Integrand size = 21, antiderivative size = 126 \[ \int (d \cos (a+b x))^{9/2} \sin ^2(a+b x) \, dx=\frac {28 d^4 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{195 b \sqrt {\cos (a+b x)}}+\frac {28 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{585 b}+\frac {4 d (d \cos (a+b x))^{7/2} \sin (a+b x)}{117 b}-\frac {2 (d \cos (a+b x))^{11/2} \sin (a+b x)}{13 b d} \] Output:
28/195*d^4*(d*cos(b*x+a))^(1/2)*EllipticE(sin(1/2*a+1/2*b*x),2^(1/2))/b/co s(b*x+a)^(1/2)+28/585*d^3*(d*cos(b*x+a))^(3/2)*sin(b*x+a)/b+4/117*d*(d*cos (b*x+a))^(7/2)*sin(b*x+a)/b-2/13*(d*cos(b*x+a))^(11/2)*sin(b*x+a)/b/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.48 \[ \int (d \cos (a+b x))^{9/2} \sin ^2(a+b x) \, dx=\frac {d^2 (d \cos (a+b x))^{5/2} \sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {3}{2},\frac {5}{2},\sin ^2(a+b x)\right ) \tan ^3(a+b x)}{3 b} \] Input:
Integrate[(d*Cos[a + b*x])^(9/2)*Sin[a + b*x]^2,x]
Output:
(d^2*(d*Cos[a + b*x])^(5/2)*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[-7/4, 3/2, 5/2, Sin[a + b*x]^2]*Tan[a + b*x]^3)/(3*b)
Time = 0.97 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3048, 3042, 3115, 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.
| \(\displaystyle \int \sin ^2(a+b x) (d \cos (a+b x))^{9/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (a+b x)^2 (d \cos (a+b x))^{9/2}dx\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {2}{13} \int (d \cos (a+b x))^{9/2}dx-\frac {2 \sin (a+b x) (d \cos (a+b x))^{11/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{13} \int \left (d \sin \left (a+b x+\frac {\pi }{2}\right )\right )^{9/2}dx-\frac {2 \sin (a+b x) (d \cos (a+b x))^{11/2}}{13 b d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {2}{13} \left (\frac {7}{9} d^2 \int (d \cos (a+b x))^{5/2}dx+\frac {2 d \sin (a+b x) (d \cos (a+b x))^{7/2}}{9 b}\right )-\frac {2 \sin (a+b x) (d \cos (a+b x))^{11/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{13} \left (\frac {7}{9} d^2 \int \left (d \sin \left (a+b x+\frac {\pi }{2}\right )\right )^{5/2}dx+\frac {2 d \sin (a+b x) (d \cos (a+b x))^{7/2}}{9 b}\right )-\frac {2 \sin (a+b x) (d \cos (a+b x))^{11/2}}{13 b d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {2}{13} \left (\frac {7}{9} d^2 \left (\frac {3}{5} d^2 \int \sqrt {d \cos (a+b x)}dx+\frac {2 d \sin (a+b x) (d \cos (a+b x))^{3/2}}{5 b}\right )+\frac {2 d \sin (a+b x) (d \cos (a+b x))^{7/2}}{9 b}\right )-\frac {2 \sin (a+b x) (d \cos (a+b x))^{11/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{13} \left (\frac {7}{9} d^2 \left (\frac {3}{5} d^2 \int \sqrt {d \sin \left (a+b x+\frac {\pi }{2}\right )}dx+\frac {2 d \sin (a+b x) (d \cos (a+b x))^{3/2}}{5 b}\right )+\frac {2 d \sin (a+b x) (d \cos (a+b x))^{7/2}}{9 b}\right )-\frac {2 \sin (a+b x) (d \cos (a+b x))^{11/2}}{13 b d}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {2}{13} \left (\frac {7}{9} d^2 \left (\frac {3 d^2 \sqrt {d \cos (a+b x)} \int \sqrt {\cos (a+b x)}dx}{5 \sqrt {\cos (a+b x)}}+\frac {2 d \sin (a+b x) (d \cos (a+b x))^{3/2}}{5 b}\right )+\frac {2 d \sin (a+b x) (d \cos (a+b x))^{7/2}}{9 b}\right )-\frac {2 \sin (a+b x) (d \cos (a+b x))^{11/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{13} \left (\frac {7}{9} d^2 \left (\frac {3 d^2 \sqrt {d \cos (a+b x)} \int \sqrt {\sin \left (a+b x+\frac {\pi }{2}\right )}dx}{5 \sqrt {\cos (a+b x)}}+\frac {2 d \sin (a+b x) (d \cos (a+b x))^{3/2}}{5 b}\right )+\frac {2 d \sin (a+b x) (d \cos (a+b x))^{7/2}}{9 b}\right )-\frac {2 \sin (a+b x) (d \cos (a+b x))^{11/2}}{13 b d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2}{13} \left (\frac {7}{9} d^2 \left (\frac {6 d^2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{5 b \sqrt {\cos (a+b x)}}+\frac {2 d \sin (a+b x) (d \cos (a+b x))^{3/2}}{5 b}\right )+\frac {2 d \sin (a+b x) (d \cos (a+b x))^{7/2}}{9 b}\right )-\frac {2 \sin (a+b x) (d \cos (a+b x))^{11/2}}{13 b d}\) |
Input:
Int[(d*Cos[a + b*x])^(9/2)*Sin[a + b*x]^2,x]
Output:
(-2*(d*Cos[a + b*x])^(11/2)*Sin[a + b*x])/(13*b*d) + (2*((2*d*(d*Cos[a + b *x])^(7/2)*Sin[a + b*x])/(9*b) + (7*d^2*((6*d^2*Sqrt[d*Cos[a + b*x]]*Ellip ticE[(a + b*x)/2, 2])/(5*b*Sqrt[Cos[a + b*x]]) + (2*d*(d*Cos[a + b*x])^(3/ 2)*Sin[a + b*x])/(5*b)))/9))/13
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(248\) vs. \(2(110)=220\).
