Integrand size = 18, antiderivative size = 83 \[ \int x \cos ^{\frac {5}{2}}(a+b x) \sin (a+b x) \, dx=-\frac {2 x \cos ^{\frac {7}{2}}(a+b x)}{7 b}+\frac {20 \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{147 b^2}+\frac {20 \sqrt {\cos (a+b x)} \sin (a+b x)}{147 b^2}+\frac {4 \cos ^{\frac {5}{2}}(a+b x) \sin (a+b x)}{49 b^2} \] Output:
-2/7*x*cos(b*x+a)^(7/2)/b+20/147*InverseJacobiAM(1/2*a+1/2*b*x,2^(1/2))/b^ 2+20/147*cos(b*x+a)^(1/2)*sin(b*x+a)/b^2+4/49*cos(b*x+a)^(5/2)*sin(b*x+a)/ b^2
Time = 0.39 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.88 \[ \int x \cos ^{\frac {5}{2}}(a+b x) \sin (a+b x) \, dx=\frac {40 \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )+\sqrt {\cos (a+b x)} (-63 b x \cos (a+b x)-21 b x \cos (3 (a+b x))+46 \sin (a+b x)+6 \sin (3 (a+b x)))}{294 b^2} \] Input:
Integrate[x*Cos[a + b*x]^(5/2)*Sin[a + b*x],x]
Output:
(40*EllipticF[(a + b*x)/2, 2] + Sqrt[Cos[a + b*x]]*(-63*b*x*Cos[a + b*x] - 21*b*x*Cos[3*(a + b*x)] + 46*Sin[a + b*x] + 6*Sin[3*(a + b*x)]))/(294*b^2 )
Time = 0.37 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3925, 3042, 3115, 3042, 3115, 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.
\(\displaystyle \int x \sin (a+b x) \cos ^{\frac {5}{2}}(a+b x) \, dx\) |
\(\Big \downarrow \) 3925 |
\(\displaystyle \frac {2 \int \cos ^{\frac {7}{2}}(a+b x)dx}{7 b}-\frac {2 x \cos ^{\frac {7}{2}}(a+b x)}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \int \sin \left (a+b x+\frac {\pi }{2}\right )^{7/2}dx}{7 b}-\frac {2 x \cos ^{\frac {7}{2}}(a+b x)}{7 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {2 \left (\frac {5}{7} \int \cos ^{\frac {3}{2}}(a+b x)dx+\frac {2 \sin (a+b x) \cos ^{\frac {5}{2}}(a+b x)}{7 b}\right )}{7 b}-\frac {2 x \cos ^{\frac {7}{2}}(a+b x)}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (\frac {5}{7} \int \sin \left (a+b x+\frac {\pi }{2}\right )^{3/2}dx+\frac {2 \sin (a+b x) \cos ^{\frac {5}{2}}(a+b x)}{7 b}\right )}{7 b}-\frac {2 x \cos ^{\frac {7}{2}}(a+b x)}{7 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {2 \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (a+b x)}}dx+\frac {2 \sin (a+b x) \sqrt {\cos (a+b x)}}{3 b}\right )+\frac {2 \sin (a+b x) \cos ^{\frac {5}{2}}(a+b x)}{7 b}\right )}{7 b}-\frac {2 x \cos ^{\frac {7}{2}}(a+b x)}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (a+b x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (a+b x) \sqrt {\cos (a+b x)}}{3 b}\right )+\frac {2 \sin (a+b x) \cos ^{\frac {5}{2}}(a+b x)}{7 b}\right )}{7 b}-\frac {2 x \cos ^{\frac {7}{2}}(a+b x)}{7 b}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 \left (\frac {2 \sin (a+b x) \cos ^{\frac {5}{2}}(a+b x)}{7 b}+\frac {5}{7} \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{3 b}+\frac {2 \sin (a+b x) \sqrt {\cos (a+b x)}}{3 b}\right )\right )}{7 b}-\frac {2 x \cos ^{\frac {7}{2}}(a+b x)}{7 b}\) |
Input:
Int[x*Cos[a + b*x]^(5/2)*Sin[a + b*x],x]
Output:
(-2*x*Cos[a + b*x]^(7/2))/(7*b) + (2*((2*Cos[a + b*x]^(5/2)*Sin[a + b*x])/ (7*b) + (5*((2*EllipticF[(a + b*x)/2, 2])/(3*b) + (2*Sqrt[Cos[a + b*x]]*Si n[a + b*x])/(3*b)))/7))/(7*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[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[Cos[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^( n_.)], x_Symbol] :> Simp[(-x^(m - n + 1))*(Cos[a + b*x^n]^(p + 1)/(b*n*(p + 1))), x] + Simp[(m - n + 1)/(b*n*(p + 1)) Int[x^(m - n)*Cos[a + b*x^n]^( p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
\[\int x \cos \left (b x +a \right )^{\frac {5}{2}} \sin \left (b x +a \right )d x\]
Input:
int(x*cos(b*x+a)^(5/2)*sin(b*x+a),x)
Output:
int(x*cos(b*x+a)^(5/2)*sin(b*x+a),x)
Exception generated. \[ \int x \cos ^{\frac {5}{2}}(a+b x) \sin (a+b x) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*cos(b*x+a)^(5/2)*sin(b*x+a),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
Timed out. \[ \int x \cos ^{\frac {5}{2}}(a+b x) \sin (a+b x) \, dx=\text {Timed out} \] Input:
integrate(x*cos(b*x+a)**(5/2)*sin(b*x+a),x)
Output:
Timed out
\[ \int x \cos ^{\frac {5}{2}}(a+b x) \sin (a+b x) \, dx=\int { x \cos \left (b x + a\right )^{\frac {5}{2}} \sin \left (b x + a\right ) \,d x } \] Input:
integrate(x*cos(b*x+a)^(5/2)*sin(b*x+a),x, algorithm="maxima")
Output:
integrate(x*cos(b*x + a)^(5/2)*sin(b*x + a), x)
\[ \int x \cos ^{\frac {5}{2}}(a+b x) \sin (a+b x) \, dx=\int { x \cos \left (b x + a\right )^{\frac {5}{2}} \sin \left (b x + a\right ) \,d x } \] Input:
integrate(x*cos(b*x+a)^(5/2)*sin(b*x+a),x, algorithm="giac")
Output:
integrate(x*cos(b*x + a)^(5/2)*sin(b*x + a), x)
Timed out. \[ \int x \cos ^{\frac {5}{2}}(a+b x) \sin (a+b x) \, dx=\int x\,{\cos \left (a+b\,x\right )}^{5/2}\,\sin \left (a+b\,x\right ) \,d x \] Input:
int(x*cos(a + b*x)^(5/2)*sin(a + b*x),x)
Output:
int(x*cos(a + b*x)^(5/2)*sin(a + b*x), x)
\[ \int x \cos ^{\frac {5}{2}}(a+b x) \sin (a+b x) \, dx=\int \sqrt {\cos \left (b x +a \right )}\, \cos \left (b x +a \right )^{2} \sin \left (b x +a \right ) x d x \] Input:
int(x*cos(b*x+a)^(5/2)*sin(b*x+a),x)
Output:
int(sqrt(cos(a + b*x))*cos(a + b*x)**2*sin(a + b*x)*x,x)