Integrand size = 25, antiderivative size = 446 \[ \int \frac {(e \cos (c+d x))^{9/2}}{a+b \sin (c+d x)} \, dx=\frac {\left (-a^2+b^2\right )^{7/4} e^{9/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{9/2} d}-\frac {\left (-a^2+b^2\right )^{7/4} e^{9/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{9/2} d}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}-\frac {2 a \left (5 a^2-8 b^2\right ) e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^4 d \sqrt {\cos (c+d x)}}+\frac {a \left (a^2-b^2\right )^2 e^5 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{b^5 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {a \left (a^2-b^2\right )^2 e^5 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{b^5 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {2 e^3 (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^3 d} \]
(-a^2+b^2)^(7/4)*e^(9/2)*arctan(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^(1 /4)/e^(1/2))/b^(9/2)/d-(-a^2+b^2)^(7/4)*e^(9/2)*arctanh(b^(1/2)*(e*cos(d*x +c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/b^(9/2)/d+2/7*e*(e*cos(d*x+c))^(7/2)/ b/d-2/15*e^3*(e*cos(d*x+c))^(3/2)*(5*a^2-5*b^2-3*a*b*sin(d*x+c))/b^3/d+a*( a^2-b^2)^2*e^5*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi( sin(1/2*d*x+1/2*c),2*b/(b-(-a^2+b^2)^(1/2)),2^(1/2))*cos(d*x+c)^(1/2)/b^5/ d/(b-(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)+a*(a^2-b^2)^2*e^5*(cos(1/2*d*x +1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(b+( -a^2+b^2)^(1/2)),2^(1/2))*cos(d*x+c)^(1/2)/b^5/d/(b+(-a^2+b^2)^(1/2))/(e*c os(d*x+c))^(1/2)-2/5*a*(5*a^2-8*b^2)*e^4*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos( 1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x+c))^(1/2)/ b^4/d/cos(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 37.44 (sec) , antiderivative size = 757, normalized size of antiderivative = 1.70 \[ \int \frac {(e \cos (c+d x))^{9/2}}{a+b \sin (c+d x)} \, dx=\frac {(e \cos (c+d x))^{9/2} \left (\frac {\cos ^{\frac {3}{2}}(c+d x) \left (-70 a^2+85 b^2+15 b^2 \cos (2 (c+d x))+42 a b \sin (c+d x)\right )}{21 b^3}+\frac {\sin (c+d x) \left (-\frac {a \left (5 a^2-8 b^2\right ) \csc (c+d x) \left (8 b^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)+3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )\right )\right )}{a^2-b^2}+\frac {(1+i) b^2 \left (-2 a^2+5 b^2\right ) \left ((4-4 i) a \sqrt {b} \sqrt [4]{-a^2+b^2} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)+3 \left (a^2-b^2\right ) \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )\right )\right )}{\left (-a^2+b^2\right )^{5/4} \sqrt {\sin ^2(c+d x)}}\right ) \left (a+b \sqrt {\sin ^2(c+d x)}\right )}{12 b^{9/2} (a+b \sin (c+d x))}\right )}{5 d \cos ^{\frac {9}{2}}(c+d x)} \]
((e*Cos[c + d*x])^(9/2)*((Cos[c + d*x]^(3/2)*(-70*a^2 + 85*b^2 + 15*b^2*Co s[2*(c + d*x)] + 42*a*b*Sin[c + d*x]))/(21*b^3) + (Sin[c + d*x]*(-((a*(5*a ^2 - 8*b^2)*Csc[c + d*x]*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d* x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a* (a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b ^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[ Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a ^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]])))/(a^2 - b^2)) + ((1 + I)*b^2*(-2*a^2 + 5*b^2)*((4 - 4*I)*a*Sqrt[b]*(-a^2 + b^2)^(1/4)*AppellF 1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos [c + d*x]^(3/2) + 3*(a^2 - b^2)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[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[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x ]])))/((-a^2 + b^2)^(5/4)*Sqrt[Sin[c + d*x]^2]))*(a + b*Sqrt[Sin[c + d*x]^ 2]))/(12*b^(9/2)*(a + b*Sin[c + d*x]))))/(5*d*Cos[c + d*x]^(9/2))
Time = 2.13 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 3174, 3042, 3344, 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.
