Integrand size = 27, antiderivative size = 534 \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^2} \, dx=\frac {d \left (6 a b c d-5 a^2 d^2-b^2 \left (3 c^2-2 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\left (29 a^2 b c d^2-15 a^3 d^3+b^3 \left (3 c^3-20 c d^2\right )-a b^2 \left (9 c^2 d-12 d^3\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 b^3 \left (a^2-b^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (24 a^3 b c d^3-15 a^4 d^4-12 a b^3 c d \left (c^2+3 d^2\right )+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 b^4 \left (a^2-b^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-a d)^3 \left (2 a b c+5 a^2 d-7 b^2 d\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(a-b) b^4 (a+b)^2 f \sqrt {c+d \sin (e+f x)}} \] Output:
1/3*d*(6*a*b*c*d-5*a^2*d^2-b^2*(3*c^2-2*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^ (1/2)/b^2/(a^2-b^2)/f+(-a*d+b*c)^2*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/b/(a^ 2-b^2)/f/(a+b*sin(f*x+e))-1/3*(29*a^2*b*c*d^2-15*a^3*d^3+b^3*(3*c^3-20*c*d ^2)-a*b^2*(9*c^2*d-12*d^3))*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)/b^3/(a^2-b^2)/f/((c+d*sin(f*x+e))/(c +d))^(1/2)-1/3*(24*a^3*b*c*d^3-15*a^4*d^4-12*a*b^3*c*d*(c^2+3*d^2)+2*a^2*b ^2*d^2*(c^2+8*d^2)+b^4*(3*c^4+16*c^2*d^2+2*d^4))*InverseJacobiAM(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)/b^4/(a ^2-b^2)/f/(c+d*sin(f*x+e))^(1/2)-(-a*d+b*c)^3*(5*a^2*d+2*a*b*c-7*b^2*d)*El lipticPi(cos(1/2*e+1/4*Pi+1/2*f*x),2*b/(a+b),2^(1/2)*(d/(c+d))^(1/2))*((c+ d*sin(f*x+e))/(c+d))^(1/2)/(a-b)/b^4/(a+b)^2/f/(c+d*sin(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 10.82 (sec) , antiderivative size = 1109, normalized size of antiderivative = 2.08 \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^2} \, dx =\text {Too large to display} \] Input:
Integrate[(c + d*Sin[e + f*x])^(7/2)/(a + b*Sin[e + f*x])^2,x]
Output:
(Sqrt[c + d*Sin[e + f*x]]*((-2*d^3*Cos[e + f*x])/(3*b^2) + (-(b^3*c^3*Cos[ e + f*x]) + 3*a*b^2*c^2*d*Cos[e + f*x] - 3*a^2*b*c*d^2*Cos[e + f*x] + a^3* d^3*Cos[e + f*x])/(b^2*(-a^2 + b^2)*(a + b*Sin[e + f*x]))))/f - ((-2*(-12* a*b^2*c^4 + 39*b^3*c^3*d - 45*a*b^2*c^2*d^2 + a^2*b*c*d^3 + 20*b^3*c*d^3 + 5*a^3*d^4 - 8*a*b^2*d^4)*EllipticPi[(2*b)/(a + b), (-e + Pi/2 - f*x)/2, ( 2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*Sqrt[c + d*Sin[ e + f*x]]) - ((2*I)*(-12*a*b^2*c^3*d - 36*a^2*b*c^2*d^2 + 72*b^3*c^2*d^2 + 20*a^3*c*d^3 - 56*a*b^2*c*d^3 + 8*a^2*b*d^4 + 4*b^3*d^4)*Cos[e + f*x]*((b *c - a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]] ], (c + d)/(c - d)] + a*d*EllipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sq rt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d *Sin[e + f*x])/(c + d)]*Sqrt[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a* d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b *Sin[e + f*x])*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin [e + f*x]) + (c + d*Sin[e + f*x])^2)/d^2)]) - ((2*I)*(3*b^3*c^3*d - 9*a*b^ 2*c^2*d^2 + 29*a^2*b*c*d^3 - 20*b^3*c*d^3 - 15*a^3*d^4 + 12*a*b^2*d^4)*Cos [e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sq rt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[ e + f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(...
Time = 4.68 (sec) , antiderivative size = 528, normalized size of antiderivative = 0.99, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {3042, 3271, 27, 3042, 3528, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}
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.
