Integrand size = 25, antiderivative size = 544 \[ \int \frac {(e \sin (c+d x))^{11/2}}{a+b \cos (c+d x)} \, dx=\frac {\left (-a^2+b^2\right )^{9/4} e^{11/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{11/2} d}+\frac {\left (-a^2+b^2\right )^{9/4} e^{11/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{11/2} d}+\frac {2 a \left (21 a^4-49 a^2 b^2+33 b^4\right ) e^6 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 b^6 d \sqrt {e \sin (c+d x)}}-\frac {a \left (a^2-b^2\right )^3 e^6 \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{b^6 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \sin (c+d x)}}-\frac {a \left (a^2-b^2\right )^3 e^6 \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{b^6 \left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 e^5 \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{21 b^5 d}+\frac {2 e^3 \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right ) (e \sin (c+d x))^{5/2}}{35 b^3 d}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d} \]
(-a^2+b^2)^(9/4)*e^(11/2)*arctan(b^(1/2)*(e*sin(d*x+c))^(1/2)/(-a^2+b^2)^( 1/4)/e^(1/2))/b^(11/2)/d+(-a^2+b^2)^(9/4)*e^(11/2)*arctanh(b^(1/2)*(e*sin( d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/b^(11/2)/d+2/35*e^3*(7*a^2-7*b^2-5 *a*b*cos(d*x+c))*(e*sin(d*x+c))^(5/2)/b^3/d-2/9*e*(e*sin(d*x+c))^(9/2)/b/d -2/21*a*(21*a^4-49*a^2*b^2+33*b^4)*e^6*(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))*si n(d*x+c)^(1/2)/b^6/d/(e*sin(d*x+c))^(1/2)+a*(a^2-b^2)^3*e^6*(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*P i+1/2*d*x),2*b/(b-(-a^2+b^2)^(1/2)),2^(1/2))*sin(d*x+c)^(1/2)/b^6/d/(a^2-b *(b-(-a^2+b^2)^(1/2)))/(e*sin(d*x+c))^(1/2)+a*(a^2-b^2)^3*e^6*(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^6/d/(a^2 -b*(b+(-a^2+b^2)^(1/2)))/(e*sin(d*x+c))^(1/2)-2/21*e^5*(21*(a^2-b^2)^2-a*b *(7*a^2-12*b^2)*cos(d*x+c))*(e*sin(d*x+c))^(1/2)/b^5/d
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 48.77 (sec) , antiderivative size = 2035, normalized size of antiderivative = 3.74 \[ \int \frac {(e \sin (c+d x))^{11/2}}{a+b \cos (c+d x)} \, dx=\text {Result too large to show} \]
(((a*(28*a^2 - 51*b^2)*Cos[c + d*x])/(42*b^4) + ((-9*a^2 + 14*b^2)*Cos[2*( c + d*x)])/(45*b^3) + (a*Cos[3*(c + d*x)])/(14*b^2) - Cos[4*(c + d*x)]/(36 *b))*Csc[c + d*x]^5*(e*Sin[c + d*x])^(11/2))/d - ((e*Sin[c + d*x])^(11/2)* ((2*(392*a^3*b - 722*a*b^3)*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2] )*((a*(-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)] - 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]] + 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)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^ 2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[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*(-280*a^4 + 636*a^2*b^2 - 721*b^4)*Cos[c + d*x]*(a + b*Sqrt[ 1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[b]*(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]*S qrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)...
Time = 2.83 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.02, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.960, Rules used = {3042, 3174, 25, 3042, 3344, 27, 3042, 3344, 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.
