Integrand size = 25, antiderivative size = 530 \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx=-\frac {7 a b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{11/4} d e^{5/2}}-\frac {7 a b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{11/4} d e^{5/2}}-\frac {b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)}{3 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{3/2}}+\frac {\left (2 a^2+5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 \left (a^2-b^2\right )^2 d e^2 \sqrt {e \sin (c+d x)}}-\frac {7 a^2 b^2 \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 \left (a^2-b^2\right )^2 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d e^2 \sqrt {e \sin (c+d x)}}-\frac {7 a^2 b^2 \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 \left (a^2-b^2\right )^2 \left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) d e^2 \sqrt {e \sin (c+d x)}} \]
-7/2*a*b^(5/2)*arctan(b^(1/2)*(e*sin(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2 ))/(-a^2+b^2)^(11/4)/d/e^(5/2)-7/2*a*b^(5/2)*arctanh(b^(1/2)*(e*sin(d*x+c) )^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/(-a^2+b^2)^(11/4)/d/e^(5/2)-b/(a^2-b^2)/ d/e/(a+b*cos(d*x+c))/(e*sin(d*x+c))^(3/2)+1/3*(7*a*b-(2*a^2+5*b^2)*cos(d*x +c))/(a^2-b^2)^2/d/e/(e*sin(d*x+c))^(3/2)-1/3*(2*a^2+5*b^2)*(sin(1/2*c+1/4 *Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticF(cos(1/2*c+1/4*Pi +1/2*d*x),2^(1/2))*sin(d*x+c)^(1/2)/(a^2-b^2)^2/d/e^2/(e*sin(d*x+c))^(1/2) +7/2*a^2*b^2*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x) *EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*b/(b-(-a^2+b^2)^(1/2)),2^(1/2))*si n(d*x+c)^(1/2)/(a^2-b^2)^2/d/e^2/(a^2-b*(b-(-a^2+b^2)^(1/2)))/(e*sin(d*x+c ))^(1/2)+7/2*a^2*b^2*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+ 1/2*d*x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*b/(b+(-a^2+b^2)^(1/2)),2^( 1/2))*sin(d*x+c)^(1/2)/(a^2-b^2)^2/d/e^2/(a^2-b*(b+(-a^2+b^2)^(1/2)))/(e*s in(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.81 (sec) , antiderivative size = 1257, normalized size of antiderivative = 2.37 \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx=\frac {\left (\frac {b^3}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))}-\frac {2 \left (-2 a b+a^2 \cos (c+d x)+b^2 \cos (c+d x)\right ) \csc ^2(c+d x)}{3 \left (a^2-b^2\right )^2}\right ) \sin ^3(c+d x)}{d (e \sin (c+d x))^{5/2}}+\frac {\sin ^{\frac {5}{2}}(c+d x) \left (\frac {2 \left (2 a^2 b+5 b^3\right ) \cos ^2(c+d x) \left (a+b \sqrt {1-\sin ^2(c+d x)}\right ) \left (\frac {a \left (-2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )\right )}{4 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4}}+\frac {5 b \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{2},1,\frac {5}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sqrt {\sin (c+d x)} \sqrt {1-\sin ^2(c+d x)}}{\left (-5 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{2},1,\frac {5}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right )+2 \left (2 b^2 \operatorname {AppellF1}\left (\frac {5}{4},-\frac {1}{2},2,\frac {9}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right )+\left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right )\right ) \sin ^2(c+d x)\right ) \left (a^2+b^2 \left (-1+\sin ^2(c+d x)\right )\right )}\right )}{(a+b \cos (c+d x)) \left (1-\sin ^2(c+d x)\right )}+\frac {2 \left (2 a^3-16 a b^2\right ) \cos (c+d x) \left (a+b \sqrt {1-\sin ^2(c+d x)}\right ) \left (-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \sqrt {b} \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )+\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )-\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )\right )}{\left (-a^2+b^2\right )^{3/4}}+\frac {5 a \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sqrt {\sin (c+d x)}}{\sqrt {1-\sin ^2(c+d x)} \left (5 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right )-2 \left (2 b^2 \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right )+\left (-a^2+b^2\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right )\right ) \sin ^2(c+d x)\right ) \left (a^2+b^2 \left (-1+\sin ^2(c+d x)\right )\right )}\right )}{(a+b \cos (c+d x)) \sqrt {1-\sin ^2(c+d x)}}\right )}{6 (a-b)^2 (a+b)^2 d (e \sin (c+d x))^{5/2}} \]
((b^3/((a^2 - b^2)^2*(a + b*Cos[c + d*x])) - (2*(-2*a*b + a^2*Cos[c + d*x] + b^2*Cos[c + d*x])*Csc[c + d*x]^2)/(3*(a^2 - b^2)^2))*Sin[c + d*x]^3)/(d *(e*Sin[c + d*x])^(5/2)) + (Sin[c + d*x]^(5/2)*((2*(2*a^2*b + 5*b^3)*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqr t[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b ]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*S qrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a ^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*Appell F1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*S qrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b ^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[ c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2 )))))/((a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(2*a^3 - 16*a*b^2)* Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[b]*(2*Ar cTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcT an[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt [-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + ...
