Integrand size = 27, antiderivative size = 400 \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=-\frac {3 \left (c^2-10 c d+73 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} (c-d)^5 f}+\frac {3 d^{5/2} \left (21 c^2+30 c d+13 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{4 a^{5/2} (c-d)^5 (c+d)^{5/2} f}-\frac {\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 c-19 d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \] Output:
-3/32*(c^2-10*c*d+73*d^2)*arctanh(1/2*a^(1/2)*cos(f*x+e)*2^(1/2)/(a+a*sin( f*x+e))^(1/2))*2^(1/2)/a^(5/2)/(c-d)^5/f+3/4*d^(5/2)*(21*c^2+30*c*d+13*d^2 )*arctanh(a^(1/2)*d^(1/2)*cos(f*x+e)/(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*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*c-19*d)*cos(f*x+e)/a/(c-d)^2/f/(a+a*sin(f*x+e) )^(3/2)/(c+d*sin(f*x+e))^2-1/16*d*(3*c^2-20*c*d-31*d^2)*cos(f*x+e)/a^2/(c- d)^3/(c+d)/f/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^2-3/16*d*(c+3*d)*(c^2 -10*c*d-7*d^2)*cos(f*x+e)/a^2/(c-d)^4/(c+d)^2/f/(a+a*sin(f*x+e))^(1/2)/(c+ d*sin(f*x+e))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 12.70 (sec) , antiderivative size = 1045, normalized size of antiderivative = 2.61 \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:
Integrate[1/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3),x]
Output:
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((8*Sin[(e + f*x)/2])/(c - d)^3 - ( 4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(c - d)^3 + (6*(c - 9*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)/(c - d)^4 - (3*(c - 9*d) *(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/(c - d)^4 + ((3 + 3*I)*(-1)^(3/4 )*(c^2 - 10*c*d + 73*d^2)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f* x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/(c - d)^5 + (3*d^(5/2)*(2 1*c^2 + 30*c*d + 13*d^2)*(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])^4)/((c - d)^5*(c + d)^(5/2)) + (3*d^(5/2)*(21*c^2 + 30*c*d + 13*d^2)*(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 + ...
Time = 2.59 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.14, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.704, Rules used = {3042, 3245, 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.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^3}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle -\frac {\int -\frac {3 a (c-4 d)+7 a d \sin (e+f x)}{2 (\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^3}dx}{4 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 a (c-4 d)+7 a d \sin (e+f x)}{(\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^3}dx}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 a (c-4 d)+7 a d \sin (e+f x)}{(\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^3}dx}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {-\frac {\int -\frac {\left (3 c^2-15 d c+124 d^2\right ) a^2+5 (3 c-19 d) d \sin (e+f x) a^2}{2 \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3}dx}{2 a^2 (c-d)}-\frac {a (3 c-19 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (3 c^2-15 d c+124 d^2\right ) a^2+5 (3 c-19 d) d \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3}dx}{4 a^2 (c-d)}-\frac {a (3 c-19 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\left (3 c^2-15 d c+124 d^2\right ) a^2+5 (3 c-19 d) d \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^3}dx}{4 a^2 (c-d)}-\frac {a (3 c-19 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {6 \left (\left (c^3-6 d c^2+43 d^2 c+42 d^3\right ) a^3+d \left (3 c^2-20 d c-31 d^2\right ) \sin (e+f x) a^3\right )}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{2 a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 c-19 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {\left (c^3-6 d c^2+43 d^2 c+42 d^3\right ) a^3+d \left (3 c^2-20 d c-31 d^2\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 c-19 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {\left (c^3-6 d c^2+43 d^2 c+42 d^3\right ) a^3+d \left (3 c^2-20 d c-31 d^2\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2}dx}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 c-19 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3463 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\int -\frac {\left (c^4-7 d c^3+47 d^2 c^2+99 d^3 c+52 d^4\right ) a^4+d (c+3 d) \left (c^2-10 d c-7 d^2\right ) \sin (e+f x) a^4}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {2 a^3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}\right )}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 c-19 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \frac {\left (c^4-7 d c^3+47 d^2 c^2+99 d^3 c+52 d^4\right ) a^4+d (c+3 d) \left (c^2-10 d c-7 d^2\right ) \sin (e+f x) a^4}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {2 a^3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}\right )}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 c-19 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \frac {\left (c^4-7 d c^3+47 d^2 c^2+99 d^3 c+52 d^4\right ) a^4+d (c+3 d) \left (c^2-10 d c-7 d^2\right ) \sin (e+f x) a^4}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))}dx}{a \left (c^2-d^2\right )}-\frac {2 a^3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}\right )}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 c-19 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3464 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {a^4 (c+d)^2 \left (c^2-10 c d+73 d^2\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{c-d}-\frac {4 a^3 d^3 \left (21 c^2+30 c d+13 d^2\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}\right )}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 c-19 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {a^4 (c+d)^2 \left (c^2-10 c d+73 d^2\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a}}dx}{c-d}-\frac {4 a^3 d^3 \left (21 c^2+30 c d+13 d^2\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}\right )}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 c-19 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {-\frac {4 a^3 d^3 \left (21 c^2+30 c d+13 d^2\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}-\frac {2 a^4 (c+d)^2 \left (c^2-10 c d+73 d^2\right ) \int \frac {1}{2 a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}\right )}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 c-19 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {-\frac {4 a^3 d^3 \left (21 c^2+30 c d+13 d^2\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{c+d \sin (e+f x)}dx}{c-d}-\frac {\sqrt {2} a^{7/2} (c+d)^2 \left (c^2-10 c d+73 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}\right )}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 c-19 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {8 a^4 d^3 \left (21 c^2+30 c d+13 d^2\right ) \int \frac {1}{a (c+d)-\frac {a^2 d \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f (c-d)}-\frac {\sqrt {2} a^{7/2} (c+d)^2 \left (c^2-10 c d+73 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}\right )}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 c-19 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {8 a^{7/2} d^{5/2} \left (21 c^2+30 c d+13 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d) \sqrt {c+d}}-\frac {\sqrt {2} a^{7/2} (c+d)^2 \left (c^2-10 c d+73 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{f (c-d)}}{a \left (c^2-d^2\right )}-\frac {2 a^3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}\right )}{a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}}{4 a^2 (c-d)}-\frac {a (3 c-19 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}\) |
Input:
Int[1/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3),x]
Output:
-1/4*Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x ])^2) + (-1/2*(a*(3*c - 19*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*(3*c^2 - 20*c*d - 31*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) + (3*((-((Sqrt[2]*a^(7/2)*(c + d)^2*(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^(5/2)*(21*c^2 + 30*c*d + 13*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^3*d*(c + 3*d)*(c^2 - 10*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)))/(8*a^2*(c - d))
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[((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*((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*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ [a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && !GtQ[n, 0] && (Intege rsQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 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. \(3534\) vs. \(2(357)=714\).