\(\displaystyle \int \frac {(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^2}dx\) |
\(\Big \downarrow \) 3271 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (3 a^2 d^3+7 b^2 c^2 d+\left (-\left (\left (3 c^2-2 d^2\right ) b^2\right )+6 a c d b-5 a^2 d^2\right ) \sin ^2(e+f x) d-2 \left (c d a^2+b \left (c^2+d^2\right ) a-3 b^2 c d\right ) \sin (e+f x) d-2 a b c \left (c^2+4 d^2\right )\right )}{2 (a+b \sin (e+f x))}dx}{b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (3 a^2 d^3+7 b^2 c^2 d+\left (-\left (\left (3 c^2-2 d^2\right ) b^2\right )+6 a c d b-5 a^2 d^2\right ) \sin ^2(e+f x) d-2 \left (c d a^2+b \left (c^2+d^2\right ) a-3 b^2 c d\right ) \sin (e+f x) d-2 a b c \left (c^2+4 d^2\right )\right )}{a+b \sin (e+f x)}dx}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (3 a^2 d^3+7 b^2 c^2 d+\left (-\left (\left (3 c^2-2 d^2\right ) b^2\right )+6 a c d b-5 a^2 d^2\right ) \sin (e+f x)^2 d-2 \left (c d a^2+b \left (c^2+d^2\right ) a-3 b^2 c d\right ) \sin (e+f x) d-2 a b c \left (c^2+4 d^2\right )\right )}{a+b \sin (e+f x)}dx}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3528 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {2 \int \frac {-5 a^3 d^4+15 a^2 b c d^3+21 b^3 c^3 d-\left (\left (3 c^3-20 c d^2\right ) b^3-a \left (9 c^2 d-12 d^3\right ) b^2+29 a^2 c d^2 b-15 a^3 d^3\right ) \sin ^2(e+f x) d+2 \left (5 c d^2 a^3-b \left (9 c^2 d-2 d^3\right ) a^2-b^2 c \left (3 c^2+14 d^2\right ) a+b^3 d \left (18 c^2+d^2\right )\right ) \sin (e+f x) d-a b^2 \left (6 c^4+27 d^2 c^2-2 d^4\right )}{2 (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\int \frac {-5 a^3 d^4+15 a^2 b c d^3+21 b^3 c^3 d-\left (\left (3 c^3-20 c d^2\right ) b^3-a \left (9 c^2 d-12 d^3\right ) b^2+29 a^2 c d^2 b-15 a^3 d^3\right ) \sin ^2(e+f x) d+2 \left (5 c d^2 a^3-b \left (9 c^2 d-2 d^3\right ) a^2-b^2 c \left (3 c^2+14 d^2\right ) a+b^3 d \left (18 c^2+d^2\right )\right ) \sin (e+f x) d-a b^2 \left (6 c^4+27 d^2 c^2-2 d^4\right )}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\int \frac {-5 a^3 d^4+15 a^2 b c d^3+21 b^3 c^3 d-\left (\left (3 c^3-20 c d^2\right ) b^3-a \left (9 c^2 d-12 d^3\right ) b^2+29 a^2 c d^2 b-15 a^3 d^3\right ) \sin (e+f x)^2 d+2 \left (5 c d^2 a^3-b \left (9 c^2 d-2 d^3\right ) a^2-b^2 c \left (3 c^2+14 d^2\right ) a+b^3 d \left (18 c^2+d^2\right )\right ) \sin (e+f x) d-a b^2 \left (6 c^4+27 d^2 c^2-2 d^4\right )}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3538 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {-\frac {\left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{b}-\frac {\int -\frac {d \left (-15 c d^3 a^4+b d^2 \left (29 c^2-5 d^2\right ) a^3-9 b^2 c d \left (c^2-3 d^2\right ) a^2-b^3 \left (3 c^4+47 d^2 c^2-2 d^4\right ) a+21 b^4 c^3 d\right )+d \left (\left (3 c^4+16 d^2 c^2+2 d^4\right ) b^4-12 a c d \left (c^2+3 d^2\right ) b^3+2 a^2 d^2 \left (c^2+8 d^2\right ) b^2+24 a^3 c d^3 b-15 a^4 d^4\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\int \frac {d \left (-15 c d^3 a^4+b d^2 \left (29 c^2-5 d^2\right ) a^3-9 b^2 c d \left (c^2-3 d^2\right ) a^2-b^3 \left (3 c^4+47 d^2 c^2-2 d^4\right ) a+21 b^4 c^3 d\right )+d \left (\left (3 c^4+16 d^2 c^2+2 d^4\right ) b^4-12 a c d \left (c^2+3 d^2\right ) b^3+2 a^2 d^2 \left (c^2+8 d^2\right ) b^2+24 a^3 c d^3 b-15 a^4 d^4\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {\left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{b}}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\int \frac {d \left (-15 c d^3 a^4+b d^2 \left (29 c^2-5 d^2\right ) a^3-9 b^2 c d \left (c^2-3 d^2\right ) a^2-b^3 \left (3 c^4+47 d^2 c^2-2 d^4\right ) a+21 b^4 c^3 d\right )+d \left (\left (3 c^4+16 d^2 c^2+2 d^4\right ) b^4-12 a c d \left (c^2+3 d^2\right ) b^3+2 a^2 d^2 \left (c^2+8 d^2\right ) b^2+24 a^3 c d^3 b-15 a^4 d^4\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {\left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{b}}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\int \frac {d \left (-15 c d^3 a^4+b d^2 \left (29 c^2-5 d^2\right ) a^3-9 b^2 c d \left (c^2-3 d^2\right ) a^2-b^3 \left (3 c^4+47 d^2 c^2-2 d^4\right ) a+21 b^4 c^3 d\right )+d \left (\left (3 c^4+16 d^2 c^2+2 d^4\right ) b^4-12 a c d \left (c^2+3 d^2\right ) b^3+2 a^2 d^2 \left (c^2+8 d^2\right ) b^2+24 a^3 c d^3 b-15 a^4 d^4\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {\left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{b \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\int \frac {d \left (-15 c d^3 a^4+b d^2 \left (29 c^2-5 d^2\right ) a^3-9 b^2 c d \left (c^2-3 d^2\right ) a^2-b^3 \left (3 c^4+47 d^2 c^2-2 d^4\right ) a+21 b^4 c^3 d\right )+d \left (\left (3 c^4+16 d^2 c^2+2 d^4\right ) b^4-12 a c d \left (c^2+3 d^2\right ) b^3+2 a^2 d^2 \left (c^2+8 d^2\right ) b^2+24 a^3 c d^3 b-15 a^4 d^4\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {\left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{b \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\int \frac {d \left (-15 c d^3 a^4+b d^2 \left (29 c^2-5 d^2\right ) a^3-9 b^2 c d \left (c^2-3 d^2\right ) a^2-b^3 \left (3 c^4+47 d^2 c^2-2 d^4\right ) a+21 b^4 c^3 d\right )+d \left (\left (3 c^4+16 d^2 c^2+2 d^4\right ) b^4-12 a c d \left (c^2+3 d^2\right ) b^3+2 a^2 d^2 \left (c^2+8 d^2\right ) b^2+24 a^3 c d^3 b-15 a^4 d^4\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {2 \left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3481 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\frac {d \left (-15 a^4 d^4+24 a^3 b c d^3+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )-12 a b^3 c d \left (c^2+3 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {3 d (b c-a d)^3 \left (5 a^2 d+2 a b c-7 b^2 d\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}-\frac {2 \left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\frac {d \left (-15 a^4 d^4+24 a^3 b c d^3+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )-12 a b^3 c d \left (c^2+3 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {3 d (b c-a d)^3 \left (5 a^2 d+2 a b c-7 b^2 d\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}-\frac {2 \left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\frac {d \left (-15 a^4 d^4+24 a^3 b c d^3+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )-12 a b^3 c d \left (c^2+3 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}-\frac {3 d (b c-a d)^3 \left (5 a^2 d+2 a b c-7 b^2 d\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}-\frac {2 \left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\frac {d \left (-15 a^4 d^4+24 a^3 b c d^3+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )-12 a b^3 c d \left (c^2+3 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}-\frac {3 d (b c-a d)^3 \left (5 a^2 d+2 a b c-7 b^2 d\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}-\frac {2 \left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\frac {2 d \left (-15 a^4 d^4+24 a^3 b c d^3+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )-12 a b^3 c d \left (c^2+3 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f \sqrt {c+d \sin (e+f x)}}-\frac {3 d (b c-a d)^3 \left (5 a^2 d+2 a b c-7 b^2 d\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}-\frac {2 \left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3286 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\frac {2 d \left (-15 a^4 d^4+24 a^3 b c d^3+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )-12 a b^3 c d \left (c^2+3 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f \sqrt {c+d \sin (e+f x)}}-\frac {3 d (b c-a d)^3 \left (5 a^2 d+2 a b c-7 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 \left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\frac {2 d \left (-15 a^4 d^4+24 a^3 b c d^3+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )-12 a b^3 c d \left (c^2+3 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f \sqrt {c+d \sin (e+f x)}}-\frac {3 d (b c-a d)^3 \left (5 a^2 d+2 a b c-7 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 \left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\frac {2 d \left (-15 a^4 d^4+24 a^3 b c d^3+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )-12 a b^3 c d \left (c^2+3 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f \sqrt {c+d \sin (e+f x)}}-\frac {6 d (b c-a d)^3 \left (5 a^2 d+2 a b c-7 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f (a+b) \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 \left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
Input:
Int[(c + d*Sin[e + f*x])^(7/2)/(a + b*Sin[e + f*x])^2,x]
Output:
((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(b*(a^2 - b^2)*f*( a + b*Sin[e + f*x])) - ((-2*d*(6*a*b*c*d - 5*a^2*d^2 - b^2*(3*c^2 - 2*d^2) )*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*b*f) + ((-2*(29*a^2*b*c*d^2 - 15*a^3*d^3 + b^3*(3*c^3 - 20*c*d^2) - a*b^2*(9*c^2*d - 12*d^3))*EllipticE[ (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(b*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((2*d*(24*a^3*b*c*d^3 - 15*a^4*d^4 - 12*a*b^ 3*c*d*(c^2 + 3*d^2) + 2*a^2*b^2*d^2*(c^2 + 8*d^2) + b^4*(3*c^4 + 16*c^2*d^ 2 + 2*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*f*Sqrt[c + d*Sin[e + f*x]]) - (6*d*(b*c - a*d)^3*(2* a*b*c + 5*a^2*d - 7*b^2*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, ( 2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*(a + b)*f*Sqrt[c + d* Sin[e + f*x]]))/(b*d))/(3*b))/(2*b*(a^2 - b^2))
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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f* (n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin [e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x] , x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*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] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1870\) vs. \(2(521)=1042\).
