Integrand size = 37, antiderivative size = 402 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx=-\frac {(A (c-13 d)+3 B (c+3 d)) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} (c-d)^4 f}-\frac {\sqrt {d} \left (A d \left (35 c^2+42 c d+19 d^2\right )-3 B \left (5 c^3+10 c^2 d+13 c d^2+4 d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{4 a^{3/2} (c-d)^4 (c+d)^{5/2} f}-\frac {(A-B) \cos (e+f x)}{2 (c-d) f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}+\frac {d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{2 a (c-d)^2 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}+\frac {d \left (3 B \left (3 c^2+3 c d+2 d^2\right )-A \left (2 c^2+15 c d+7 d^2\right )\right ) \cos (e+f x)}{4 a (c-d)^3 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \]
-1/2*(A-B)*cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^2-1/ 4*(A*(c-13*d)+3*B*(c+3*d))*arctanh(1/2*cos(f*x+e)*a^(1/2)*2^(1/2)/(a+a*sin (f*x+e))^(1/2))/a^(3/2)/(c-d)^4/f*2^(1/2)-1/4*(A*d*(35*c^2+42*c*d+19*d^2)- 3*B*(5*c^3+10*c^2*d+13*c*d^2+4*d^3))*arctanh(cos(f*x+e)*a^(1/2)*d^(1/2)/(c +d)^(1/2)/(a+a*sin(f*x+e))^(1/2))*d^(1/2)/a^(3/2)/(c-d)^4/(c+d)^(5/2)/f+1/ 2*d*(B*(2*c+d)-A*(c+2*d))*cos(f*x+e)/a/(c-d)^2/(c+d)/f/(c+d*sin(f*x+e))^2/ (a+a*sin(f*x+e))^(1/2)+1/4*d*(3*B*(3*c^2+3*c*d+2*d^2)-A*(2*c^2+15*c*d+7*d^ 2))*cos(f*x+e)/a/(c-d)^3/(c+d)^2/f/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 12.80 (sec) , antiderivative size = 1757, normalized size of antiderivative = 4.37 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx =\text {Too large to display} \]
((1 + I)*(A*c + 3*B*c - 13*A*d + 9*B*d)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec [(e + f*x)/4]*(Cos[(e + f*x)/4] - Sin[(e + f*x)/4])]*(Cos[(e + f*x)/2] + S in[(e + f*x)/2])^3)/((2*(-1)^(1/4)*c^4 - 8*(-1)^(1/4)*c^3*d + 12*(-1)^(1/4 )*c^2*d^2 - 8*(-1)^(1/4)*c*d^3 + 2*(-1)^(1/4)*d^4)*f*(a*(1 + Sin[e + f*x]) )^(3/2)) + (Sqrt[d]*(-(A*d*(35*c^2 + 42*c*d + 19*d^2)) + 3*B*(5*c^3 + 10*c ^2*d + 13*c*d^2 + 4*d^3))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + RootSum[c + 4*d*#1 + 2*c*#1^2 - 4*d*#1^3 + c*#1^4 & , (-(d*Log[-#1 + Tan[(e + f*x)/ 4]]) + Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]] - c*Log[-#1 + Tan[( e + f*x)/4]]*#1 + 2*Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]]*#1 + 3 *d*Log[-#1 + Tan[(e + f*x)/4]]*#1^2 - Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]]*#1^2 - c*Log[-#1 + Tan[(e + f*x)/4]]*#1^3)/(-d - c*#1 + 3*d*#1 ^2 - c*#1^3) & ])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/(16*(c - d)^4*( c + d)^(5/2)*f*(a*(1 + Sin[e + f*x]))^(3/2)) - (Sqrt[d]*(-(A*d*(35*c^2 + 4 2*c*d + 19*d^2)) + 3*B*(5*c^3 + 10*c^2*d + 13*c*d^2 + 4*d^3))*(e + f*x - 2 *Log[Sec[(e + f*x)/4]^2] + RootSum[c + 4*d*#1 + 2*c*#1^2 - 4*d*#1^3 + c*#1 ^4 & , (-(d*Log[-#1 + Tan[(e + f*x)/4]]) - Sqrt[d]*Sqrt[c + d]*Log[-#1 + T an[(e + f*x)/4]] - c*Log[-#1 + Tan[(e + f*x)/4]]*#1 - 2*Sqrt[d]*Sqrt[c + d ]*Log[-#1 + Tan[(e + f*x)/4]]*#1 + 3*d*Log[-#1 + Tan[(e + f*x)/4]]*#1^2 + Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]]*#1^2 - c*Log[-#1 + Tan[(e + f*x)/4]]*#1^3)/(-d - c*#1 + 3*d*#1^2 - c*#1^3) & ])*(Cos[(e + f*x)/2]...
