^2 \sqrt {i \sinh (a+b x)}}-\frac {4 \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b} \] Output:
12/25*I*EllipticE(cos(1/2*I*a+1/4*Pi+1/2*I*b*x),2^(1/2))*sinh(b*x+a)^(1/2) /b^2/(I*sinh(b*x+a))^(1/2)-4/25*cosh(b*x+a)*sinh(b*x+a)^(3/2)/b^2+2/5*x*si nh(b*x+a)^(5/2)/b
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.73 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.46 \[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=\frac {e^{-3 (a+b x)} \left (\left (-1+e^{2 (a+b x)}\right ) \left (2+5 b x+e^{2 (a+b x)} (24-10 b x)+e^{4 (a+b x)} (-2+5 b x)\right )+48 e^{2 (a+b x)} \sqrt {1-e^{2 (a+b x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},e^{2 (a+b x)}\right )\right )}{50 \sqrt {2} b^2 \sqrt {-e^{-a-b x}+e^{a+b x}}} \] Input:
Integrate[x*Cosh[a + b*x]*Sinh[a + b*x]^(3/2),x]
Output:
((-1 + E^(2*(a + b*x)))*(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))*Sqrt[1 - E^(2*(a + b*x))]*Hy pergeometric2F1[-1/4, 1/2, 3/4, E^(2*(a + b*x))])/(50*Sqrt[2]*b^2*E^(3*(a + b*x))*Sqrt[-E^(-a - b*x) + E^(a + b*x)])
Time = 0.43 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5895, 3042, 3115, 3042, 3121, 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.
\(\displaystyle \int x \sinh ^{\frac {3}{2}}(a+b x) \cosh (a+b x) \, dx\) |
\(\Big \downarrow \) 5895 |
\(\displaystyle \frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {2 \int \sinh ^{\frac {5}{2}}(a+b x)dx}{5 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {2 \int (-i \sin (i a+i b x))^{5/2}dx}{5 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {2 \left (\frac {2 \sinh ^{\frac {3}{2}}(a+b x) \cosh (a+b x)}{5 b}-\frac {3}{5} \int \sqrt {\sinh (a+b x)}dx\right )}{5 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {2 \left (\frac {2 \sinh ^{\frac {3}{2}}(a+b x) \cosh (a+b x)}{5 b}-\frac {3}{5} \int \sqrt {-i \sin (i a+i b x)}dx\right )}{5 b}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {2 \left (\frac {2 \sinh ^{\frac {3}{2}}(a+b x) \cosh (a+b x)}{5 b}-\frac {3 \sqrt {\sinh (a+b x)} \int \sqrt {i \sinh (a+b x)}dx}{5 \sqrt {i \sinh (a+b x)}}\right )}{5 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {2 \left (\frac {2 \sinh ^{\frac {3}{2}}(a+b x) \cosh (a+b x)}{5 b}-\frac {3 \sqrt {\sinh (a+b x)} \int \sqrt {\sin (i a+i b x)}dx}{5 \sqrt {i \sinh (a+b x)}}\right )}{5 b}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {2 \left (\frac {2 \sinh ^{\frac {3}{2}}(a+b x) \cosh (a+b x)}{5 b}+\frac {6 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{5 b \sqrt {i \sinh (a+b x)}}\right )}{5 b}\) |
Input:
Int[x*Cosh[a + b*x]*Sinh[a + b*x]^(3/2),x]
Output:
(2*x*Sinh[a + b*x]^(5/2))/(5*b) - (2*((((6*I)/5)*EllipticE[(I*a - Pi/2 + I *b*x)/2, 2]*Sqrt[Sinh[a + b*x]])/(b*Sqrt[I*Sinh[a + b*x]]) + (2*Cosh[a + b *x]*Sinh[a + b*x]^(3/2))/(5*b)))/(5*b)
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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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]
\[\int x \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{\frac {3}{2}}d x\]
Input:
int(x*cosh(b*x+a)*sinh(b*x+a)^(3/2),x)
Output:
int(x*cosh(b*x+a)*sinh(b*x+a)^(3/2),x)
Exception generated. \[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*cosh(b*x+a)*sinh(b*x+a)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
Timed out. \[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=\text {Timed out} \] Input:
integrate(x*cosh(b*x+a)*sinh(b*x+a)**(3/2),x)
Output:
Timed out
\[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=\int { x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(x*cosh(b*x+a)*sinh(b*x+a)^(3/2),x, algorithm="maxima")
Output:
integrate(x*cosh(b*x + a)*sinh(b*x + a)^(3/2), x)
\[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=\int { x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(x*cosh(b*x+a)*sinh(b*x+a)^(3/2),x, algorithm="giac")
Output:
integrate(x*cosh(b*x + a)*sinh(b*x + a)^(3/2), x)
Timed out. \[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=\int x\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^{3/2} \,d x \] Input:
int(x*cosh(a + b*x)*sinh(a + b*x)^(3/2),x)
Output:
int(x*cosh(a + b*x)*sinh(a + b*x)^(3/2), x)
\[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=\int \sqrt {\sinh \left (b x +a \right )}\, \cosh \left (b x +a \right ) \sinh \left (b x +a \right ) x d x \] Input:
int(x*cosh(b*x+a)*sinh(b*x+a)^(3/2),x)
Output:
int(sqrt(sinh(a + b*x))*cosh(a + b*x)*sinh(a + b*x)*x,x)