| \(\displaystyle \int \frac {(e \cos (c+d x))^{9/2}}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(e \cos (c+d x))^{9/2}}{a+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3174 |
| \(\displaystyle \frac {e^2 \int \frac {(e \cos (c+d x))^{5/2} (b+a \sin (c+d x))}{a+b \sin (c+d x)}dx}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \int \frac {(e \cos (c+d x))^{5/2} (b+a \sin (c+d x))}{a+b \sin (c+d x)}dx}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e^2 \int -\frac {\sqrt {e \cos (c+d x)} \left (b \left (2 a^2-5 b^2\right )+a \left (5 a^2-8 b^2\right ) \sin (c+d x)\right )}{2 (a+b \sin (c+d x))}dx}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \int \frac {\sqrt {e \cos (c+d x)} \left (b \left (2 a^2-5 b^2\right )+a \left (5 a^2-8 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \int \frac {\sqrt {e \cos (c+d x)} \left (b \left (2 a^2-5 b^2\right )+a \left (5 a^2-8 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 3346 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \left (\frac {a \left (5 a^2-8 b^2\right ) \int \sqrt {e \cos (c+d x)}dx}{b}-\frac {5 \left (a^2-b^2\right )^2 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{b}\right )}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \left (\frac {a \left (5 a^2-8 b^2\right ) \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {5 \left (a^2-b^2\right )^2 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{b}\right )}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \left (\frac {a \left (5 a^2-8 b^2\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{b \sqrt {\cos (c+d x)}}-\frac {5 \left (a^2-b^2\right )^2 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{b}\right )}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \left (\frac {a \left (5 a^2-8 b^2\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b \sqrt {\cos (c+d x)}}-\frac {5 \left (a^2-b^2\right )^2 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{b}\right )}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \left (\frac {2 a \left (5 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {5 \left (a^2-b^2\right )^2 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{b}\right )}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 3180 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \left (\frac {2 a \left (5 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {5 \left (a^2-b^2\right )^2 \left (\frac {b e \int \frac {\sqrt {e \cos (c+d x)}}{b^2 \cos ^2(c+d x) e^2+\left (a^2-b^2\right ) e^2}d(e \cos (c+d x))}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{b}\right )}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \left (\frac {2 a \left (5 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {5 \left (a^2-b^2\right )^2 \left (\frac {2 b e \int \frac {e^2 \cos ^2(c+d x)}{b^2 e^4 \cos ^4(c+d x)+\left (a^2-b^2\right ) e^2}d\sqrt {e \cos (c+d x)}}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{b}\right )}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \left (\frac {2 a \left (5 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {5 \left (a^2-b^2\right )^2 \left (\frac {2 b e \left (\frac {\int \frac {1}{b e^2 \cos ^2(c+d x)+\sqrt {b^2-a^2} e}d\sqrt {e \cos (c+d x)}}{2 b}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \cos ^2(c+d x)}d\sqrt {e \cos (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{b}\right )}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \left (\frac {2 a \left (5 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {5 \left (a^2-b^2\right )^2 \left (\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \cos ^2(c+d x)}d\sqrt {e \cos (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{b}\right )}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \left (\frac {2 a \left (5 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {5 \left (a^2-b^2\right )^2 \left (-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{b}\right )}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \left (\frac {2 a \left (5 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {5 \left (a^2-b^2\right )^2 \left (-\frac {a e \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \sin \left (c+d x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 b}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{b}\right )}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \left (\frac {2 a \left (5 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {5 \left (a^2-b^2\right )^2 \left (-\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{b}\right )}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \left (\frac {2 a \left (5 