Integrand size = 29, antiderivative size = 338 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx=-\frac {\left (3 c^2-26 c d+163 d^2\right ) \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {c-d} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{144 \sqrt {6} (c-d)^{9/2} f}-\frac {\cos (e+f x)}{4 (c-d) f (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}}-\frac {(3 c-17 d) \cos (e+f x)}{48 (c-d)^2 f (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}-\frac {d \left (9 c^2-54 c d-95 d^2\right ) \cos (e+f x)}{432 (c-d)^3 (c+d) f \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {d \left (9 c^3-57 c^2 d-493 c d^2-299 d^3\right ) \cos (e+f x)}{432 (c-d)^4 (c+d)^2 f \sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]
-1/4*cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2)-1/16 *(3*c-17*d)*cos(f*x+e)/a/(c-d)^2/f/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e)) ^(3/2)-1/32*(3*c^2-26*c*d+163*d^2)*arctanh(1/2*cos(f*x+e)*a^(1/2)*(c-d)^(1 /2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2))/a^(5/2)/(c-d)^( 9/2)/f*2^(1/2)-1/48*d*(9*c^2-54*c*d-95*d^2)*cos(f*x+e)/a^2/(c-d)^3/(c+d)/f /(c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)-1/48*d*(9*c^3-57*c^2*d-493* c*d^2-299*d^3)*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))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(717\) vs. \(2(338)=676\).
Time = 12.59 (sec) , antiderivative size = 717, normalized size of antiderivative = 2.12 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \sqrt {c+d \sin (e+f x)} \left (\frac {\sin \left (\frac {1}{2} (e+f x)\right )}{2 (c-d)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {1}{4 (c-d)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {-3 c+25 d}{16 (c-d)^4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {3 c \sin \left (\frac {1}{2} (e+f x)\right )-25 d \sin \left (\frac {1}{2} (e+f x)\right )}{8 (c-d)^4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {2 \left (d^3 \cos \left (\frac {1}{2} (e+f x)\right )-d^3 \sin \left (\frac {1}{2} (e+f x)\right )\right )}{3 (c-d)^3 (c+d) (c+d \sin (e+f x))^2}+\frac {2 \left (11 c d^3 \cos \left (\frac {1}{2} (e+f x)\right )+7 d^4 \cos \left (\frac {1}{2} (e+f x)\right )-11 c d^3 \sin \left (\frac {1}{2} (e+f x)\right )-7 d^4 \sin \left (\frac {1}{2} (e+f x)\right )\right )}{3 (c-d)^4 (c+d)^2 (c+d \sin (e+f x))}\right )}{f (3+3 \sin (e+f x))^{5/2}}+\frac {\left (3 c^2-26 c d+163 d^2\right ) \left (\log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}{32 (c-d)^4 f (3+3 \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \left (\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right )}{2+2 \tan \left (\frac {1}{2} (e+f x)\right )}-\frac {-\frac {1}{2} (c-d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {\sqrt {c-d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} (d+d \cos (e+f x)+c \sin (e+f x))}{\sqrt {c+d \sin (e+f x)}}}{c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}\right )} \]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sqrt[c + d*Sin[e + f*x]]*(Sin[(e + f*x)/2]/(2*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - 1/(4*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3) + (-3*c + 25*d)/(16*(c - d )^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) + (3*c*Sin[(e + f*x)/2] - 25*d* Sin[(e + f*x)/2])/(8*(c - d)^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) + (2*(d^3*Cos[(e + f*x)/2] - d^3*Sin[(e + f*x)/2]))/(3*(c - d)^3*(c + d)*(c + d*Sin[e + f*x])^2) + (2*(11*c*d^3*Cos[(e + f*x)/2] + 7*d^4*Cos[(e + f*x) /2] - 11*c*d^3*Sin[(e + f*x)/2] - 7*d^4*Sin[(e + f*x)/2]))/(3*(c - d)^4*(c + d)^2*(c + d*Sin[e + f*x]))))/(f*(3 + 3*Sin[e + f*x])^(5/2)) + ((3*c^2 - 26*c*d + 163*d^2)*(Log[1 + Tan[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]* Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/(32*(c - d)^4*f*(3 + 3 *Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]]*(Sec[(e + f*x)/2]^2/(2 + 2*T an[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqr t[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2])))
Time = 1.88 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.22, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {3042, 3245, 27, 3042, 3457, 27, 3042, 3463, 27, 3042, 3463, 27, 3042, 3261, 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.