Time = 11.27 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.98
| method | result | size |
| default | \(\frac {4 \sqrt {d \left (2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, d^{5} \left (2880 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{15}-11520 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{13}+19280 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{11}-17520 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{9}+9284 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{7}-2808 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}+425 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}+21 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {-2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-21 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{585 \sqrt {-d \left (2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}-\sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {d \left (2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )}\, b}\) | \(249\) |
Input:
int((d*cos(b*x+a))^(9/2)*sin(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
4/585*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)*d^5*(2880* cos(1/2*b*x+1/2*a)^15-11520*cos(1/2*b*x+1/2*a)^13+19280*cos(1/2*b*x+1/2*a) ^11-17520*cos(1/2*b*x+1/2*a)^9+9284*cos(1/2*b*x+1/2*a)^7-2808*cos(1/2*b*x+ 1/2*a)^5+425*cos(1/2*b*x+1/2*a)^3+21*(sin(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cos( 1/2*b*x+1/2*a)^2+1)^(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))-21*cos(1/2 *b*x+1/2*a))/(-d*(2*sin(1/2*b*x+1/2*a)^4-sin(1/2*b*x+1/2*a)^2))^(1/2)/sin( 1/2*b*x+1/2*a)/(d*(2*cos(1/2*b*x+1/2*a)^2-1))^(1/2)/b
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.94 \[ \int (d \cos (a+b x))^{9/2} \sin ^2(a+b x) \, dx=-\frac {2 \, {\left (-42 i \, \sqrt {\frac {1}{2}} d^{\frac {9}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 42 i \, \sqrt {\frac {1}{2}} d^{\frac {9}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + {\left (45 \, d^{4} \cos \left (b x + a\right )^{5} - 10 \, d^{4} \cos \left (b x + a\right )^{3} - 14 \, d^{4} \cos \left (b x + a\right )\right )} \sqrt {d \cos \left (b x + a\right )} \sin \left (b x + a\right )\right )}}{585 \, b} \] Input:
integrate((d*cos(b*x+a))^(9/2)*sin(b*x+a)^2,x, algorithm="fricas")
Output:
-2/585*(-42*I*sqrt(1/2)*d^(9/2)*weierstrassZeta(-4, 0, weierstrassPInverse (-4, 0, cos(b*x + a) + I*sin(b*x + a))) + 42*I*sqrt(1/2)*d^(9/2)*weierstra ssZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x + a) - I*sin(b*x + a))) + (45*d^4*cos(b*x + a)^5 - 10*d^4*cos(b*x + a)^3 - 14*d^4*cos(b*x + a))*sqr t(d*cos(b*x + a))*sin(b*x + a))/b
Timed out. \[ \int (d \cos (a+b x))^{9/2} \sin ^2(a+b x) \, dx=\text {Timed out} \] Input:
integrate((d*cos(b*x+a))**(9/2)*sin(b*x+a)**2,x)
Output:
Timed out
\[ \int (d \cos (a+b x))^{9/2} \sin ^2(a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \sin \left (b x + a\right )^{2} \,d x } \] Input:
integrate((d*cos(b*x+a))^(9/2)*sin(b*x+a)^2,x, algorithm="maxima")
Output:
integrate((d*cos(b*x + a))^(9/2)*sin(b*x + a)^2, x)
\[ \int (d \cos (a+b x))^{9/2} \sin ^2(a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \sin \left (b x + a\right )^{2} \,d x } \] Input:
integrate((d*cos(b*x+a))^(9/2)*sin(b*x+a)^2,x, algorithm="giac")
Output:
integrate((d*cos(b*x + a))^(9/2)*sin(b*x + a)^2, x)
Timed out. \[ \int (d \cos (a+b x))^{9/2} \sin ^2(a+b x) \, dx=\int {\sin \left (a+b\,x\right )}^2\,{\left (d\,\cos \left (a+b\,x\right )\right )}^{9/2} \,d x \] Input:
int(sin(a + b*x)^2*(d*cos(a + b*x))^(9/2),x)
Output:
int(sin(a + b*x)^2*(d*cos(a + b*x))^(9/2), x)
\[ \int (d \cos (a+b x))^{9/2} \sin ^2(a+b x) \, dx=\sqrt {d}\, \left (\int \sqrt {\cos \left (b x +a \right )}\, \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{2}d x \right ) d^{4} \] Input:
int((d*cos(b*x+a))^(9/2)*sin(b*x+a)^2,x)
Output:
sqrt(d)*int(sqrt(cos(a + b*x))*cos(a + b*x)**4*sin(a + b*x)**2,x)*d**4