Time = 2.42 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.10, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.432, Rules used = {3042, 3457, 27, 3042, 3463, 27, 3042, 3463, 27, 3042, 3464, 3042, 3128, 219, 3252, 221}
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.
\(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}dx\) |
\(\Big \downarrow \) 3457 |
\(\displaystyle -\frac {\int -\frac {a (A c+3 B c-8 A d+4 B d)+5 a (A-B) d \sin (e+f x)}{2 \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3}dx}{2 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {a (A c+3 B c-8 A d+4 B d)+5 a (A-B) d \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3}dx}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {a (A c+3 B c-8 A d+4 B d)+5 a (A-B) d \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3}dx}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3463 |
\(\displaystyle \frac {\frac {2 a d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}-\frac {\int -\frac {2 \left (a^2 \left (A \left (c^2-9 d c-7 d^2\right )+3 B \left (c^2+2 d c+2 d^2\right )\right )-3 a^2 d (B (2 c+d)-A (c+2 d)) \sin (e+f x)\right )}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{2 a \left (c^2-d^2\right )}}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\int \frac {a^2 \left (A \left (c^2-9 d c-7 d^2\right )+3 B \left (c^2+2 d c+2 d^2\right )\right )-3 a^2 d (B (2 c+d)-A (c+2 d)) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{a \left (c^2-d^2\right )}+\frac {2 a d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {a^2 \left (A \left (c^2-9 d c-7 d^2\right )+3 B \left (c^2+2 d c+2 d^2\right )\right )-3 a^2 d (B (2 c+d)-A (c+2 d)) \sin (e+f x)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{a \left (c^2-d^2\right )}+\frac {2 a d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3463 |
\(\displaystyle \frac {\frac {\frac {a^2 d \left (3 B \left (3 c^2+3 c d+2 d^2\right )-A \left (2 c^2+15 c d+7 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {\int -\frac {a^3 \left (A \left (2 c^3-20 d c^2-35 d^2 c-19 d^3\right )+3 B \left (2 c^3+7 d c^2+11 d^2 c+4 d^3\right )\right )-a^3 d \left (3 B \left (3 c^2+3 d c+2 d^2\right )-A \left (2 c^2+15 d c+7 d^2\right )\right ) \sin (e+f x)}{2 \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}}{a \left (c^2-d^2\right )}+\frac {2 a d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {\int \frac {\left (A \left (2 c^3-20 d c^2-35 d^2 c-19 d^3\right )+3 B \left (2 c^3+7 d c^2+11 d^2 c+4 d^3\right )\right ) a^3+d \left (2 A c^2-9 B c^2+15 A d c-9 B d c+7 A d^2-6 B d^2\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}+\frac {a^2 d \left (3 B \left (3 c^2+3 c d+2 d^2\right )-A \left (2 c^2+15 c d+7 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}+\frac {2 a d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {\int \frac {\left (A \left (2 c^3-20 d c^2-35 d^2 c-19 d^3\right )+3 B \left (2 c^3+7 d c^2+11 d^2 c+4 d^3\right )\right ) a^3+d \left (2 A c^2-9 B c^2+15 A d c-9 B d c+7 A d^2-6 B d^2\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{2 a \left (c^2-d^2\right )}+\frac {a^2 d \left (3 B \left (3 c^2+3 c d+2 d^2\right )-A \left (2 c^2+15 c d+7 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}+\frac {2 a d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3464 |
\(\displaystyle \frac {\frac {\frac {\frac {2 a^3 (c+d)^2 (A (c-13 d)+3 B (c+3 d)) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{c-d}+\frac {a^2 d \left (A d \left (35 c^2+42 c d+19 d^2\right )-3 B \left (5 c^3+10 c^2 d+13 c d^2+4 d^3\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}}{2 a \left (c^2-d^2\right )}+\frac {a^2 d \left (3 B \left (3 c^2+3 c d+2 d^2\right )-A \left (2 c^2+15 c d+7 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}+\frac {2 a d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {\frac {2 a^3 (c+d)^2 (A (c-13 d)+3 B (c+3 d)) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{c-d}+\frac {a^2 d \left (A d \left (35 c^2+42 c d+19 d^2\right )-3 B \left (5 c^3+10 c^2 d+13 c d^2+4 d^3\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}}{2 a \left (c^2-d^2\right )}+\frac {a^2 d \left (3 B \left (3 c^2+3 c d+2 d^2\right )-A \left (2 c^2+15 c d+7 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}+\frac {2 a d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3128 |