Time = 2.42 (sec) , antiderivative size = 513, normalized size of antiderivative = 0.97, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.840, Rules used = {3042, 3173, 27, 3042, 3345, 27, 3042, 3346, 3042, 3121, 3042, 3120, 3181, 266, 756, 218, 221, 3042, 3286, 3042, 3284}
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.
| \(\displaystyle \int \frac {1}{(e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{5/2} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 3173 |
| \(\displaystyle -\frac {\int -\frac {2 a-5 b \cos (c+d x)}{2 (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}dx}{a^2-b^2}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 a-5 b \cos (c+d x)}{(a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}dx}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {2 a+5 b \sin \left (c+d x-\frac {\pi }{2}\right )}{\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{5/2} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}-\frac {2 \int -\frac {2 a \left (a^2-8 b^2\right )+b \left (2 a^2+5 b^2\right ) \cos (c+d x)}{2 (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}dx}{3 e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {2 a \left (a^2-8 b^2\right )+b \left (2 a^2+5 b^2\right ) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}dx}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {2 a \left (a^2-8 b^2\right )-b \left (2 a^2+5 b^2\right ) \sin \left (c+d x-\frac {\pi }{2}\right )}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3346 |
| \(\displaystyle \frac {\frac {\left (2 a^2+5 b^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}}dx-21 a b^2 \int \frac {1}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}dx}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (2 a^2+5 b^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}}dx-21 a b^2 \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {\frac {\frac {\left (2 a^2+5 b^2\right ) \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{\sqrt {e \sin (c+d x)}}-21 a b^2 \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\left (2 a^2+5 b^2\right ) \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{\sqrt {e \sin (c+d x)}}-21 a b^2 \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 a^2+5 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {e \sin (c+d x)}}-21 a b^2 \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3181 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 a^2+5 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {e \sin (c+d x)}}-21 a b^2 \left (-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b^2 \sin ^2(c+d x) e^2+\left (a^2-b^2\right ) e^2\right )}d(e \sin (c+d x))}{d}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 a^2+5 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {e \sin (c+d x)}}-21 a b^2 \left (-\frac {2 b e \int \frac {1}{b^2 e^4 \sin ^4(c+d x)+\left (a^2-b^2\right ) e^2}d\sqrt {e \sin (c+d x)}}{d}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 a^2+5 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {e \sin (c+d x)}}-21 a b^2 \left (-\frac {2 b e \left (-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 e \sqrt {b^2-a^2}}-\frac {\int \frac {1}{b e^2 \sin ^2(c+d x)+\sqrt {b^2-a^2} e}d\sqrt {e \sin (c+d x)}}{2 e \sqrt {b^2-a^2}}\right )}{d}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 a^2+5 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {e \sin (c+d x)}}-21 a b^2 \left (-\frac {2 b e \left (-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 e \sqrt {b^2-a^2}}-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 a^2+5 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {e \sin (c+d x)}}-21 a b^2 \left (-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 a^2+5 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {e \sin (c+d x)}}-21 a b^2 \left (-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 a^2+5 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {e \sin (c+d x)}}-21 a b^2 \left (-\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \sin (c+d x)}}-\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \sin (c+d x)}}-\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 a^2+5 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {e \sin (c+d x)}}-21 a b^2 \left (-\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \sin (c+d x)}}-\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \sin (c+d