Integrand size = 29, antiderivative size = 1039 \[ \int (3+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx=\frac {\sqrt {3+b} (c-d) \sqrt {c+d} \left (513 b c d^2-243 d^3+3 b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{192 b^2 (b c-3 d) d f}-\frac {\sqrt {c+d} \left (540 b c d^3-243 d^4-180 b^3 c d \left (c^2+4 d^2\right )-54 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{64 b^3 \sqrt {3+b} d^2 f}-\frac {\left (513 b c d^2-243 d^3+3 b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{192 b d f \sqrt {3+b \sin (e+f x)}}-\frac {\left (162 b c d-81 d^2+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{96 b f}-\frac {(17 b c-9 d) d \cos (e+f x) (3+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{24 b f}-\frac {d^2 \cos (e+f x) (3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}+\frac {(3+b)^{3/2} \left (243 d^3-27 b d^2 (17 c+6 d)+9 b^2 d \left (73 c^2+36 c d+28 d^2\right )+b^3 \left (15 c^3+118 c^2 d+284 c d^2+72 d^3\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{192 b^3 d \sqrt {c+d} f} \]
-1/64*(20*a^3*b*c*d^3-3*a^4*d^4-60*a*b^3*c*d*(c^2+4*d^2)-6*a^2*b^2*d^2*(15 *c^2+4*d^2)+b^4*(5*c^4-120*c^2*d^2-48*d^4))*EllipticPi((a+b)^(1/2)*(c+d*si n(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e))^(1/2),b*(c+d)/(a+b)/d,((a-b)* (c+d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*(c+d)^(1/2)*(-(-a*d+ b*c)*(1-sin(f*x+e))/(c+d)/(a+b*sin(f*x+e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e ))/(c-d)/(a+b*sin(f*x+e)))^(1/2)/b^3/d^2/f/(a+b)^(1/2)+1/192*(c-d)*(57*a^2 *b*c*d^2-9*a^3*d^3+a*b^2*d*(337*c^2+156*d^2)+b^3*(15*c^3+284*c*d^2))*Ellip ticE((a+b)^(1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e))^(1/2) ,((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*(a+b)^(1/2)* (c+d)^(1/2)*(-(-a*d+b*c)*(1-sin(f*x+e))/(c+d)/(a+b*sin(f*x+e)))^(1/2)*((-a *d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+e)))^(1/2)/b^2/d/(-a*d+b*c)/f+1/ 192*(a+b)^(3/2)*(9*a^3*d^3-3*a^2*b*d^2*(17*c+6*d)+3*a*b^2*d*(73*c^2+36*c*d +28*d^2)+b^3*(15*c^3+118*c^2*d+284*c*d^2+72*d^3))*EllipticF((c+d)^(1/2)*(a +b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sin(f*x+e))^(1/2),((a+b)*(c-d)/(a-b) /(c+d))^(1/2))*sec(f*x+e)*(c+d*sin(f*x+e))*((-a*d+b*c)*(1-sin(f*x+e))/(a+b )/(c+d*sin(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+sin(f*x+e))/(a-b)/(c+d*sin(f*x+e )))^(1/2)/b^3/d/f/(c+d)^(1/2)-1/24*d*(-3*a*d+17*b*c)*cos(f*x+e)*(a+b*sin(f *x+e))^(3/2)*(c+d*sin(f*x+e))^(1/2)/b/f-1/4*d^2*cos(f*x+e)*(a+b*sin(f*x+e) )^(5/2)*(c+d*sin(f*x+e))^(1/2)/b/f-1/192*(57*a^2*b*c*d^2-9*a^3*d^3+a*b^2*d *(337*c^2+156*d^2)+b^3*(15*c^3+284*c*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^...
Time = 8.72 (sec) , antiderivative size = 2036, normalized size of antiderivative = 1.96 \[ \int (3+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx=\text {Result too large to show} \]
-1/384*((-4*(-(b*c) + 3*d)*(-3456*b*c^3 - 133*b^3*c^3 - 2913*b^2*c^2*d - 4 059*b*c*d^2 - 356*b^3*c*d^2 + 81*d^3 - 684*b^2*d^3)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi /2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[( (c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*S qrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3 *d)])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + 3*d)*(-1596*b^2*c^3 - 5976*b*c^2*d - 644*b^3*c^2*d + 324*c*d ^2 - 3480*b^2*c*d^2 - 2052*b*d^3 - 144*b^3*d^3)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqr t[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d )])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + 3*d)/((3 + b)*d), ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d* Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + P...