\(\displaystyle \frac {\frac {\frac {\frac {a^2 d \left (A d \left (35 c^2+42 c d+19 d^2\right )-3 B \left (5 c^3+10 c^2 d+13 c d^2+4 d^3\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}-\frac {4 a^3 (c+d)^2 (A (c-13 d)+3 B (c+3 d)) \int \frac {1}{2 a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f (c-d)}}{2 a \left (c^2-d^2\right )}+\frac {a^2 d \left (3 B \left (3 c^2+3 c d+2 d^2\right )-A \left (2 c^2+15 c d+7 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}+\frac {2 a d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {\frac {\frac {a^2 d \left (A d \left (35 c^2+42 c d+19 d^2\right )-3 B \left (5 c^3+10 c^2 d+13 c d^2+4 d^3\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}-\frac {2 \sqrt {2} a^{5/2} (c+d)^2 (A (c-13 d)+3 B (c+3 d)) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{2 a \left (c^2-d^2\right )}+\frac {a^2 d \left (3 B \left (3 c^2+3 c d+2 d^2\right )-A \left (2 c^2+15 c d+7 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}+\frac {2 a d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3252 |
\(\displaystyle \frac {\frac {\frac {-\frac {2 a^3 d \left (A d \left (35 c^2+42 c d+19 d^2\right )-3 B \left (5 c^3+10 c^2 d+13 c d^2+4 d^3\right )\right ) \int \frac {1}{a (c+d)-\frac {a^2 d \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f (c-d)}-\frac {2 \sqrt {2} a^{5/2} (c+d)^2 (A (c-13 d)+3 B (c+3 d)) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{2 a \left (c^2-d^2\right )}+\frac {a^2 d \left (3 B \left (3 c^2+3 c d+2 d^2\right )-A \left (2 c^2+15 c d+7 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}+\frac {2 a d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {\frac {-\frac {2 a^{5/2} \sqrt {d} \left (A d \left (35 c^2+42 c d+19 d^2\right )-3 B \left (5 c^3+10 c^2 d+13 c d^2+4 d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d) \sqrt {c+d}}-\frac {2 \sqrt {2} a^{5/2} (c+d)^2 (A (c-13 d)+3 B (c+3 d)) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{2 a \left (c^2-d^2\right )}+\frac {a^2 d \left (3 B \left (3 c^2+3 c d+2 d^2\right )-A \left (2 c^2+15 c d+7 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}+\frac {2 a d (B (2 c+d)-A (c+2 d)) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}\) |
-1/2*((A - B)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*S in[e + f*x])^2) + ((2*a*d*(B*(2*c + d) - A*(c + 2*d))*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) + (((-2*Sqrt[2]* a^(5/2)*(c + d)^2*(A*(c - 13*d) + 3*B*(c + 3*d))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/((c - d)*f) - (2*a^(5/2)*Sqrt[d ]*(A*d*(35*c^2 + 42*c*d + 19*d^2) - 3*B*(5*c^3 + 10*c^2*d + 13*c*d^2 + 4*d ^3))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/((c - d)*Sqrt[c + d]*f))/(2*a*(c^2 - d^2)) + (a^2*d*(3*B*(3*c^2 + 3*c*d + 2*d^2) - A*(2*c^2 + 15*c*d + 7*d^2))*Cos[e + f*x])/((c^2 - d^2) *f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])))/(a*(c^2 - d^2)))/(4*a^2 *(c - d))
3.4.20.3.1 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_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)]), x_Symbol] :> Simp[-2*(b/f) Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && Eq Q[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 0])
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A *b - a*B)/(b*c - a*d) Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Simp[(B*c - A*d)/(b*c - a*d) Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(4706\) vs. \(2(363)=726\).
