Integrand size = 37, antiderivative size = 519 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=-\frac {\left (B \left (5 c^2-82 c d-115 d^2\right )+3 A \left (c^2-10 c d+73 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} (c-d)^5 f}+\frac {d^{3/2} \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{4 a^{5/2} (c-d)^5 (c+d)^{5/2} f}-\frac {(A-B) \cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 A c+5 B c-19 A d+11 B d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \]
1/4*d^(3/2)*(3*A*d*(21*c^2+30*c*d+13*d^2)-B*(35*c^3+70*c^2*d+67*c*d^2+20*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) )/a^(5/2)/(c-d)^5/(c+d)^(5/2)/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))^2-1/16*(3*A*c-19*A*d+5*B*c+11*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))^2-1/32*(B*(5*c^2-82*c*d -115*d^2)+3*A*(c^2-10*c*d+73*d^2))*arctanh(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)^5/f*2^(1/2)-1/16*d*(A*(3*c^2-20*c*d- 31*d^2)+B*(5*c^2+28*c*d+15*d^2))*cos(f*x+e)/a^2/(c-d)^3/(c+d)/f/(c+d*sin(f *x+e))^2/(a+a*sin(f*x+e))^(1/2)-1/16*d*(3*A*(c^3-7*c^2*d-37*c*d^2-21*d^3)+ B*(5*c^3+73*c^2*d+79*c*d^2+35*d^3))*cos(f*x+e)/a^2/(c-d)^4/(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 = 16.01 (sec) , antiderivative size = 2465, normalized size of antiderivative = 4.75 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Result too large to show} \]
((1 + I)*(3*A*c^2 + 5*B*c^2 - 30*A*c*d - 82*B*c*d + 219*A*d^2 - 115*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^5 - 80*(-1)^(1/4)*c^4*d + 160*(-1)^(1/4)*c^3*d^2 - 160*(-1)^(1/4)*c^2*d^3 + 80*(-1)^(1/4)*c*d^4 - 16*(-1)^(1/4)*d^5)*f*(a*(1 + Sin[e + f*x]))^(5/2)) - (d^(3/2)*(-3*A*d*(21*c^2 + 30*c*d + 13*d^2) + B*(35*c^3 + 70*c^2*d + 67* c*d^2 + 20*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]]) + Sq rt[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)/(16*(c - d)^5*(c + d)^(5 /2)*f*(a*(1 + Sin[e + f*x]))^(5/2)) + (d^(3/2)*(-3*A*d*(21*c^2 + 30*c*d + 13*d^2) + B*(35*c^3 + 70*c^2*d + 67*c*d^2 + 20*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]*S qrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]]*#1^2 - c*Log[-#1 + Tan[(e + f*x)...
Time = 3.30 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.11, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.514, Rules used = {3042, 3457, 27, 3042, 3457, 27, 3042, 3463, 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.
\(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^3}dx\) |
\(\Big \downarrow \) 3457 |
\(\displaystyle -\frac {\int -\frac {a (3 A c+5 B c-12 A d+4 B d)+7 a (A-B) d \sin (e+f x)}{2 (\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^3}dx}{4 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {a (3 A c+5 B c-12 A d+4 B d)+7 a (A-B) d \sin (e+f x)}{(\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^3}dx}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {a (3 A c+5 B c-12 A d+4 B d)+7 a (A-B) d \sin (e+f x)}{(\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^3}dx}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3457 |
\(\displaystyle \frac {-\frac {\int -\frac {\left (B \left (5 c^2-57 d c-60 d^2\right )+A \left (3 c^2-15 d c+124 d^2\right )\right ) a^2+5 d (3 A c+5 B c-19 A d+11 B d) \sin (e+f x) a^2}{2 \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3}dx}{2 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\int \frac {\left (B \left (5 c^2-57 d c-60 d^2\right )+A \left (3 c^2-15 d c+124 d^2\right )\right ) a^2+5 d (3 A c+5 B c-19 A d+11 B d) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3}dx}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {\left (B \left (5 c^2-57 d c-60 d^2\right )+A \left (3 c^2-15 d c+124 d^2\right )\right ) a^2+5 d (3 A c+5 B c-19 A d+11 B d) \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3}dx}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3463 |