Time = 5.98 (sec) , antiderivative size = 1108, normalized size of antiderivative = 1.07, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.759, Rules used = {3042, 3272, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3540, 25, 3042, 3532, 25, 25, 3042, 3290, 3477, 3042, 3297, 3475}
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.
| \(\displaystyle \int (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 3272 |
| \(\displaystyle \frac {\int \frac {(a+b \sin (e+f x))^{3/2} \left (8 b c^3+5 b d^2 c+a d^3+d^2 (17 b c-3 a d) \sin ^2(e+f x)-2 d \left (a c d-3 b \left (4 c^2+d^2\right )\right ) \sin (e+f x)\right )}{2 \sqrt {c+d \sin (e+f x)}}dx}{4 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \sin (e+f x))^{3/2} \left (8 b c^3+5 b d^2 c+a d^3+d^2 (17 b c-3 a d) \sin ^2(e+f x)-2 d \left (a c d-3 b \left (4 c^2+d^2\right )\right ) \sin (e+f x)\right )}{\sqrt {c+d \sin (e+f x)}}dx}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a+b \sin (e+f x))^{3/2} \left (8 b c^3+5 b d^2 c+a d^3+d^2 (17 b c-3 a d) \sin (e+f x)^2-2 d \left (a c d-3 b \left (4 c^2+d^2\right )\right ) \sin (e+f x)\right )}{\sqrt {c+d \sin (e+f x)}}dx}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {a+b \sin (e+f x)} \left (d^2 \left (\left (59 c^2+36 d^2\right ) b^2+54 a c d b-9 a^2 d^2\right ) \sin ^2(e+f x)-2 d \left (-\left (\left (24 c^3+49 d^2 c\right ) b^2\right )-5 a d \left (11 c^2+3 d^2\right ) b+3 a^2 c d^2\right ) \sin (e+f x)+d \left (3 a^2 d^3+51 b^2 c^2 d+a b \left (48 c^3+38 d^2 c\right )\right )\right )}{2 \sqrt {c+d \sin (e+f x)}}dx}{3 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {a+b \sin (e+f x)} \left (d^2 \left (\left (59 c^2+36 d^2\right ) b^2+54 a c d b-9 a^2 d^2\right ) \sin ^2(e+f x)-2 d \left (-\left (\left (24 c^3+49 d^2 c\right ) b^2\right )-5 a d \left (11 c^2+3 d^2\right ) b+3 a^2 c d^2\right ) \sin (e+f x)+d \left (3 a^2 d^3+51 b^2 c^2 d+a b \left (48 c^3+38 d^2 c\right )\right )\right )}{\sqrt {c+d \sin (e+f x)}}dx}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {a+b \sin (e+f x)} \left (d^2 \left (\left (59 c^2+36 d^2\right ) b^2+54 a c d b-9 a^2 d^2\right ) \sin (e+f x)^2-2 d \left (-\left (\left (24 c^3+49 d^2 c\right ) b^2\right )-5 a d \left (11 c^2+3 d^2\right ) b+3 a^2 c d^2\right ) \sin (e+f x)+d \left (3 a^2 d^3+51 b^2 c^2 d+a b \left (48 c^3+38 d^2 c\right )\right )\right )}{\sqrt {c+d \sin (e+f x)}}dx}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (\left (15 c^3+284 d^2 c\right ) b^3+a d \left (337 c^2+156 d^2\right ) b^2+57 a^2 c d^2 b-9 a^3 d^3\right ) \sin ^2(e+f x) d^2+\left (c \left (59 c^2+36 d^2\right ) b^3+a d \left (317 c^2+36 d^2\right ) b^2+a^2 c \left (192 c^2+197 d^2\right ) b+3 a^3 d^3\right ) d^2-2 \left (3 c d^2 a^3-b d \left (166 c^2+57 d^2\right ) a^2-b^2 c \left (133 c^2+290 d^2\right ) a-b^3 d \left (161 c^2+36 d^2\right )\right ) \sin (e+f x) d^2}{2 \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {d \left (-9 a^2 d^2+54 a b c d+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (\left (15 c^3+284 d^2 c\right ) b^3+a d \left (337 c^2+156 d^2\right ) b^2+57 a^2 c d^2 b-9 a^3 d^3\right ) \sin ^2(e+f x) d^2+\left (c \left (59 c^2+36 d^2\right ) b^3+a d \left (317 c^2+36 d^2\right ) b^2+a^2 c \left (192 c^2+197 d^2\right ) b+3 a^3 d^3\right ) d^2-2 \left (3 c d^2 a^3-b d \left (166 c^2+57 d^2\right ) a^2-b^2 c \left (133 c^2+290 d^2\right ) a-b^3 d \left (161 c^2+36 d^2\right )\right ) \sin (e+f x) d^2}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{4 d}-\frac {d \left (-9 a^2 d^2+54 a b c d+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (\left (15 c^3+284 d^2 c\right ) b^3+a d \left (337 c^2+156 d^2\right ) b^2+57 a^2 c d^2 b-9 a^3 d^3\right ) \sin (e+f x)^2 d^2+\left (c \left (59 c^2+36 