Integrand size = 27, antiderivative size = 298 \[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\frac {4 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{35 d f}+\frac {4 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {4 a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{35 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{35 d^2 f \sqrt {c+d \sin (e+f x)}} \] Output:
4/35*a^2*(c^2-7*c*d-10*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d/f+4/35*a^2 *(c-7*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d/f-2/7*a^2*cos(f*x+e)*(c+d*sin (f*x+e))^(5/2)/d/f+4/35*a^2*(c+3*d)*(c^2-10*c*d-7*d^2)*EllipticE(cos(1/2*e +1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/d^2/f/((c +d*sin(f*x+e))/(c+d))^(1/2)+4/35*a^2*(c^2-7*c*d-10*d^2)*(c^2-d^2)*InverseJ acobiAM(1/2*e-1/4*Pi+1/2*f*x,2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c +d))^(1/2)/d^2/f/(c+d*sin(f*x+e))^(1/2)
Time = 3.91 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.88 \[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\frac {a^2 \left (8 \left (c^4-6 c^3 d-44 c^2 d^2-58 c d^3-21 d^4\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-8 \left (c^4-7 c^3 d-11 c^2 d^2+7 c d^3+10 d^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d \cos (e+f x) \left (-4 c^3-112 c^2 d-106 c d^2-28 d^3+2 d^2 (13 c+14 d) \cos (2 (e+f x))-d \left (36 c^2+168 c d+95 d^2\right ) \sin (e+f x)+5 d^3 \sin (3 (e+f x))\right )\right )}{70 d^2 f \sqrt {c+d \sin (e+f x)}} \] Input:
Integrate[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2),x]
Output:
(a^2*(8*(c^4 - 6*c^3*d - 44*c^2*d^2 - 58*c*d^3 - 21*d^4)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - 8*(c^4 - 7*c^3*d - 11*c^2*d^2 + 7*c*d^3 + 10*d^4)*EllipticF[(-2*e + Pi - 2*f*x)/ 4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + d*Cos[e + f*x]*(-4* c^3 - 112*c^2*d - 106*c*d^2 - 28*d^3 + 2*d^2*(13*c + 14*d)*Cos[2*(e + f*x) ] - d*(36*c^2 + 168*c*d + 95*d^2)*Sin[e + f*x] + 5*d^3*Sin[3*(e + f*x)]))) /(70*d^2*f*Sqrt[c + d*Sin[e + f*x]])
Time = 1.51 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.03, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {3042, 3242, 3042, 3232, 27, 3042, 3232, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}dx\) |
| \(\Big \downarrow \) 3242 |
| \(\displaystyle \frac {2 \int \left (6 a^2 d-a^2 (c-7 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}dx}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int \left (6 a^2 d-a^2 (c-7 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}dx}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2 \left (\frac {2}{5} \int \frac {3}{2} \sqrt {c+d \sin (e+f x)} \left (a^2 d (9 c+7 d)-a^2 \left (c^2-7 d c-10 d^2\right ) \sin (e+f x)\right )dx+\frac {2 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {3}{5} \int \sqrt {c+d \sin (e+f x)} \left (a^2 d (9 c+7 d)-a^2 \left (c^2-7 d c-10 d^2\right ) \sin (e+f x)\right )dx+\frac {2 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {3}{5} \int \sqrt {c+d \sin (e+f x)} \left (a^2 d (9 c+7 d)-a^2 \left (c^2-7 d c-10 d^2\right ) \sin (e+f x)\right )dx+\frac {2 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2 \left (\frac {3}{5} \left (\frac {2}{3} \int \frac {2 a^2 d \left (13 c^2+14 d c+5 d^2\right )-a^2 (c+3 d) \left (c^2-10 d c-7 d^2\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx+\frac {2 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {3}{5} \left (\frac {1}{3} \int \frac {2 a^2 d \left (13 c^2+14 d c+5 d^2\right )-a^2 (c+3 d) \left (c^2-10 d c-7 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx+\frac {2 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {3}{5} \left (\frac {1}{3} \int \frac {2 a^2 d \left (13 c^2+14 d c+5 d^2\right )-a^2 (c+3 d) \left (c^2-10 d c-7 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx+\frac {2 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {2 \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )+\frac {2 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )+\frac {2 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2 \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2 \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 \left (\frac {3}{5} \left (\frac {2 a^2 \left (c^2-7 c d-10 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}+\frac {1}{3} \left (\frac {2 a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )\right )+\frac {2 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )}{7 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
Input:
Int[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2),x]
Output:
(-2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*d*f) + (2*((2*a^2*(c - 7*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*f) + (3*((2*a^2*(c^2 - 7 *c*d - 10*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f) + ((-2*a^2*(c + 3*d)*(c^2 - 10*c*d - 7*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)] *Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*a ^2*(c^2 - 7*c*d - 10*d^2)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/ (c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]] ))/3))/5))/(7*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c* (m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !LtQ[ n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[ c, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(1315\) vs. \(2(279)=558\).