x)}}-\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 a^2+5 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {e \sin (c+d x)}}-21 a b^2 \left (-\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}+\frac {a \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {b^2-a^2} \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \sin (c+d x)}}-\frac {a \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {b^2-a^2} \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \sin (c+d x)}}\right )}{3 e^2 \left (a^2-b^2\right )}+\frac {2 \left (7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)\right )}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}}{2 \left (a^2-b^2\right )}-\frac {b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}\) |
-(b/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))) + ((2*( 7*a*b - (2*a^2 + 5*b^2)*Cos[c + d*x]))/(3*(a^2 - b^2)*d*e*(e*Sin[c + d*x]) ^(3/2)) + ((2*(2*a^2 + 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(d*Sqrt[e*Sin[c + d*x]]) - 21*a*b^2*((-2*b*e*(-1/2*ArcTan[(Sqrt[b ]*Sqrt[e]*Sin[c + d*x])/(-a^2 + b^2)^(1/4)]/(Sqrt[b]*(-a^2 + b^2)^(3/4)*e^ (3/2)) - ArcTanh[(Sqrt[b]*Sqrt[e]*Sin[c + d*x])/(-a^2 + b^2)^(1/4)]/(2*Sqr t[b]*(-a^2 + b^2)^(3/4)*e^(3/2))))/d + (a*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(Sqrt[-a^2 + b^2]*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) - (a*EllipticPi[(2*b)/(b + Sqrt [-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(Sqrt[-a^2 + b^2 ]*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]])))/(3*(a^2 - b^2)*e^2))/(2 *(a^2 - b^2))
3.1.76.3.1 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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + p + 2)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b ^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]*((a_) + (b_.)*sin[(e_.) + (f_.)* (x_)])), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Simp[-a/(2*q) Int[1/( Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (Simp[b*(g/f) Subst[ Int[1/(Sqrt[x]*(g^2*(a^2 - b^2) + b^2*x^2)), x], x, g*Cos[e + f*x]], x] - S imp[a/(2*q) Int[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x], x])] / ; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt Q[p, -1] && IntegerQ[2*m]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((c_.) + (d_.)*sin[(e_.) + (f_.)* (x_)]))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d/b Int [(g*Cos[e + f*x])^p, x], x] + Simp[(b*c - a*d)/b Int[(g*Cos[e + f*x])^p/( a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1473\) vs. \(2(556)=1112\).
Time = 6.30 (sec) , antiderivative size = 1474, normalized size of antiderivative = 2.78
(-4*e^3*a*b*(-1/e^4/(a-b)^2/(a+b)^2*b^2*(1/4*(e*sin(d*x+c))^(1/2)/(-b^2*co s(d*x+c)^2*e^2+a^2*e^2)+7/32*(e^2*(a^2-b^2)/b^2)^(1/4)/(a^2*e^2-b^2*e^2)*2 ^(1/2)*(ln((e*sin(d*x+c)+(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)*2^ (1/2)+(e^2*(a^2-b^2)/b^2)^(1/2))/(e*sin(d*x+c)-(e^2*(a^2-b^2)/b^2)^(1/4)*( e*sin(d*x+c))^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2)))+2*arctan(2^(1/2)/( e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)+1)+2*arctan(2^(1/2)/(e^2*(a^ 2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)-1)))-1/3/e^4/(a^2-b^2)^2/(e*sin(d*x +c))^(3/2))-(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/e^2*(1/3*(-a^2-b^2)/(a^2-b^2 )^2/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(cos(d*x+c)^2-1)*((1-sin(d*x+c))^(1/ 2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(5/2)*EllipticF((1-sin(d*x+c))^(1/2), 1/2*2^(1/2))+2*cos(d*x+c)^2*sin(d*x+c))+2*a^2*b^2/(a-b)/(a+b)*(1/2*b^2/e/a ^2/(a^2-b^2)*(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(-b^2*cos(d*x+c)^2+a^2)+1/4 /a^2/(a^2-b^2)*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2 )/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^( 1/2))-5/8/(a^2-b^2)/(-a^2+b^2)^(1/2)/b*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+ 2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(1-(-a^2+b^2)^ (1/2)/b)*EllipticPi((1-sin(d*x+c))^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*2^(1 /2))+1/4/a^2/(a^2-b^2)/(-a^2+b^2)^(1/2)*b*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+ c)+2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(1-(-a^2+b^ 2)^(1/2)/b)*EllipticPi((1-sin(d*x+c))^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/...
Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx=\int \frac {1}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2} \,d x \]