Integrand size = 25, antiderivative size = 149 \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx=-\frac {22 a b (e \cos (c+d x))^{7/2}}{63 d e}+\frac {2 \left (9 a^2+2 b^2\right ) e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {2 \left (9 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e} \]
-22/63*a*b*(e*cos(d*x+c))^(7/2)/d/e+2/45*(9*a^2+2*b^2)*e*(e*cos(d*x+c))^(3 /2)*sin(d*x+c)/d-2/9*b*(e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))/d/e+2/15*(9*a ^2+2*b^2)*e^2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(si n(1/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(1/2)
Time = 0.97 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.76 \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx=\frac {(e \cos (c+d x))^{5/2} \left (84 \left (9 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\cos ^{\frac {3}{2}}(c+d x) \left (-180 a b \cos (2 (c+d x))+21 \left (12 a^2+b^2\right ) \sin (c+d x)-5 b (36 a+7 b \sin (3 (c+d x)))\right )\right )}{630 d \cos ^{\frac {5}{2}}(c+d x)} \]
((e*Cos[c + d*x])^(5/2)*(84*(9*a^2 + 2*b^2)*EllipticE[(c + d*x)/2, 2] + Co s[c + d*x]^(3/2)*(-180*a*b*Cos[2*(c + d*x)] + 21*(12*a^2 + b^2)*Sin[c + d* x] - 5*b*(36*a + 7*b*Sin[3*(c + d*x)]))))/(630*d*Cos[c + d*x]^(5/2))
Time = 0.65 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 3171, 27, 3042, 3148, 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.
| \(\displaystyle \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{9} \int \frac {1}{2} (e \cos (c+d x))^{5/2} \left (9 a^2+11 b \sin (c+d x) a+2 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int (e \cos (c+d x))^{5/2} \left (9 a^2+11 b \sin (c+d x) a+2 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int (e \cos (c+d x))^{5/2} \left (9 a^2+11 b \sin (c+d x) a+2 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{9} \left (\left (9 a^2+2 b^2\right ) \int (e \cos (c+d x))^{5/2}dx-\frac {22 a b (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\left (9 a^2+2 b^2\right ) \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}dx-\frac {22 a b (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{9} \left (\left (9 a^2+2 b^2\right ) \left (\frac {3}{5} e^2 \int \sqrt {e \cos (c+d x)}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )-\frac {22 a b (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\left (9 a^2+2 b^2\right ) \left (\frac {3}{5} e^2 \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )-\frac {22 a b (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{9} \left (\left (9 a^2+2 b^2\right ) \left (\frac {3 e^2 \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{5 \sqrt {\cos (c+d x)}}+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )-\frac {22 a b (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\left (9 a^2+2 b^2\right ) \left (\frac {3 e^2 \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 \sqrt {\cos (c+d x)}}+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )-\frac {22 a b (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\left (9 a^2+2 b^2\right ) \left (\frac {6 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )-\frac {22 a b (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}\) |
(-2*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]))/(9*d*e) + ((-22*a*b*(e* Cos[c + d*x])^(7/2))/(7*d*e) + (9*a^2 + 2*b^2)*((6*e^2*Sqrt[e*Cos[c + d*x] ]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*e*(e*Cos[c + d* x])^(3/2)*Sin[c + d*x])/(5*d)))/9
3.6.48.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[1/(m + p) Int[(g*Cos[e + f*x])^p* (a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1) *Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs. \(2(157)=314\).
Time = 13.06 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.74
| method | result | size |
| default | \(-\frac {2 e^{3} \left (1120 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-2240 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}+1440 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -504 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}+1568 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-2880 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b +504 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}-448 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}+2160 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -126 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}+42 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-189 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}-42 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}-720 a b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+90 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a b \right )}{315 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(408\) |
| parts | \(-\frac {2 a^{2} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e^{3} \left (-8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{5 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}+\frac {4 b^{2} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e^{3} \left (80 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-240 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+272 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-144 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{45 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}-\frac {4 a b \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d e}\) | \(462\) |
-2/315/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^3*(1120*co s(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10*b^2-2240*cos(1/2*d*x+1/2*c)*sin(1/2 *d*x+1/2*c)^8*b^2+1440*sin(1/2*d*x+1/2*c)^9*a*b-504*cos(1/2*d*x+1/2*c)*sin (1/2*d*x+1/2*c)^6*a^2+1568*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6*b^2-288 0*sin(1/2*d*x+1/2*c)^7*a*b+504*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*a^2 -448*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*b^2+2160*sin(1/2*d*x+1/2*c)^5 *a*b-126*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a^2+42*cos(1/2*d*x+1/2*c) *sin(1/2*d*x+1/2*c)^2*b^2-189*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+ 1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2-42*(sin(1/2*d* x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1 /2*c),2^(1/2))*b^2-720*a*b*sin(1/2*d*x+1/2*c)^3+90*sin(1/2*d*x+1/2*c)*a*b) /d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.08 \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx=\frac {21 i \, \sqrt {2} {\left (9 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (9 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (90 \, a b e^{2} \cos \left (d x + c\right )^{3} + 7 \, {\left (5 \, b^{2} e^{2} \cos \left (d x + c\right )^{3} - {\left (9 \, a^{2} + 2 \, b^{2}\right )} e^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{315 \, d} \]
1/315*(21*I*sqrt(2)*(9*a^2 + 2*b^2)*e^(5/2)*weierstrassZeta(-4, 0, weierst rassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 21*I*sqrt(2)*(9*a^2 + 2*b^2)*e^(5/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(90*a*b*e^2*cos(d*x + c)^3 + 7*(5*b^2*e^2*cos (d*x + c)^3 - (9*a^2 + 2*b^2)*e^2*cos(d*x + c))*sin(d*x + c))*sqrt(e*cos(d *x + c)))/d
Timed out. \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]
\[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
\[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2 \,d x \]