Time = 4.49 (sec) , antiderivative size = 1316, normalized size of antiderivative = 4.42
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1316\) |
| parts | \(\text {Expression too large to display}\) | \(2398\) |
Input:
int((a+sin(f*x+e)*a)^2*(c+d*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/35*a^2*(-13*sin(f*x+e)^4*c*d^4-15*d^5*sin(f*x+e)^3+14*d^5*sin(f*x+e)^2+ 20*d^5*sin(f*x+e)-sin(f*x+e)^2*c^3*d^2-9*sin(f*x+e)^3*c^2*d^3-2*((c+d*sin( f*x+e))/(c-d))^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/( c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))* c^5-42*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^(1/2)*(-d *(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c- d)/(c+d))^(1/2))*d^5+62*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+sin(f*x+e)) /(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/ (c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^5+2*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d *(-1+sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF((( c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^4*d-68*((c+d*sin(f*x+e ))/(c-d))^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d)) ^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^3*d ^2-64*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^(1/2)*(-d* (1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d )/(c+d))^(1/2))*c^2*d^3+68*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+sin(f*x+ e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e ))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c*d^4+14*((c+d*sin(f*x+e))/(c-d))^(1/ 2)*(-d*(-1+sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*Ellipt icE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^4*d+76*((c+d*...
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.06 \[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx =\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^(3/2),x, algorithm="fricas")
Output:
2/105*(4*(a^2*c^4 - 7*a^2*c^3*d + 2*a^2*c^2*d^2 + 21*a^2*c*d^3 + 15*a^2*d^ 4)*sqrt(1/2*I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I* c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/ d) + 4*(a^2*c^4 - 7*a^2*c^3*d + 2*a^2*c^2*d^2 + 21*a^2*c*d^3 + 15*a^2*d^4) *sqrt(-1/2*I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I* c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/ d) - 6*(-I*a^2*c^3*d + 7*I*a^2*c^2*d^2 + 37*I*a^2*c*d^3 + 21*I*a^2*d^4)*sq rt(1/2*I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I *c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) - 6*(I*a^2*c^3*d - 7*I*a^2*c^2*d^2 - 37*I*a^2*c*d^3 - 21*I*a^2*d^4)*sqrt(- 1/2*I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c *d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d)) + 3*(5*a^2*d^4*cos(f*x + e)^3 - 2*(4*a^2*c*d^3 + 7*a^2*d^4)*cos(f*x + e)*si n(f*x + e) - (a^2*c^2*d^2 + 28*a^2*c*d^3 + 25*a^2*d^4)*cos(f*x + e))*sqrt( d*sin(f*x + e) + c))/(d^3*f)
\[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=a^{2} \left (\int c \sqrt {c + d \sin {\left (e + f x \right )}}\, dx + \int 2 c \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int c \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 2 d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx\right ) \] Input:
integrate((a+a*sin(f*x+e))**2*(c+d*sin(f*x+e))**(3/2),x)
Output:
a**2*(Integral(c*sqrt(c + d*sin(e + f*x)), x) + Integral(2*c*sqrt(c + d*si n(e + f*x))*sin(e + f*x), x) + Integral(c*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2, x) + Integral(d*sqrt(c + d*sin(e + f*x))*sin(e + f*x), x) + Inte gral(2*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2, x) + Integral(d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3, x))
\[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^(3/2),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^2*(d*sin(f*x + e) + c)^(3/2), x)
\[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^(3/2),x, algorithm="giac")
Output:
integrate((a*sin(f*x + e) + a)^2*(d*sin(f*x + e) + c)^(3/2), x)
Timed out. \[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \] Input:
int((a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^(3/2),x)
Output:
int((a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^(3/2), x)
\[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=a^{2} \left (\left (\int \sqrt {\sin \left (f x +e \right ) d +c}d x \right ) c +\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}d x \right ) d +\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) c +2 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) d +2 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )d x \right ) c +\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )d x \right ) d \right ) \] Input:
int((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^(3/2),x)
Output:
a**2*(int(sqrt(sin(e + f*x)*d + c),x)*c + int(sqrt(sin(e + f*x)*d + c)*sin (e + f*x)**3,x)*d + int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**2,x)*c + 2* int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**2,x)*d + 2*int(sqrt(sin(e + f*x )*d + c)*sin(e + f*x),x)*c + int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x),x)* d)