\(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{5/2}}dx\) |
\(\Big \downarrow \) 3245 |
\(\displaystyle -\frac {\int -\frac {a (3 c-11 d)+6 a d \sin (e+f x)}{2 (\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^{5/2}}dx}{4 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {a (3 c-11 d)+6 a d \sin (e+f x)}{(\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^{5/2}}dx}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {a (3 c-11 d)+6 a d \sin (e+f x)}{(\sin (e+f x) a+a)^{3/2} (c+d \sin (e+f x))^{5/2}}dx}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3457 |
\(\displaystyle \frac {-\frac {\int -\frac {\left (3 c^2-14 d c+95 d^2\right ) a^2+4 (3 c-17 d) d \sin (e+f x) a^2}{2 \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{5/2}}dx}{2 a^2 (c-d)}-\frac {a (3 c-17 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\int \frac {\left (3 c^2-14 d c+95 d^2\right ) a^2+4 (3 c-17 d) d \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{5/2}}dx}{4 a^2 (c-d)}-\frac {a (3 c-17 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {\left (3 c^2-14 d c+95 d^2\right ) a^2+4 (3 c-17 d) d \sin (e+f x) a^2}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{5/2}}dx}{4 a^2 (c-d)}-\frac {a (3 c-17 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3463 |
\(\displaystyle \frac {\frac {-\frac {2 \int -\frac {\left (9 c^3-51 d c^2+303 d^2 c+299 d^3\right ) a^3+2 d \left (9 c^2-54 d c-95 d^2\right ) \sin (e+f x) a^3}{2 \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{3/2}}dx}{3 a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (9 c^2-54 c d-95 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}}{4 a^2 (c-d)}-\frac {a (3 c-17 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {\int \frac {\left (9 c^3-51 d c^2+303 d^2 c+299 d^3\right ) a^3+2 d \left (9 c^2-54 d c-95 d^2\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{3/2}}dx}{3 a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (9 c^2-54 c d-95 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}}{4 a^2 (c-d)}-\frac {a (3 c-17 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {\int \frac {\left (9 c^3-51 d c^2+303 d^2 c+299 d^3\right ) a^3+2 d \left (9 c^2-54 d c-95 d^2\right ) \sin (e+f x) a^3}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{3/2}}dx}{3 a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (9 c^2-54 c d-95 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}}{4 a^2 (c-d)}-\frac {a (3 c-17 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3463 |
\(\displaystyle \frac {\frac {\frac {-\frac {2 \int -\frac {3 a^4 (c+d)^2 \left (3 c^2-26 d c+163 d^2\right )}{2 \sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}dx}{a \left (c^2-d^2\right )}-\frac {2 a^3 d \left (9 c^3-57 c^2 d-493 c d^2-299 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}}{3 a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (9 c^2-54 c d-95 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}}{4 a^2 (c-d)}-\frac {a (3 c-17 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {\frac {3 a^3 (c+d)^2 \left (3 c^2-26 c d+163 d^2\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (9 c^3-57 c^2 d-493 c d^2-299 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}}{3 a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (9 c^2-54 c d-95 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}}{4 a^2 (c-d)}-\frac {a (3 c-17 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {\frac {3 a^3 (c+d)^2 \left (3 c^2-26 c d+163 d^2\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (9 c^3-57 c^2 d-493 c d^2-299 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}}{3 a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (9 c^2-54 c d-95 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}}{4 a^2 (c-d)}-\frac {a (3 c-17 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3261 |
\(\displaystyle \frac {\frac {\frac {-\frac {6 a^4 (c+d)^2 \left (3 c^2-26 c d+163 d^2\right ) \int \frac {1}{2 a^2-\frac {a^3 (c-d) \cos ^2(e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}}{f \left (c^2-d^2\right )}-\frac {2 a^3 d \left (9 c^3-57 c^2 d-493 c d^2-299 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}}{3 a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (9 c^2-54 c d-95 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}}{4 a^2 (c-d)}-\frac {a (3 c-17 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {\frac {-\frac {3 \sqrt {2} a^{5/2} (c+d)^2 \left (3 c^2-26 c d+163 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{f \sqrt {c-d} \left (c^2-d^2\right )}-\frac {2 a^3 d \left (9 c^3-57 c^2 d-493 c d^2-299 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}}{3 a \left (c^2-d^2\right )}-\frac {2 a^2 d \left (9 c^2-54 c d-95 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}}{4 a^2 (c-d)}-\frac {a (3 c-17 d) \cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}}{8 a^2 (c-d)}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}\) |
-1/4*Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x ])^(3/2)) + (-1/2*(a*(3*c - 17*d)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2)) + ((-2*a^2*d*(9*c^2 - 54*c*d - 95* d^2)*Cos[e + f*x])/(3*(c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) + ((-3*Sqrt[2]*a^(5/2)*(c + d)^2*(3*c^2 - 26*c*d + 163*d^2) *ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f* x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[c - d]*(c^2 - d^2)*f) - (2*a^3*d*(9* c^3 - 57*c^2*d - 493*c*d^2 - 299*d^3)*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]))/(3*a*(c^2 - d^2)))/(4*a^2*(c - d)))/(8*a^2*(c - d))
3.7.5.3.1 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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(a/f) Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*S in[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])
Leaf count of result is larger than twice the leaf count of optimal. \(8008\) vs. \(2(314)=628\).