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {5 \left (a^2-b^2\right )^2 \left (-\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \sin \left (c+d x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{b}\right )}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {e^2 \left (-\frac {e^2 \left (\frac {2 a \left (5 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {5 \left (a^2-b^2\right )^2 \left (\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}+\frac {a e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{b d \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {a e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{b d \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}\right )}{b}\right )}{5 b^2}-\frac {2 e (e \cos (c+d x))^{3/2} \left (5 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{15 b^2 d}\right )}{b}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 b d}\) |
(2*e*(e*Cos[c + d*x])^(7/2))/(7*b*d) + (e^2*(-1/5*(e^2*((2*a*(5*a^2 - 8*b^ 2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b*d*Sqrt[Cos[c + d*x]] ) - (5*(a^2 - b^2)^2*((2*b*e*(ArcTan[(Sqrt[b]*Sqrt[e]*Cos[c + d*x])/(-a^2 + b^2)^(1/4)]/(2*b^(3/2)*(-a^2 + b^2)^(1/4)*Sqrt[e]) - ArcTanh[(Sqrt[b]*Sq rt[e]*Cos[c + d*x])/(-a^2 + b^2)^(1/4)]/(2*b^(3/2)*(-a^2 + b^2)^(1/4)*Sqrt [e])))/d + (a*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]) , (c + d*x)/2, 2])/(b*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (a* e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]])))/b))/b^2 - (2*e*(e *Cos[c + d*x])^(3/2)*(5*(a^2 - b^2) - 3*a*b*Sin[c + d*x]))/(15*b^2*d)))/b
3.6.75.3.1 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 + p))), x] + Simp[g^2*((p - 1)/(b*(m + p))) Int[(g*Cos [e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*(b + a*Sin[e + f*x]), x], x] /; F reeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && IntegersQ[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[g*(g* Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d* p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Simp[g^2*( (p - 1)/(b^2*(m + p)*(m + p + 1))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Si n[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1) - d*(a^ 2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1 , 0] && 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.35 (sec) , antiderivative size = 1203, normalized size of antiderivative = 2.70
(4*e^5*b*(-1/42/b^4/e*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*(24*sin(1/2*d*x+ 1/2*c)^6*b^2-36*sin(1/2*d*x+1/2*c)^4*b^2-14*sin(1/2*d*x+1/2*c)^2*a^2+32*si n(1/2*d*x+1/2*c)^2*b^2+7*a^2-10*b^2)+1/16/b^6/(e^2*(a^2-b^2)/b^2)^(1/4)*(a +b)^2*2^(1/2)*(ln((2*e*cos(1/2*d*x+1/2*c)^2-e-(e^2*(a^2-b^2)/b^2)^(1/4)*(2 *e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2))/(2*e*c os(1/2*d*x+1/2*c)^2-e+(e^2*(a^2-b^2)/b^2)^(1/4)*(2*e*cos(1/2*d*x+1/2*c)^2- e)^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2)))+2*arctan((2^(1/2)*(2*e*cos(1/ 2*d*x+1/2*c)^2-e)^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/4))/(e^2*(a^2-b^2)/b^2)^(1/ 4))-2*arctan((-2^(1/2)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)+(e^2*(a^2-b^2)/b ^2)^(1/4))/(e^2*(a^2-b^2)/b^2)^(1/4)))*(a-b)^2)-1/40*(e*(2*cos(1/2*d*x+1/2 *c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*e^5*(128*cos(1/2*d*x+1/2*c)^7*a^2*b^4 -256*cos(1/2*d*x+1/2*c)^5*a^2*b^4+160*cos(1/2*d*x+1/2*c)^3*a^2*b^4+80*Elli pticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2 *d*x+1/2*c)^2+1)^(1/2)*a^4*b^2-128*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*( sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*a^2*b^4-32*a ^2*b^4*cos(1/2*d*x+1/2*c)+5*sum((a^4-2*a^2*b^2+b^4)/_alpha*(8*(sin(1/2*d*x +1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticPi(cos(1/2*d*x+ 1/2*c),-4*b^2/a^2*(_alpha^2-1),2^(1/2))*(e*(2*_alpha^2*b^2+a^2-2*b^2)/b^2) ^(1/2)*_alpha^3*b^2-8*b^2*_alpha*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2* d*x+1/2*c)^2+1)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-4*b^2/a^2*(_alpha^...
Timed out. \[ \int \frac {(e \cos (c+d x))^{9/2}}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{9/2}}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e \cos (c+d x))^{9/2}}{a+b \sin (c+d x)} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}}{b \sin \left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {(e \cos (c+d x))^{9/2}}{a+b \sin (c+d x)} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}}{b \sin \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{9/2}}{a+b \sin (c+d x)} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{9/2}}{a+b\,\sin \left (c+d\,x\right )} \,d x \]