Integrand size = 25, antiderivative size = 520 \[ \int \frac {(e \sin (c+d x))^{5/2}}{(a+b \cos (c+d x))^3} \, dx=-\frac {3 \left (a^2-2 b^2\right ) e^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{5/2} \left (-a^2+b^2\right )^{5/4} d}+\frac {3 \left (a^2-2 b^2\right ) e^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{5/2} \left (-a^2+b^2\right )^{5/4} d}-\frac {3 a \left (a^2-2 b^2\right ) e^3 \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 b^3 \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {3 a \left (a^2-2 b^2\right ) e^3 \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 b^3 \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {3 a e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{4 b^2 \left (a^2-b^2\right ) d \sqrt {\sin (c+d x)}}+\frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 a e (e \sin (c+d x))^{3/2}}{4 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))} \]
-3/8*(a^2-2*b^2)*e^(5/2)*arctan(b^(1/2)*(e*sin(d*x+c))^(1/2)/(-a^2+b^2)^(1 /4)/e^(1/2))/b^(5/2)/(-a^2+b^2)^(5/4)/d+3/8*(a^2-2*b^2)*e^(5/2)*arctanh(b^ (1/2)*(e*sin(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/b^(5/2)/(-a^2+b^2)^(5 /4)/d+1/2*e*(e*sin(d*x+c))^(3/2)/b/d/(a+b*cos(d*x+c))^2-3/4*a*e*(e*sin(d*x +c))^(3/2)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))+3/8*a*(a^2-2*b^2)*e^3*(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)/b^3/(a^2 -b^2)/d/(b-(-a^2+b^2)^(1/2))/(e*sin(d*x+c))^(1/2)+3/8*a*(a^2-2*b^2)*e^3*(s in(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)/ b^3/(a^2-b^2)/d/(b+(-a^2+b^2)^(1/2))/(e*sin(d*x+c))^(1/2)-3/4*a*e^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)*EllipticE(cos(1/2* c+1/4*Pi+1/2*d*x),2^(1/2))*(e*sin(d*x+c))^(1/2)/b^2/(a^2-b^2)/d/sin(d*x+c) ^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 14.24 (sec) , antiderivative size = 831, normalized size of antiderivative = 1.60 \[ \int \frac {(e \sin (c+d x))^{5/2}}{(a+b \cos (c+d x))^3} \, dx=\frac {\csc ^2(c+d x) (e \sin (c+d x))^{5/2} \left (\frac {\sin (c+d x)}{2 b (a+b \cos (c+d x))^2}+\frac {3 a \sin (c+d x)}{4 b \left (-a^2+b^2\right ) (a+b \cos (c+d x))}\right )}{d}+\frac {3 (e \sin (c+d x))^{5/2} \left (\frac {a \cos ^2(c+d x) \left (3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )\right )+8 b^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)\right ) \left (a+b \sqrt {1-\sin ^2(c+d x)}\right )}{12 b^{3/2} \left (-a^2+b^2\right ) (a+b \cos (c+d x)) \left (1-\sin ^2(c+d x)\right )}+\frac {4 b \cos (c+d x) \left (\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}+\frac {a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}\right ) \left (a+b \sqrt {1-\sin ^2(c+d x)}\right )}{(a+b \cos (c+d x)) \sqrt {1-\sin ^2(c+d x)}}\right )}{8 (a-b) b (a+b) d \sin ^{\frac {5}{2}}(c+d x)} \]
(Csc[c + d*x]^2*(e*Sin[c + d*x])^(5/2)*(Sin[c + d*x]/(2*b*(a + b*Cos[c + d *x])^2) + (3*a*Sin[c + d*x])/(4*b*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d + (3*(e*Sin[c + d*x])^(5/2)*((a*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/ 4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[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)] - Lo g[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]] + 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]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2) )*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12*b^(3/2)*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (4*b*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[ 1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[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]] + I*b*Sin[ c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt [Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*App ellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] *Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(( a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(8*(a - b)*b*(a + b)*d*Sin [c + d*x]^(5/2))
Time = 2.15 (sec) , antiderivative size = 464, normalized size of antiderivative = 0.89, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.840, Rules used = {3042, 3172, 25, 3042, 3343, 27, 3042, 3346, 3042, 3121, 3042, 3119, 3180, 266, 827, 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.