Time = 22.98 (sec) , antiderivative size = 1871, normalized size of antiderivative = 3.50
Input:
int((c+d*sin(f*x+e))^(7/2)/(a+b*sin(f*x+e))^2,x,method=_RETURNVERBOSE)
Output:
(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*(d^2/b^4*(-4*b*d*(a*d-2*b*c)*(c/d- 1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+si n(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*((-c/d-1)*E llipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c +d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))+d^2*b^2*(-2/3/d*(-(-c-d* sin(f*x+e))*cos(f*x+e)^2)^(1/2)+2/3*(c/d-1)*((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)/(-(-c-d*si n(f*x+e))*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c -d)/(c+d))^(1/2))-4/3*c/d*(c/d-1)*((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)/(-(-c-d*sin(f*x+e))* cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c -d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^( 1/2))))+6*a^2*d^2*(c/d-1)*((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)/(-(-c-d*sin(f*x+e))*cos(f*x+ e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+ 12*b^2*c^2*(c/d-1)*((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)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^( 1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-16*a*b* c*d*(c/d-1)*((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)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)...
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^2} \, dx=\text {Timed out} \] Input:
integrate((c+d*sin(f*x+e))^(7/2)/(a+b*sin(f*x+e))^2,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^2} \, dx=\text {Timed out} \] Input:
integrate((c+d*sin(f*x+e))**(7/2)/(a+b*sin(f*x+e))**2,x)
Output:
Timed out
\[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^2} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((c+d*sin(f*x+e))^(7/2)/(a+b*sin(f*x+e))^2,x, algorithm="maxima")
Output:
integrate((d*sin(f*x + e) + c)^(7/2)/(b*sin(f*x + e) + a)^2, x)
\[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^2} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((c+d*sin(f*x+e))^(7/2)/(a+b*sin(f*x+e))^2,x, algorithm="giac")
Output:
integrate((d*sin(f*x + e) + c)^(7/2)/(b*sin(f*x + e) + a)^2, x)
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^2} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2} \,d x \] Input:
int((c + d*sin(e + f*x))^(7/2)/(a + b*sin(e + f*x))^2,x)
Output:
int((c + d*sin(e + f*x))^(7/2)/(a + b*sin(e + f*x))^2, x)
\[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^2} \, dx=\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{2} b^{2}+2 \sin \left (f x +e \right ) a b +a^{2}}d x \right ) c^{3}+\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )^{2} b^{2}+2 \sin \left (f x +e \right ) a b +a^{2}}d x \right ) d^{3}+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{2} b^{2}+2 \sin \left (f x +e \right ) a b +a^{2}}d x \right ) c \,d^{2}+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{2} b^{2}+2 \sin \left (f x +e \right ) a b +a^{2}}d x \right ) c^{2} d \] Input:
int((c+d*sin(f*x+e))^(7/2)/(a+b*sin(f*x+e))^2,x)
Output:
int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)**2*b**2 + 2*sin(e + f*x)*a*b + a**2),x)*c**3 + int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**3)/(sin(e + f* x)**2*b**2 + 2*sin(e + f*x)*a*b + a**2),x)*d**3 + 3*int((sqrt(sin(e + f*x) *d + c)*sin(e + f*x)**2)/(sin(e + f*x)**2*b**2 + 2*sin(e + f*x)*a*b + a**2 ),x)*c*d**2 + 3*int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x))/(sin(e + f*x)* *2*b**2 + 2*sin(e + f*x)*a*b + a**2),x)*c**2*d