Time = 2.07 (sec) , antiderivative size = 4707, normalized size of antiderivative = 11.71
-1/4*(-a*(sin(f*x+e)-1))^(1/2)*(-2*B*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d) ^(1/2)*a^(3/2)*sin(f*x+e)^2*c^2*d^3+2*B*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d) *d)^(1/2)*a^(3/2)*sin(f*x+e)^2*c*d^4-11*A*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+ d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*c^2*d^3+15*B*(a*(c+d)*d)^(1/2)*2^(1/2)*arct anh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*a^2*c^4*d+9*B*(a*(c+d)* d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*si n(f*x+e)^3*a^2*d^5-13*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x +e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^2*a^2*d^5+9*B*(a*(c+d)*d)^(1/2)* 2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^ 2*a^2*d^5+4*A*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+ e)*c^4*d+17*A*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+ e)*c^3*d^2-A*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e )*c^2*d^3-17*A*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x +e)*c*d^4-13*B*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x +e)*c^4*d-7*B*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+ e)*c^3*d^2+6*A*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x +e)*c*d^4+A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2 )*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2*c^5-13*A*(a*(c+d)*d)^(1/2)*2^(1/2)*arcta nh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)^3*a^2*d^5-11* A*(a*(c+d)*d)^(1/2)*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/...
Leaf count of result is larger than twice the leaf count of optimal. 2789 vs. \(2 (362) = 724\).
Time = 8.46 (sec) , antiderivative size = 5864, normalized size of antiderivative = 14.59 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1263 vs. \(2 (362) = 724\).
Time = 0.60 (sec) , antiderivative size = 1263, normalized size of antiderivative = 3.14 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]
1/8*(sqrt(2)*(15*sqrt(2)*B*sqrt(a)*c^3*d - 35*sqrt(2)*A*sqrt(a)*c^2*d^2 + 30*sqrt(2)*B*sqrt(a)*c^2*d^2 - 42*sqrt(2)*A*sqrt(a)*c*d^3 + 39*sqrt(2)*B*s qrt(a)*c*d^3 - 19*sqrt(2)*A*sqrt(a)*d^4 + 12*sqrt(2)*B*sqrt(a)*d^4)*arctan (sqrt(2)*d*sin(-1/4*pi + 1/2*f*x + 1/2*e)/sqrt(-c*d - d^2))/((a^2*c^6*sgn( cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 2*a^2*c^5*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - a^2*c^4*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 4*a^2*c^3*d^3* sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - a^2*c^2*d^4*sgn(cos(-1/4*pi + 1/2*f* x + 1/2*e)) - 2*a^2*c*d^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + a^2*d^6*sg n(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sqrt(-c*d - d^2)) + 2*(A*sqrt(a)*c + 3* B*sqrt(a)*c - 13*A*sqrt(a)*d + 9*B*sqrt(a)*d)*log(sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(sqrt(2)*a^2*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 4*sqrt( 2)*a^2*c^3*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 6*sqrt(2)*a^2*c^2*d^2*s gn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 4*sqrt(2)*a^2*c*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + sqrt(2)*a^2*d^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - 2*(A*sqrt(a)*c + 3*B*sqrt(a)*c - 13*A*sqrt(a)*d + 9*B*sqrt(a)*d)*log(-si n(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(sqrt(2)*a^2*c^4*sgn(cos(-1/4*pi + 1/2*f *x + 1/2*e)) - 4*sqrt(2)*a^2*c^3*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 6 *sqrt(2)*a^2*c^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 4*sqrt(2)*a^2*c *d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + sqrt(2)*a^2*d^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - 4*(A*sqrt(a)*sin(-1/4*pi + 1/2*f*x + 1/2*e) - B...
Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx=\int \frac {A+B\,\sin \left (e+f\,x\right )}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]