Time = 5.06 (sec) , antiderivative size = 8009, normalized size of antiderivative = 23.70
Leaf count of result is larger than twice the leaf count of optimal. 2312 vs. \(2 (314) = 628\).
Time = 1.96 (sec) , antiderivative size = 4858, normalized size of antiderivative = 14.37 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
[1/384*(3*(12*c^6 - 56*c^5*d + 308*c^4*d^2 + 2032*c^3*d^3 + 3508*c^2*d^4 + 2504*c*d^5 + 652*d^6 + (3*c^4*d^2 - 20*c^3*d^3 + 114*c^2*d^4 + 300*c*d^5 + 163*d^6)*cos(f*x + e)^5 + (6*c^5*d - 31*c^4*d^2 + 168*c^3*d^3 + 942*c^2* d^4 + 1226*c*d^5 + 489*d^6)*cos(f*x + e)^4 - (3*c^6 - 8*c^5*d + 43*c^4*d^2 + 696*c^3*d^3 + 1705*c^2*d^4 + 1552*c*d^5 + 489*d^6)*cos(f*x + e)^3 - (9* c^6 - 30*c^5*d + 163*c^4*d^2 + 1900*c^3*d^3 + 4287*c^2*d^4 + 3730*c*d^5 + 1141*d^6)*cos(f*x + e)^2 + 2*(3*c^6 - 14*c^5*d + 77*c^4*d^2 + 508*c^3*d^3 + 877*c^2*d^4 + 626*c*d^5 + 163*d^6)*cos(f*x + e) + (12*c^6 - 56*c^5*d + 3 08*c^4*d^2 + 2032*c^3*d^3 + 3508*c^2*d^4 + 2504*c*d^5 + 652*d^6 + (3*c^4*d ^2 - 20*c^3*d^3 + 114*c^2*d^4 + 300*c*d^5 + 163*d^6)*cos(f*x + e)^4 - 2*(3 *c^5*d - 17*c^4*d^2 + 94*c^3*d^3 + 414*c^2*d^4 + 463*c*d^5 + 163*d^6)*cos( f*x + e)^3 - (3*c^6 - 2*c^5*d + 9*c^4*d^2 + 884*c^3*d^3 + 2533*c^2*d^4 + 2 478*c*d^5 + 815*d^6)*cos(f*x + e)^2 + 2*(3*c^6 - 14*c^5*d + 77*c^4*d^2 + 5 08*c^3*d^3 + 877*c^2*d^4 + 626*c*d^5 + 163*d^6)*cos(f*x + e))*sin(f*x + e) )*sqrt(2*a*c - 2*a*d)*log(((a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^3 - 4*a*c^2 - 8*a*c*d - 4*a*d^2 - (13*a*c^2 - 22*a*c*d - 3*a*d^2)*cos(f*x + e) ^2 - 4*((c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos (f*x + e) + 4*c - 4*d)*sin(f*x + e) - 4*c + 4*d)*sqrt(2*a*c - 2*a*d)*sqrt( a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) - 2*(9*a*c^2 - 14*a*c*d + 9*a *d^2)*cos(f*x + e) - (4*a*c^2 + 8*a*c*d + 4*a*d^2 - (a*c^2 - 14*a*c*d +...
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]