d^2\right ) b^3+a d \left (317 c^2+36 d^2\right ) b^2+a^2 c \left (192 c^2+197 d^2\right ) b+3 a^3 d^3\right ) d^2-2 \left (3 c d^2 a^3-b d \left (166 c^2+57 d^2\right ) a^2-b^2 c \left (133 c^2+290 d^2\right ) a-b^3 d \left (161 c^2+36 d^2\right )\right ) \sin (e+f x) d^2}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{4 d}-\frac {d \left (-9 a^2 d^2+54 a b c d+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int -\frac {3 \left (\left (5 c^4-120 d^2 c^2-48 d^4\right ) b^4-60 a c d \left (c^2+4 d^2\right ) b^3-6 a^2 d^2 \left (15 c^2+4 d^2\right ) b^2+20 a^3 c d^3 b-3 a^4 d^4\right ) \sin ^2(e+f x) d^2+\left (\left (15 c^4+284 d^2 c^2\right ) b^4+4 a c d \left (51 c^2-50 d^2\right ) b^3-2 a^2 d^2 \left (457 c^2+114 d^2\right ) b^2-4 a^3 c d \left (96 c^2+115 d^2\right ) b+3 a^4 d^4\right ) d^2-2 \left (3 c d^3 a^4+b d^2 \left (275 c^2+117 d^2\right ) a^3+b^2 c d \left (121 c^2+621 d^2\right ) a^2-b^3 \left (15 c^4-355 d^2 c^2-108 d^4\right ) a+b^4 c d \left (59 c^2+36 d^2\right )\right ) \sin (e+f x) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {d \left (-9 a^3 d^3+57 a^2 b c d^2+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d \left (-9 a^2 d^2+54 a b c d+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {3 \left (\left (5 c^4-120 d^2 c^2-48 d^4\right ) b^4-60 a c d \left (c^2+4 d^2\right ) b^3-6 a^2 d^2 \left (15 c^2+4 d^2\right ) b^2+20 a^3 c d^3 b-3 a^4 d^4\right ) \sin ^2(e+f x) d^2+\left (\left (15 c^4+284 d^2 c^2\right ) b^4+4 a c d \left (51 c^2-50 d^2\right ) b^3-2 a^2 d^2 \left (457 c^2+114 d^2\right ) b^2-4 a^3 c d \left (96 c^2+115 d^2\right ) b+3 a^4 d^4\right ) d^2-2 \left (3 c d^3 a^4+b d^2 \left (275 c^2+117 d^2\right ) a^3+b^2 c d \left (121 c^2+621 d^2\right ) a^2-b^3 \left (15 c^4-355 d^2 c^2-108 d^4\right ) a+b^4 c d \left (59 c^2+36 d^2\right )\right ) \sin (e+f x) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {d \left (-9 a^3 d^3+57 a^2 b c d^2+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d \left (-9 a^2 d^2+54 a b c d+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {3 \left (\left (5 c^4-120 d^2 c^2-48 d^4\right ) b^4-60 a c d \left (c^2+4 d^2\right ) b^3-6 a^2 d^2 \left (15 c^2+4 d^2\right ) b^2+20 a^3 c d^3 b-3 a^4 d^4\right ) \sin (e+f x)^2 d^2+\left (\left (15 c^4+284 d^2 c^2\right ) b^4+4 a c d \left (51 c^2-50 d^2\right ) b^3-2 a^2 d^2 \left (457 c^2+114 d^2\right ) b^2-4 a^3 c d \left (96 c^2+115 d^2\right ) b+3 a^4 d^4\right ) d^2-2 \left (3 c d^3 a^4+b d^2 \left (275 c^2+117 d^2\right ) a^3+b^2 c d \left (121 c^2+621 d^2\right ) a^2-b^3 \left (15 c^4-355 d^2 c^2-108 d^4\right ) a+b^4 c d \left (59 c^2+36 d^2\right )\right ) \sin (e+f x) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {d \left (-9 a^3 d^3+57 a^2 b c d^2+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d \left (-9 a^2 d^2+54 a b c d+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {\int -\frac {\left (a^2-b^2\right ) d^2 (b c-a d) \left (15 c^3 b^3+284 c d^2 b^3+84 a d^3 b^2+219 a c^2 d b^2-51 a^2 c d^2 b+9 a^3 d^3\right )-2 b \left (a^2-b^2\right ) d^3 (b c-a d) \left (59 b^2 c^2+54 a b d c-9 a^2 d^2+36 b^2 d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {3 d^2 \left (-3 a^4 d^4+20 a^3 b c d^3-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )-60 a b^3 c d \left (c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {d \left (-9 a^3 d^3+57 a^2 b c d^2+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d \left (-9 a^2 d^2+54 a b c d+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {3 d^2 \left (-3 a^4 d^4+20 a^3 b c d^3-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )-60 a b^3 c d \left (c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {\int -\frac {2 b \left (a^2-b^2\right ) (b c-a d) \left (\left (59 c^2+36 