| \(\displaystyle \int \frac {(e \sin (c+d x))^{5/2}}{(a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{5/2}}{\left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3172 |
| \(\displaystyle \frac {3 e^2 \int -\frac {\cos (c+d x) \sqrt {e \sin (c+d x)}}{(a+b \cos (c+d x))^2}dx}{4 b}+\frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \int \frac {\cos (c+d x) \sqrt {e \sin (c+d x)}}{(a+b \cos (c+d x))^2}dx}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \int \frac {\sqrt {-e \cos \left (c+d x+\frac {\pi }{2}\right )} \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 b}\) |
| \(\Big \downarrow \) 3343 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {(2 b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}}{2 (a+b \cos (c+d x))}dx}{a^2-b^2}\right )}{4 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {(2 b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}}{a+b \cos (c+d x)}dx}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {\sqrt {-e \cos \left (c+d x+\frac {\pi }{2}\right )} \left (2 b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
| \(\Big \downarrow \) 3346 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {a \int \sqrt {e \sin (c+d x)}dx}{b}-\frac {\left (a^2-2 b^2\right ) \int \frac {\sqrt {e \sin (c+d x)}}{a+b \cos (c+d x)}dx}{b}}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {a \int \sqrt {e \sin (c+d x)}dx}{b}-\frac {\left (a^2-2 b^2\right ) \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{b}}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {a \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{b \sqrt {\sin (c+d x)}}-\frac {\left (a^2-2 b^2\right ) \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{b}}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {a \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{b \sqrt {\sin (c+d x)}}-\frac {\left (a^2-2 b^2\right ) \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{b}}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{b d \sqrt {\sin (c+d x)}}-\frac {\left (a^2-2 b^2\right ) \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{b}}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
| \(\Big \downarrow \) 3180 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{b d \sqrt {\sin (c+d x)}}-\frac {\left (a^2-2 b^2\right ) \left (-\frac {b e \int \frac {\sqrt {e \sin (c+d x)}}{b^2 \sin ^2(c+d x) e^2+\left (a^2-b^2\right ) e^2}d(e \sin (c+d x))}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{b}}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{b d \sqrt {\sin (c+d x)}}-\frac {\left (a^2-2 b^2\right ) \left (-\frac {2 b e \int \frac {e^2 \sin ^2(c+d x)}{b^2 e^4 \sin ^4(c+d x)+\left (a^2-b^2\right ) e^2}d\sqrt {e \sin (c+d x)}}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{b}}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{b d \sqrt {\sin (c+d x)}}-\frac {\left (a^2-2 b^2\right ) \left (-\frac {2 b e \left (\frac {\int \frac {1}{b e^2 \sin ^2(c+d x)+\sqrt {b^2-a^2} e}d\sqrt {e \sin (c+d x)}}{2 b}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{b}}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{b d \sqrt {\sin (c+d x)}}-\frac {\left (a^2-2 b^2\right ) \left (-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{b}}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{b d \sqrt {\sin (c+d x)}}-\frac {\left (a^2-2 b^2\right ) \left (-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{b}}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{b d \sqrt {\sin (c+d x)}}-\frac {\left (a^2-2 b^2\right ) \left (-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{b}}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{b d \sqrt {\sin (c+d x)}}-\frac {\left (a^2-2 b^2\right ) \left (-\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b \sqrt {e \sin (c+d x)}}+\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \sin (c+d x)}}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{b}}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{b d \sqrt {\sin (c+d x)}}-\frac {\left (a^2-2 b^2\right ) \left (-\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b \sqrt {e \sin (c+d x)}}+\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \sin (c+d x)}}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{b}}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {3 e^2 \left (\frac {a (e \sin (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{b d \sqrt {\sin (c+d x)}}-\frac {\left (a^2-2 b^2\right ) \left (-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}+\frac {a e \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \sin (c+d x)}}+\frac {a e \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \sin (c+d x)}}\right )}{b}}{2 \left (a^2-b^2\right )}\right )}{4 b}\) |
(e*(e*Sin[c + d*x])^(3/2))/(2*b*d*(a + b*Cos[c + d*x])^2) - (3*e^2*((a*(e* Sin[c + d*x])^(3/2))/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x])) - ((2*a*Ellipt icE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(b*d*Sqrt[Sin[c + d*x]]) - ((a^2 - 2*b^2)*((-2*b*e*(ArcTan[(Sqrt[b]*Sqrt[e]*Sin[c + d*x])/(-a^2 + b ^2)^(1/4)]/(2*b^(3/2)*(-a^2 + b^2)^(1/4)*Sqrt[e]) - ArcTanh[(Sqrt[b]*Sqrt[ e]*Sin[c + d*x])/(-a^2 + b^2)^(1/4)]/(2*b^(3/2)*(-a^2 + b^2)^(1/4)*Sqrt[e] )))/d + (a*e*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (a*e*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt [Sin[c + d*x]])/(b*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]])))/b)/(2* (a^2 - b^2))))/(4*b)
3.1.82.3.1 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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + 1))), x] + Simp[g^2*((p - 1)/(b*(m + 1))) Int[(g*Cos [e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; Fre eQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && I ntegersQ[2*m, 2*p]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_ )]), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Simp[a*(g/(2*b)) Int[1/(S qrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (-Simp[a*(g/(2*b)) In t[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x], x] + Simp[b*(g/f) Su bst[Int[Sqrt[x]/(g^2*(a^2 - b^2) + b^2*x^2), x], x, g*Cos[e + f*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[(-(b*c - a*d))*(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)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ [a^2 - b^2, 0] && LtQ[m, -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. \(2611\) vs. \(2(544)=1088\).
Time = 98.23 (sec) , antiderivative size = 2612, normalized size of antiderivative = 5.02
(e^3*b*(-1/4*(e*sin(d*x+c))^(3/2)*e^2*(-5*a^2*b^2*cos(d*x+c)^2+2*b^4*cos(d *x+c)^2+a^4+2*a^2*b^2)/b^2/(a^2-b^2)/(-b^2*cos(d*x+c)^2*e^2+a^2*e^2)^2+3/3 2*(a^2-2*b^2)/b^4/(a^2-b^2)/(e^2*(a^2-b^2)/b^2)^(1/4)*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)))-(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)*e^3*a*(3/b^2*(- 1/2/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)/(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))-1/2/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)/(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)))-(7*a^2-3*b^2)/b^2*(1/2*b^2/e/a^2/(a^2-b^2)*sin(d*x+ c)*(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(-b^2*cos(d*x+c)^2+a^2)-1/2/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)*EllipticE((1-sin(d*x+c))^(1/2),1/2*2^(1/2))+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))-3/8/(a^2-b^2)/b^2*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(...
Timed out. \[ \int \frac {(e \sin (c+d x))^{5/2}}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e \sin (c+d x))^{5/2}}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {(e \sin (c+d x))^{5/2}}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
\[ \int \frac {(e \sin (c+d x))^{5/2}}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(e \sin (c+d x))^{5/2}}{(a+b \cos (c+d x))^3} \, dx=\int \frac {{\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]