Integrand size = 37, antiderivative size = 395 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=-\frac {\left (B \left (5 c^2-58 c d-43 d^2\right )+A \left (3 c^2-22 c d+115 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} (c-d)^4 f}+\frac {d^{3/2} \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{a^{5/2} (c-d)^4 (c+d)^{3/2} f}-\frac {(A-B) \cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))}-\frac {(3 A c+5 B c-15 A d+7 B d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))}-\frac {d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \]
d^(3/2)*(A*d*(7*c+5*d)-B*(5*c^2+5*c*d+2*d^2))*arctanh(cos(f*x+e)*a^(1/2)*d ^(1/2)/(c+d)^(1/2)/(a+a*sin(f*x+e))^(1/2))/a^(5/2)/(c-d)^4/(c+d)^(3/2)/f-1 /4*(A-B)*cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))-1/16*( 3*A*c-15*A*d+5*B*c+7*B*d)*cos(f*x+e)/a/(c-d)^2/f/(a+a*sin(f*x+e))^(3/2)/(c +d*sin(f*x+e))-1/32*(B*(5*c^2-58*c*d-43*d^2)+A*(3*c^2-22*c*d+115*d^2))*arc tanh(1/2*cos(f*x+e)*a^(1/2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2))/a^(5/2)/(c-d)^ 4/f*2^(1/2)-1/16*d*(A*(3*c^2-16*c*d-35*d^2)+B*(5*c^2+32*c*d+11*d^2))*cos(f *x+e)/a^2/(c-d)^3/(c+d)/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 = 15.73 (sec) , antiderivative size = 1680, normalized size of antiderivative = 4.25 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx =\text {Too large to display} \]
((1 + I)*(3*A*c^2 + 5*B*c^2 - 22*A*c*d - 58*B*c*d + 115*A*d^2 - 43*B*d^2)* 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] + Sin[(e + f*x)/2])^5)/((16*(-1)^(1/4)*c^4 - 64*(-1)^(1/4)*c^3*d + 96*(-1)^(1/4)*c^2*d^2 - 64*(-1)^(1/4)*c*d^3 + 16*( -1)^(1/4)*d^4)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + (d^(3/2)*(A*d*(7*c + 5*d) - B*(5*c^2 + 5*c*d + 2*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + RootS um[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 + T an[(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 + Ta n[(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])^5)/(4*(c - d)^ 4*(c + d)^(3/2)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + (d^(3/2)*(-(A*d*(7*c + 5 *d)) + B*(5*c^2 + 5*c*d + 2*d^2))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + R ootSum[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])^5)/(4*(...
Time = 2.42 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.11, 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, 3457, 27, 3042, 3463, 25, 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.
| \(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}dx\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle -\frac {\int -\frac {a (3 A c+5 B c-10 A d+2 B d)+5 a (A-B) d \sin (e+f x)}{2 (\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^2}dx}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a (3 A c+5 B c-10 A d+2 B d)+5 a (A-B) d \sin (e+f x)}{(\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^2}dx}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (3 A c+5 B c-10 A d+2 B d)+5 a (A-B) d \sin (e+f x)}{(\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^2}dx}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {-\frac {\int -\frac {\left (B \left (5 c^2-43 d c-22 d^2\right )+A \left (3 c^2-13 d c+70 d^2\right )\right ) a^2+3 d (3 A (c-5 d)+B (5 c+7 d)) \sin (e+f x) a^2}{2 \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{2 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (B \left (5 c^2-43 d c-22 d^2\right )+A \left (3 c^2-13 d c+70 d^2\right )\right ) a^2+3 d (3 A (c-5 d)+B (5 c+7 d)) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\left (B \left (5 c^2-43 d c-22 d^2\right )+A \left (3 c^2-13 d c+70 d^2\right )\right ) a^2+3 d (3 A (c-5 d)+B (5 c+7 d)) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {\left (B \left (5 c^3-48 d c^2-69 d^2 c-32 d^3\right )+A \left (3 c^3-16 d c^2+77 d^2 c+80 d^3\right )\right ) a^3+d \left (A \left (3 c^2-16 d c-35 d^2\right )+B \left (5 c^2+32 d c+11 d^2\right )\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (B \left (5 c^3-48 d c^2-69 d^2 c-32 d^3\right )+A \left (3 c^3-16 d c^2+77 d^2 c+80 d^3\right )\right ) a^3+d \left (A \left (3 c^2-16 d c-35 d^2\right )+B \left (5 c^2+32 d c+11 d^2\right )\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (B \left (5 c^3-48 d c^2-69 d^2 c-32 d^3\right )+A \left (3 c^3-16 d c^2+77 d^2 c+80 d^3\right )\right ) a^3+d \left (A \left (3 c^2-16 d c-35 d^2\right )+B \left (5 c^2+32 d c+11 d^2\right )\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\) |
| \(\Big \downarrow \) 3464 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^3 (c+d) \left (A \left (3 c^2-22 c d+115 d^2\right )+B \left (5 c^2-58 c d-43 d^2\right )\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{c-d}-\frac {16 a^2 d^2 \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^3 (c+d) \left (A \left (3 c^2-22 c d+115 d^2\right )+B \left (5 c^2-58 c d-43 d^2\right )\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{c-d}-\frac {16 a^2 d^2 \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {\frac {\frac {-\frac {16 a^2 d^2 \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}-\frac {2 a^3 (c+d) \left (A \left (3 c^2-22 c d+115 d^2\right )+B \left (5 c^2-58 c d-43 d^2\right )\right ) \int \frac {1}{2 a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {-\frac {16 a^2 d^2 \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}-\frac {\sqrt {2} a^{5/2} (c+d) \left (A \left (3 c^2-22 c d+115 d^2\right )+B \left (5 c^2-58 c d-43 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {\frac {\frac {\frac {32 a^3 d^2 \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \int \frac {1}{a (c+d)-\frac {a^2 d \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f (c-d)}-\frac {\sqrt {2} a^{5/2} (c+d) \left (A \left (3 c^2-22 c d+115 d^2\right )+B \left (5 c^2-58 c d-43 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {\frac {32 a^{5/2} d^{3/2} \left (A d (7 c+5 d)-B \left (5 c^2+5 c d+2 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d) \sqrt {c+d}}-\frac {\sqrt {2} a^{5/2} (c+d) \left (A \left (3 c^2-22 c d+115 d^2\right )+B \left (5 c^2-58 c d-43 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-16 c d-35 d^2\right )+B \left (5 c^2+32 c d+11 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{4 a^2 (c-d)}-\frac {a (3 A c-15 A d+5 B c+7 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}\) |
-1/4*((A - B)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(c + d*S in[e + f*x])) + (-1/2*(a*(3*A*c + 5*B*c - 15*A*d + 7*B*d)*Cos[e + f*x])/(( c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])) + ((-((Sqrt[2]*a ^(5/2)*(c + d)*(B*(5*c^2 - 58*c*d - 43*d^2) + A*(3*c^2 - 22*c*d + 115*d^2) )*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/((c - d)*f)) + (32*a^(5/2)*d^(3/2)*(A*d*(7*c + 5*d) - B*(5*c^2 + 5*c*d + 2*d^2 ))*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))/(a*(c^2 - d^2)) - (2*a^2*d*(A*(3*c^2 - 1 6*c*d - 35*d^2) + B*(5*c^2 + 32*c*d + 11*d^2))*Cos[e + f*x])/((c^2 - d^2)* f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])))/(4*a^2*(c - d)))/(8*a^2* (c - d))
3.4.26.3.1 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. \(4091\) vs. \(2(358)=716\).
Time = 2.02 (sec) , antiderivative size = 4092, normalized size of antiderivative = 10.36
-1/32*(-84*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^2+6*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^4+115*A*(a*(c+d)*d)^(1/2)*2^(1/2)*ar ctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*sin(f*x+e)*a^2*d^4-43 *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)^3*a^2*d^4+3*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*c^4-43*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)*a^2*d^4-19*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))*a^2*c^3*d+93*A*(a*(c+d)*d)^(1/2)*2^(1/2)*a rctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*a^2*c^2*d^2+115*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))*a^2*c*d^3-53*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))*a^2*c^3*d-101*B*(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))*a^2*c^2*d^2-43*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))* a^2*c*d^3-10*B*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x +e)*c^3*d-22*B*(-a*(sin(f*x+e)-1))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)*sin(f*x +e)*c^2*d^2-76*B*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*sin(f *x+e)*c*d^3-32*B*(-a*(sin(f*x+e)-1))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*si...
Leaf count of result is larger than twice the leaf count of optimal. 2433 vs. \(2 (358) = 716\).
Time = 8.10 (sec) , antiderivative size = 5151, normalized size of antiderivative = 13.04 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1099 vs. \(2 (358) = 716\).
Time = 0.71 (sec) , antiderivative size = 1099, normalized size of antiderivative = 2.78 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=\text {Too large to display} \]
-1/32*(16*sqrt(2)*(5*sqrt(2)*B*sqrt(a)*c^2*d^2 - 7*sqrt(2)*A*sqrt(a)*c*d^3 + 5*sqrt(2)*B*sqrt(a)*c*d^3 - 5*sqrt(2)*A*sqrt(a)*d^4 + 2*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^3*c^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 3*a^3*c^4*d*sgn(cos(-1/4*p i + 1/2*f*x + 1/2*e)) + 2*a^3*c^3*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 2*a^3*c^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 3*a^3*c*d^4*sgn(cos( -1/4*pi + 1/2*f*x + 1/2*e)) + a^3*d^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) *sqrt(-c*d - d^2)) - (3*A*sqrt(a)*c^2 + 5*B*sqrt(a)*c^2 - 22*A*sqrt(a)*c*d - 58*B*sqrt(a)*c*d + 115*A*sqrt(a)*d^2 - 43*B*sqrt(a)*d^2)*log(sin(-1/4*p i + 1/2*f*x + 1/2*e) + 1)/(sqrt(2)*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2 *e)) - 4*sqrt(2)*a^3*c^3*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 6*sqrt(2) *a^3*c^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 4*sqrt(2)*a^3*c*d^3*sgn (cos(-1/4*pi + 1/2*f*x + 1/2*e)) + sqrt(2)*a^3*d^4*sgn(cos(-1/4*pi + 1/2*f *x + 1/2*e))) + (3*A*sqrt(a)*c^2 + 5*B*sqrt(a)*c^2 - 22*A*sqrt(a)*c*d - 58 *B*sqrt(a)*c*d + 115*A*sqrt(a)*d^2 - 43*B*sqrt(a)*d^2)*log(-sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(sqrt(2)*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 4*sqrt(2)*a^3*c^3*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 6*sqrt(2)*a^3 *c^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 4*sqrt(2)*a^3*c*d^3*sgn(cos (-1/4*pi + 1/2*f*x + 1/2*e)) + sqrt(2)*a^3*d^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) + 64*(B*sqrt(a)*c*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e) - A*sqrt...
Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx=\int \frac {A+B\,\sin \left (e+f\,x\right )}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]