d^2\right ) b^2+54 a c d b-9 a^2 d^2\right ) \sin (e+f x) d^3+\left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (15 c^3+284 d^2 c\right ) b^3\right )-3 a d \left (73 c^2+28 d^2\right ) b^2+51 a^2 c d^2 b-9 a^3 d^3\right ) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {d \left (-9 a^3 d^3+57 a^2 b c d^2+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d \left (-9 a^2 d^2+54 a b c d+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {\int \frac {2 b \left (a^2-b^2\right ) (b c-a d) \left (\left (59 c^2+36 d^2\right ) b^2+54 a c d b-9 a^2 d^2\right ) \sin (e+f x) d^3+\left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (15 c^3+284 d^2 c\right ) b^3\right )-3 a d \left (73 c^2+28 d^2\right ) b^2+51 a^2 c d^2 b-9 a^3 d^3\right ) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {3 d^2 \left (-3 a^4 d^4+20 a^3 b c d^3-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )-60 a b^3 c d \left (c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {d \left (-9 a^3 d^3+57 a^2 b c d^2+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d \left (-9 a^2 d^2+54 a b c d+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {\int \frac {2 b \left (a^2-b^2\right ) (b c-a d) \left (\left (59 c^2+36 d^2\right ) b^2+54 a c d b-9 a^2 d^2\right ) \sin (e+f x) d^3+\left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (15 c^3+284 d^2 c\right ) b^3\right )-3 a d \left (73 c^2+28 d^2\right ) b^2+51 a^2 c d^2 b-9 a^3 d^3\right ) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {3 d^2 \left (-3 a^4 d^4+20 a^3 b c d^3-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )-60 a b^3 c d \left (c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {d \left (-9 a^3 d^3+57 a^2 b c d^2+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d \left (-9 a^2 d^2+54 a b c d+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 3290 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {\int \frac {2 b \left (a^2-b^2\right ) (b c-a d) \left (\left (59 c^2+36 d^2\right ) b^2+54 a c d b-9 a^2 d^2\right ) \sin (e+f x) d^3+\left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (15 c^3+284 d^2 c\right ) b^3\right )-3 a d \left (73 c^2+28 d^2\right ) b^2+51 a^2 c d^2 b-9 a^3 d^3\right ) d^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {6 d \sqrt {c+d} \left (-3 a^4 d^4+20 a^3 b c d^3-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )-60 a b^3 c d \left (c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b^2 f \sqrt {a+b}}}{2 d}-\frac {d \left (-9 a^3 d^3+57 a^2 b c d^2+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {d \left (-9 a^2 d^2+54 a b c d+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {\frac {\frac {-\frac {d \left (\left (15 c^3+284 d^2 c\right ) b^3+a d \left (337 c^2+156 d^2\right ) b^2+57 a^2 c d^2 b-9 a^3 d^3\right ) \sqrt {c+d \sin (e+f x)} \cos (e+f x)}{f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {6 d \sqrt {c+d} \left (\left (5 c^4-120 d^2 c^2-48 d^4\right ) b^4-60 a c d \left (c^2+4 d^2\right ) b^3-6 a^2 d^2 \left (15 c^2+4 d^2\right ) b^2+20 a^3 c d^3 b-3 a^4 d^4\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^2 \sqrt {a+b} f}+\frac {b (a+b) d^2 (b c-a d) \left (\left (15 c^3+284 d^2 c\right ) b^3+a d \left (337 c^2+156 d^2\right ) b^2+57 a^2 c d^2 b-9 a^3 d^3\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx-\frac {\left (a^2-b^2\right ) d^2 (b c-a d) \left (\left (15 c^3+118 d c^2+284 d^2 c+72 d^3\right ) b^3+3 a d \left (73 c^2+36 d c+28 d^2\right ) b^2-3 a^2 d^2 (17 c+6 d) b+9 a^3 d^3\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{a-b}}{b^2}}{2 d}}{4 d}-\frac {d \left (\left (59 c^2+36 d^2\right ) b^2+54 a c d b-9 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {d \left (\left (15 c^3+284 d^2 c\right ) b^3+a d \left (337 c^2+156 d^2\right ) b^2+57 a^2 c d^2 b-9 a^3 d^3\right ) \sqrt {c+d \sin (e+f x)} \cos (e+f x)}{f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {6 d \sqrt {c+d} \left (\left (5 c^4-120 d^2 c^2-48 d^4\right ) b^4-60 a c d \left (c^2+4 d^2\right ) b^3-6 a^2 d^2 \left (15 c^2+4 d^2\right ) b^2+20 a^3 c d^3 b-3 a^4 d^4\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^2 \sqrt {a+b} f}+\frac {b (a+b) d^2 (b c-a d) \left (\left (15 c^3+284 d^2 c\right ) b^3+a d \left (337 c^2+156 d^2\right ) b^2+57 a^2 c d^2 b-9 a^3 d^3\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx-\frac {\left (a^2-b^2\right ) d^2 (b c-a d) \left (\left (15 c^3+118 d c^2+284 d^2 c+72 d^3\right ) b^3+3 a d \left (73 c^2+36 d c+28 d^2\right ) b^2-3 a^2 d^2 (17 c+6 d) b+9 a^3 d^3\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{a-b}}{b^2}}{2 d}}{4 d}-\frac {d \left (\left (59 c^2+36 d^2\right ) b^2+54 a c d b-9 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {\frac {\frac {-\frac {d \left (\left (15 c^3+284 d^2 c\right ) b^3+a d \left (337 c^2+156 d^2\right ) b^2+57 a^2 c d^2 b-9 a^3 d^3\right ) \sqrt {c+d \sin (e+f x)} \cos (e+f x)}{f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {6 d \sqrt {c+d} \left (\left (5 c^4-120 d^2 c^2-48 d^4\right ) b^4-60 a c d \left (c^2+4 d^2\right ) b^3-6 a^2 d^2 \left (15 c^2+4 d^2\right ) b^2+20 a^3 c d^3 b-3 a^4 d^4\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^2 \sqrt {a+b} f}+\frac {b (a+b) d^2 (b c-a d) \left (\left (15 c^3+284 d^2 c\right ) b^3+a d \left (337 c^2+156 d^2\right ) b^2+57 a^2 c d^2 b-9 a^3 d^3\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx-\frac {2 \sqrt {a+b} \left (a^2-b^2\right ) d^2 \left (\left (15 c^3+118 d c^2+284 d^2 c+72 d^3\right ) b^3+3 a d \left (73 c^2+36 d c+28 d^2\right ) b^2-3 a^2 d^2 (17 c+6 d) b+9 a^3 d^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(a-b) \sqrt {c+d} f}}{b^2}}{2 d}}{4 d}-\frac {d \left (\left (59 c^2+36 d^2\right ) b^2+54 a c d b-9 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {\frac {\frac {-\frac {d \left (\left (15 c^3+284 d^2 c\right ) b^3+a d \left (337 c^2+156 d^2\right ) b^2+57 a^2 c d^2 b-9 a^3 d^3\right ) \sqrt {c+d \sin (e+f x)} \cos (e+f x)}{f \sqrt {a+b \sin (e+f x)}}-\frac {\frac {6 d \sqrt {c+d} \left (\left (5 c^4-120 d^2 c^2-48 d^4\right ) b^4-60 a c d \left (c^2+4 d^2\right ) b^3-6 a^2 d^2 \left (15 c^2+4 d^2\right ) b^2+20 a^3 c d^3 b-3 a^4 d^4\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^2 \sqrt {a+b} f}+\frac {-\frac {2 b \sqrt {a+b} (c-d) \sqrt {c+d} \left (\left (15 c^3+284 d^2 c\right ) b^3+a d \left (337 c^2+156 d^2\right ) b^2+57 a^2 c d^2 b-9 a^3 d^3\right ) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x)) d^2}{(b c-a d) f}-\frac {2 \sqrt {a+b} \left (a^2-b^2\right ) \left (\left (15 c^3+118 d c^2+284 d^2 c+72 d^3\right ) b^3+3 a d \left (73 c^2+36 d c+28 d^2\right ) b^2-3 a^2 d^2 (17 c+6 d) b+9 a^3 d^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x)) d^2}{(a-b) \sqrt {c+d} f}}{b^2}}{2 d}}{4 d}-\frac {d \left (\left (59 c^2+36 d^2\right ) b^2+54 a c d b-9 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 f}}{8 b}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}{4 b f}\) |