\(\displaystyle \frac {\frac {-\frac {\int -\frac {2 \left (\left (B \left (5 c^3-62 d c^2-113 d^2 c-70 d^3\right )+3 A \left (c^3-6 d c^2+43 d^2 c+42 d^3\right )\right ) a^3+3 d \left (A \left (3 c^2-20 d c-31 d^2\right )+B \left (5 c^2+28 d c+15 d^2\right )\right ) \sin (e+f x) a^3\right )}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{2 a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {\int \frac {\left (B \left (5 c^3-62 d c^2-113 d^2 c-70 d^3\right )+3 A \left (c^3-6 d c^2+43 d^2 c+42 d^3\right )\right ) a^3+3 d \left (A \left (3 c^2-20 d c-31 d^2\right )+B \left (5 c^2+28 d c+15 d^2\right )\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {\int \frac {\left (B \left (5 c^3-62 d c^2-113 d^2 c-70 d^3\right )+3 A \left (c^3-6 d c^2+43 d^2 c+42 d^3\right )\right ) a^3+3 d \left (A \left (3 c^2-20 d c-31 d^2\right )+B \left (5 c^2+28 d c+15 d^2\right )\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3463 |
\(\displaystyle \frac {\frac {\frac {-\frac {\int -\frac {\left (B \left (5 c^4-67 d c^3-201 d^2 c^2-233 d^3 c-80 d^4\right )+3 A \left (c^4-7 d c^3+47 d^2 c^2+99 d^3 c+52 d^4\right )\right ) a^4+d \left (3 A \left (c^3-7 d c^2-37 d^2 c-21 d^3\right )+B \left (5 c^3+73 d c^2+79 d^2 c+35 d^3\right )\right ) \sin (e+f x) a^4}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\left (B \left (5 c^4-67 d c^3-201 d^2 c^2-233 d^3 c-80 d^4\right )+3 A \left (c^4-7 d c^3+47 d^2 c^2+99 d^3 c+52 d^4\right )\right ) a^4+d \left (3 A \left (c^3-7 d c^2-37 d^2 c-21 d^3\right )+B \left (5 c^3+73 d c^2+79 d^2 c+35 d^3\right )\right ) \sin (e+f x) a^4}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\left (B \left (5 c^4-67 d c^3-201 d^2 c^2-233 d^3 c-80 d^4\right )+3 A \left (c^4-7 d c^3+47 d^2 c^2+99 d^3 c+52 d^4\right )\right ) a^4+d \left (3 A \left (c^3-7 d c^2-37 d^2 c-21 d^3\right )+B \left (5 c^3+73 d c^2+79 d^2 c+35 d^3\right )\right ) \sin (e+f x) a^4}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3464 |
\(\displaystyle \frac {\frac {\frac {\frac {\frac {a^4 (c+d)^2 \left (3 A \left (c^2-10 c d+73 d^2\right )+B \left (5 c^2-82 c d-115 d^2\right )\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{c-d}-\frac {4 a^3 d^2 \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {\frac {\frac {a^4 (c+d)^2 \left (3 A \left (c^2-10 c d+73 d^2\right )+B \left (5 c^2-82 c d-115 d^2\right )\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{c-d}-\frac {4 a^3 d^2 \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3128 |
\(\displaystyle \frac {\frac {\frac {\frac {-\frac {4 a^3 d^2 \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}-\frac {2 a^4 (c+d)^2 \left (3 A \left (c^2-10 c d+73 d^2\right )+B \left (5 c^2-82 c d-115 d^2\right )\right ) \int \frac {1}{2 a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {\frac {\frac {-\frac {4 a^3 d^2 \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}-\frac {\sqrt {2} a^{7/2} (c+d)^2 \left (3 A \left (c^2-10 c d+73 d^2\right )+B \left (5 c^2-82 c d-115 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 3252 |
\(\displaystyle \frac {\frac {\frac {\frac {\frac {8 a^4 d^2 \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \int \frac {1}{a (c+d)-\frac {a^2 d \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f (c-d)}-\frac {\sqrt {2} a^{7/2} (c+d)^2 \left (3 A \left (c^2-10 c d+73 d^2\right )+B \left (5 c^2-82 c d-115 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {\frac {\frac {\frac {8 a^{7/2} d^{3/2} \left (3 A d \left (21 c^2+30 c d+13 d^2\right )-B \left (35 c^3+70 c^2 d+67 c d^2+20 d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d) \sqrt {c+d}}-\frac {\sqrt {2} a^{7/2} (c+d)^2 \left (3 A \left (c^2-10 c d+73 d^2\right )+B \left (5 c^2-82 c d-115 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (3 A \left (c^3-7 c^2 d-37 c d^2-21 d^3\right )+B \left (5 c^3+73 c^2 d+79 c d^2+35 d^3\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (A \left (3 c^2-20 c d-31 d^2\right )+B \left (5 c^2+28 c d+15 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 A c-19 A d+5 B c+11 B d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {(A-B) \cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