Time = 1.23 (sec) , antiderivative size = 3535, normalized size of antiderivative = 8.84
Input:
int(1/(a+sin(f*x+e)*a)^(5/2)/(c+d*sin(f*x+e))^3,x,method=_RETURNVERBOSE)
Output:
-1/32*(-a*(-1+sin(f*x+e)))^(1/2)*(-2064*arctanh((-a*(-1+sin(f*x+e)))^(1/2) *d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^3*c*d^6-504*arctanh((-a*(-1+sin(f *x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^2*c^4*d^3-1008*arcta nh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)*c^4* d^3-2448*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*s in(f*x+e)*c^3*d^4-2064*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1 /2))*a^(5/2)*sin(f*x+e)*c^2*d^5-624*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/( a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)*c*d^6+112*(-a*(-1+sin(f*x+e)))^(1/2)* (a*(c+d)*d)^(1/2)*a^(3/2)*sin(f*x+e)*d^6-144*(-a*(-1+sin(f*x+e)))^(3/2)*(a *(c+d)*d)^(1/2)*a^(1/2)*sin(f*x+e)*d^6+48*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c +d)*d)^(1/2)*a^(1/2)*c^5*d+60*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2) *a^(1/2)*c^4*d^2-126*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*a^(1/2)* sin(f*x+e)^2*d^6-40*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c ^3*d^3+60*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c^2*d^4+136 *(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a^(3/2)*c*d^5-2736*arctanh(( -a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^2*c^3*d^ 4-3696*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^(5/2)*sin (f*x+e)^2*c^2*d^5-1968*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1 /2))*a^(5/2)*sin(f*x+e)^2*c*d^6-504*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/( a*(c+d)*d)^(1/2))*a^(5/2)*sin(f*x+e)^4*c^2*d^5-720*arctanh((-a*(-1+sin(...
Leaf count of result is larger than twice the leaf count of optimal. 2857 vs. \(2 (357) = 714\).
Time = 1.79 (sec) , antiderivative size = 5999, normalized size of antiderivative = 15.00 \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, algorithm="fricas ")
Output:
Too large to include
Timed out. \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:
integrate(1/(a+a*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**3,x)
Output:
Timed out
Timed out. \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:
integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, algorithm="maxima ")
Output:
Timed out
Exception generated. \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3} \,d x \] Input:
int(1/((a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^3),x)
Output:
int(1/((a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^3), x)
\[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )^{6} d^{3}+3 \sin \left (f x +e \right )^{5} c \,d^{2}+3 \sin \left (f x +e \right )^{5} d^{3}+3 \sin \left (f x +e \right )^{4} c^{2} d +9 \sin \left (f x +e \right )^{4} c \,d^{2}+3 \sin \left (f x +e \right )^{4} d^{3}+\sin \left (f x +e \right )^{3} c^{3}+9 \sin \left (f x +e \right )^{3} c^{2} d +9 \sin \left (f x +e \right )^{3} c \,d^{2}+\sin \left (f x +e \right )^{3} d^{3}+3 \sin \left (f x +e \right )^{2} c^{3}+9 \sin \left (f x +e \right )^{2} c^{2} d +3 \sin \left (f x +e \right )^{2} c \,d^{2}+3 \sin \left (f x +e \right ) c^{3}+3 \sin \left (f x +e \right ) c^{2} d +c^{3}}d x \right )}{a^{3}} \] Input:
int(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x)
Output:
(sqrt(a)*int(sqrt(sin(e + f*x) + 1)/(sin(e + f*x)**6*d**3 + 3*sin(e + f*x) **5*c*d**2 + 3*sin(e + f*x)**5*d**3 + 3*sin(e + f*x)**4*c**2*d + 9*sin(e + f*x)**4*c*d**2 + 3*sin(e + f*x)**4*d**3 + sin(e + f*x)**3*c**3 + 9*sin(e + f*x)**3*c**2*d + 9*sin(e + f*x)**3*c*d**2 + sin(e + f*x)**3*d**3 + 3*sin (e + f*x)**2*c**3 + 9*sin(e + f*x)**2*c**2*d + 3*sin(e + f*x)**2*c*d**2 + 3*sin(e + f*x)*c**3 + 3*sin(e + f*x)*c**2*d + c**3),x))/a**3