-1/4*(d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]] )/(b*f) + (-1/3*(d*(17*b*c - 3*a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(3/2 )*Sqrt[c + d*Sin[e + f*x]])/f + (-1/2*(d*(54*a*b*c*d - 9*a^2*d^2 + b^2*(59 *c^2 + 36*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f *x]])/f + (-((d*(57*a^2*b*c*d^2 - 9*a^3*d^3 + a*b^2*d*(337*c^2 + 156*d^2) + b^3*(15*c^3 + 284*c*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt [a + b*Sin[e + f*x]])) - ((6*d*Sqrt[c + d]*(20*a^3*b*c*d^3 - 3*a^4*d^4 - 6 0*a*b^3*c*d*(c^2 + 4*d^2) - 6*a^2*b^2*d^2*(15*c^2 + 4*d^2) + b^4*(5*c^4 - 120*c^2*d^2 - 48*d^4))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], (( a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - S in[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[ e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b^2*Sqrt [a + b]*f) + ((-2*b*Sqrt[a + b]*(c - d)*d^2*Sqrt[c + d]*(57*a^2*b*c*d^2 - 9*a^3*d^3 + a*b^2*d*(337*c^2 + 156*d^2) + b^3*(15*c^3 + 284*c*d^2))*Ellipt icE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b* Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-( ((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b *c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((b*c - a*d)*f) - (2*Sqrt[a + b]*(a^2 - b^2)*d^2*(9*a^3*d^3 -...
3.8.71.3.1 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[((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 - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d *(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Si n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m ] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[2*((a + b*Sin[e + f*x])/(d*f*Rt[(a + b)/ (c + d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticPi[b*((c + d)/(d*(a + b))), ArcSin[Rt[(a + b)/( c + d), 2]*(Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])], (a - b)*(( c + d)/((a + b)*(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] && PosQ[(a + b)/(c + d)]
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_ .) + (f_.)*(x_)]]), x_Symbol] :> Simp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d )*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e + f*x] )/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/ ((a - b)*(c + d*Sin[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(S qrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c - d)/((a - b)*(c + d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && N eQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/(a + b)]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.) *(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Sim p[-2*A*(c - d)*((a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2 ]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticE[ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin[e + f*x]] /Sqrt[a + b*Sin[e + f*x]])], (a - b)*((c + d)/((a + b)*(c - d)))], x] /; Fr eeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e , f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
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)/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[C/b^2 Int[Sqrt[a + b*Sin[e + f*x]] /Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[1/b^2 Int[(A*b^2 - a^2*C + b*(b* B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x ]]), x], x] /; FreeQ[{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]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(Sqrt[c + d*Sin[e + f *x]]/(d*f*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[1/(2*d) Int[(1/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]))*Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - d*(A*b + a*B))*Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{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]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 26.86 (sec) , antiderivative size = 514068, normalized size of antiderivative = 494.77
Timed out. \[ \int (3+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]
Timed out. \[ \int (3+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]
\[ \int (3+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (3+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (3+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx=\int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]