-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])^2) + (-1/2*(a*(3*A*c + 5*B*c - 19*A*d + 11*B*d)*Cos[e + f*x]) /((c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2) + ((-2*a^2* d*(A*(3*c^2 - 20*c*d - 31*d^2) + B*(5*c^2 + 28*c*d + 15*d^2))*Cos[e + f*x] )/((c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) + ((-((S qrt[2]*a^(7/2)*(c + d)^2*(B*(5*c^2 - 82*c*d - 115*d^2) + 3*A*(c^2 - 10*c*d + 73*d^2))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x ]])])/((c - d)*f)) + (8*a^(7/2)*d^(3/2)*(3*A*d*(21*c^2 + 30*c*d + 13*d^2) - B*(35*c^3 + 70*c^2*d + 67*c*d^2 + 20*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) )/(a*(c^2 - d^2)) - (2*a^3*d*(3*A*(c^3 - 7*c^2*d - 37*c*d^2 - 21*d^3) + B* (5*c^3 + 73*c^2*d + 79*c*d^2 + 35*d^3))*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) ))/(8*a^2*(c - d))
3.4.27.3.1 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. \(7321\) vs. \(2(476)=952\).
Time = 3.45 (sec) , antiderivative size = 7322, normalized size of antiderivative = 14.11
Leaf count of result is larger than twice the leaf count of optimal. 4135 vs. \(2 (476) = 952\).
Time = 24.73 (sec) , antiderivative size = 8555, normalized size of antiderivative = 16.48 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2024 vs. \(2 (476) = 952\).
Time = 1.00 (sec) , antiderivative size = 2024, normalized size of antiderivative = 3.90 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]
-1/32*(4*sqrt(2)*(35*sqrt(2)*B*sqrt(a)*c^3*d^2 - 63*sqrt(2)*A*sqrt(a)*c^2* d^3 + 70*sqrt(2)*B*sqrt(a)*c^2*d^3 - 90*sqrt(2)*A*sqrt(a)*c*d^4 + 67*sqrt( 2)*B*sqrt(a)*c*d^4 - 39*sqrt(2)*A*sqrt(a)*d^5 + 20*sqrt(2)*B*sqrt(a)*d^5)* arctan(sqrt(2)*d*sin(-1/4*pi + 1/2*f*x + 1/2*e)/sqrt(-c*d - d^2))/((a^3*c^ 7*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 3*a^3*c^6*d*sgn(cos(-1/4*pi + 1/2* f*x + 1/2*e)) + a^3*c^5*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*a^3*c^ 4*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 5*a^3*c^3*d^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - a^3*c^2*d^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 3* a^3*c*d^6*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - a^3*d^7*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 - 30*A*sqrt(a)*c*d - 82*B*sqrt(a)*c*d + 219*A*sqrt(a)*d^2 - 115*B*sqrt(a)* d^2)*log(sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(sqrt(2)*a^3*c^5*sgn(cos(-1/4 *pi + 1/2*f*x + 1/2*e)) - 5*sqrt(2)*a^3*c^4*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 10*sqrt(2)*a^3*c^3*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 10* sqrt(2)*a^3*c^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*sqrt(2)*a^3*c* d^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - sqrt(2)*a^3*d^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) + (3*A*sqrt(a)*c^2 + 5*B*sqrt(a)*c^2 - 30*A*sqrt(a)*c *d - 82*B*sqrt(a)*c*d + 219*A*sqrt(a)*d^2 - 115*B*sqrt(a)*d^2)*log(-sin(-1 /4*pi + 1/2*f*x + 1/2*e) + 1)/(sqrt(2)*a^3*c^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 5*sqrt(2)*a^3*c^4*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 10...
Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\int \frac {A+B\,\sin \left (e+f\,x\right )}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]