Integrand size = 27, antiderivative size = 499 \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^2} \, dx=\frac {d \left (18 b c d-45 d^2-b^2 \left (3 c^2-2 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b^2 \left (9-b^2\right ) f}+\frac {(b c-3 d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b \left (9-b^2\right ) f (3+b \sin (e+f x))}+\frac {\left (261 b c d^2-405 d^3+b^3 \left (3 c^3-20 c d^2\right )-3 b^2 \left (9 c^2 d-12 d^3\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 b^3 \left (9-b^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (648 b c d^3-1215 d^4-36 b^3 c d \left (c^2+3 d^2\right )+18 b^2 d^2 \left (c^2+8 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 b^4 \left (9-b^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-3 d)^3 \left (6 b c+45 d-7 b^2 d\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(3-b) b^4 (3+b)^2 f \sqrt {c+d \sin (e+f x)}} \]
(-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*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-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))*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+ 1/4*Pi+1/2*f*x)*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))*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/s in(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(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))/(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)*(sin(1/2*e+1/4*Pi+1/2*f*x)^ 2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticPi(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 = 9.34 (sec) , antiderivative size = 1058, normalized size of antiderivative = 2.12 \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^2} \, dx=\frac {\sqrt {c+d \sin (e+f x)} \left (-\frac {2 d^3 \cos (e+f x)}{3 b^2}+\frac {-b^3 c^3 \cos (e+f x)+9 b^2 c^2 d \cos (e+f x)-27 b c d^2 \cos (e+f x)+27 d^3 \cos (e+f x)}{b^2 \left (-9+b^2\right ) (3+b \sin (e+f x))}\right )}{f}+\frac {-\frac {2 \left (-36 b^2 c^4+39 b^3 c^3 d-135 b^2 c^2 d^2+9 b c d^3+20 b^3 c d^3+135 d^4-24 b^2 d^4\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+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}}}{(3+b) \sqrt {c+d \sin (e+f x)}}-\frac {2 i \left (-36 b^2 c^3 d-324 b c^2 d^2+72 b^3 c^2 d^2+540 c d^3-168 b^2 c d^3+72 b d^4+4 b^3 d^4\right ) \cos (e+f x) \left ((b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )+3 d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b (b c-3 d) d^2 \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}-\frac {2 i \left (3 b^3 c^3 d-27 b^2 c^2 d^2+261 b c d^3-20 b^3 c d^3-405 d^4+36 b^2 d^4\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (b c-3 d) (c-d) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+d \left (2 (3+b) (b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )-\left (-18+b^2\right ) d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b^2 (b c-3 d) d \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \left (-2 c^2+d^2+4 c (c+d \sin (e+f x))-2 (c+d \sin (e+f x))^2\right ) \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}}{12 (-3+b) b^2 (3+b) f} \]
(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]) + 9*b^2*c^2*d*Cos[e + f*x] - 27*b*c*d^2*Cos[e + f*x] + 27*d^3*Co s[e + f*x])/(b^2*(-9 + b^2)*(3 + b*Sin[e + f*x]))))/f + ((-2*(-36*b^2*c^4 + 39*b^3*c^3*d - 135*b^2*c^2*d^2 + 9*b*c*d^3 + 20*b^3*c*d^3 + 135*d^4 - 24 *b^2*d^4)*EllipticPi[(2*b)/(3 + b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sq rt[(c + d*Sin[e + f*x])/(c + d)])/((3 + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2 *I)*(-36*b^2*c^3*d - 324*b*c^2*d^2 + 72*b^3*c^2*d^2 + 540*c*d^3 - 168*b^2* c*d^3 + 72*b*d^4 + 4*b^3*d^4)*Cos[e + f*x]*((b*c - 3*d)*EllipticF[I*ArcSin h[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + 3*d*El lipticPi[(b*(c + d))/(b*c - 3*d), I*ArcSinh[Sqrt[-(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) + 3*d + b*(c + d*Sin[e + f*x]))) /(b*(b*c - 3*d)*d^2*Sqrt[-(c + d)^(-1)]*(3 + 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 - 27*b^2*c^2*d^2 + 261*b*c*d^3 - 20* b^3*c*d^3 - 405*d^4 + 36*b^2*d^4)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(b*c - 3*d)*(c - d)*EllipticE[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(2*(3 + b)*(b*c - 3*d)*EllipticF[I*ArcSinh[Sq rt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] - (-18 + b^2 )*d*EllipticPi[(b*(c + d))/(b*c - 3*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*S...
Time = 4.41 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.06, 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 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\frac {2 d \left (-15 a^4 d^4+24 a^3 b c d^3+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )-12 a b^3 c d \left (c^2+3 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f \sqrt {c+d \sin (e+f x)}}-\frac {6 d (b c-a d)^3 \left (5 a^2 d+2 a b c-7 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f (a+b) \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 \left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 b}-\frac {2 d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}}{2 b \left (a^2-b^2\right )}\) |
((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))
3.8.51.3.1 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[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. \(1885\) vs. \(2(612)=1224\).
Time = 17.97 (sec) , antiderivative size = 1886, normalized size of antiderivative = 3.78
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(-8/b^5*d*(a^3*d^3-3*a^2*b*c*d^2+3 *a*b^2*c^2*d-b^3*c^3)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x +e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*co s(f*x+e)^2)^(1/2)/(-c/d+a/b)*EllipticPi(((c+d*sin(f*x+e))/(c-d))^(1/2),(-c /d+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2))+1/b^4*(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b ^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*(-b^2/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(-(- d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(a+b*sin(f*x+e))-a*d/(a^3*d-a^2*b*c-a* b^2*d+b^3*c)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d ))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^ 2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-b*d /(a^3*d-a^2*b*c-a*b^2*d+b^3*c)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*( 1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x +e)-c)*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)))+(3*a^2*d-2*a*b*c-b^2*d)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)/b*(c/d -1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d) *(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(-c/d+a/ b)*EllipticPi(((c+d*sin(f*x+e))/(c-d))^(1/2),(-c/d+1)/(-c/d+a/b),((c-d)/(c +d))^(1/2)))+d^2/b^4*(6*d^2*a^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d* (1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin...
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^2} \, dx=\text {Timed out} \]
\[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^2} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^2} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^2} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]