| \(\displaystyle \int \frac {(e \sin (c+d x))^{11/2}}{a+b \cos (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{11/2}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3174 |
| \(\displaystyle -\frac {e^2 \int -\frac {(b+a \cos (c+d x)) (e \sin (c+d x))^{7/2}}{a+b \cos (c+d x)}dx}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^2 \int \frac {(b+a \cos (c+d x)) (e \sin (c+d x))^{7/2}}{a+b \cos (c+d x)}dx}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \int \frac {\left (-e \cos \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e^2 \int -\frac {\left (b \left (2 a^2-7 b^2\right )+a \left (7 a^2-12 b^2\right ) \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{2 (a+b \cos (c+d x))}dx}{7 b^2}+\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \int \frac {\left (b \left (2 a^2-7 b^2\right )+a \left (7 a^2-12 b^2\right ) \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{a+b \cos (c+d x)}dx}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \int \frac {\left (-e \cos \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (b \left (2 a^2-7 b^2\right )+a \left (7 a^2-12 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e^2 \int -\frac {b \left (14 a^4-30 b^2 a^2+21 b^4\right )+a \left (21 a^4-49 b^2 a^2+33 b^4\right ) \cos (c+d x)}{2 (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}dx}{3 b^2}+\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \int \frac {b \left (14 a^4-30 b^2 a^2+21 b^4\right )+a \left (21 a^4-49 b^2 a^2+33 b^4\right ) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}dx}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \int \frac {b \left (14 a^4-30 b^2 a^2+21 b^4\right )-a \left (21 a^4-49 b^2 a^2+33 b^4\right ) \sin \left (c+d x-\frac {\pi }{2}\right )}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 3346 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {a \left (21 a^4-49 a^2 b^2+33 b^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}}dx}{b}-\frac {21 \left (a^2-b^2\right )^3 \int \frac {1}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}dx}{b}\right )}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {a \left (21 a^4-49 a^2 b^2+33 b^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}}dx}{b}-\frac {21 \left (a^2-b^2\right )^3 \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{b}\right )}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {a \left (21 a^4-49 a^2 b^2+33 b^4\right ) \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{b \sqrt {e \sin (c+d x)}}-\frac {21 \left (a^2-b^2\right )^3 \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{b}\right )}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {a \left (21 a^4-49 a^2 b^2+33 b^4\right ) \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{b \sqrt {e \sin (c+d x)}}-\frac {21 \left (a^2-b^2\right )^3 \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{b}\right )}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^4-49 a^2 b^2+33 b^4\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {21 \left (a^2-b^2\right )^3 \int \frac {1}{\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{b}\right )}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 3181 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^4-49 a^2 b^2+33 b^4\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {21 \left (a^2-b^2\right )^3 \left (-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b^2 \sin ^2(c+d x) e^2+\left (a^2-b^2\right ) e^2\right )}d(e \sin (c+d x))}{d}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^4-49 a^2 b^2+33 b^4\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {21 \left (a^2-b^2\right )^3 \left (-\frac {2 b e \int \frac {1}{b^2 e^4 \sin ^4(c+d x)+\left (a^2-b^2\right ) e^2}d\sqrt {e \sin (c+d x)}}{d}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^4-49 a^2 b^2+33 b^4\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {21 \left (a^2-b^2\right )^3 \left (-\frac {2 b e \left (-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 e \sqrt {b^2-a^2}}-\frac {\int \frac {1}{b e^2 \sin ^2(c+d x)+\sqrt {b^2-a^2} e}d\sqrt {e \sin (c+d x)}}{2 e \sqrt {b^2-a^2}}\right )}{d}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^4-49 a^2 b^2+33 b^4\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {21 \left (a^2-b^2\right )^3 \left (-\frac {2 b e \left (-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 e \sqrt {b^2-a^2}}-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^4-49 a^2 b^2+33 b^4\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {21 \left (a^2-b^2\right )^3 \left (-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^4-49 a^2 b^2+33 b^4\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {21 \left (a^2-b^2\right )^3 \left (-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^4-49 a^2 b^2+33 b^4\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {21 \left (a^2-b^2\right )^3 \left (-\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \sin (c+d x)}}-\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \sin (c+d x)}}-\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^4-49 a^2 b^2+33 b^4\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {21 \left (a^2-b^2\right )^3 \left (-\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \sin (c+d x)}}-\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \sin (c+d x)}}-\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {e^2 \left (\frac {2 e (e \sin (c+d x))^{5/2} \left (7 \left (a^2-b^2\right )-5 a b \cos (c+d x)\right )}{35 b^2 d}-\frac {e^2 \left (\frac {2 e \sqrt {e \sin (c+d x)} \left (21 \left (a^2-b^2\right )^2-a b \left (7 a^2-12 b^2\right ) \cos (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 a \left (21 a^4-49 a^2 b^2+33 b^4\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \sqrt {e \sin (c+d x)}}-\frac {21 \left (a^2-b^2\right )^3 \left (-\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}+\frac {a \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {b^2-a^2} \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \sin (c+d x)}}-\frac {a \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {b^2-a^2} \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \sin (c+d x)}}\right )}{b}\right )}{3 b^2}\right )}{7 b^2}\right )}{b}-\frac {2 e (e \sin (c+d x))^{9/2}}{9 b d}\) |
(-2*e*(e*Sin[c + d*x])^(9/2))/(9*b*d) + (e^2*((2*e*(7*(a^2 - b^2) - 5*a*b* Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(35*b^2*d) - (e^2*((2*e*(21*(a^2 - b ^2)^2 - a*b*(7*a^2 - 12*b^2)*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*b^2*d) - (e^2*((2*a*(21*a^4 - 49*a^2*b^2 + 33*b^4)*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b*d*Sqrt[e*Sin[c + d*x]]) - (21*(a^2 - b^2)^3*((- 2*b*e*(-1/2*ArcTan[(Sqrt[b]*Sqrt[e]*Sin[c + d*x])/(-a^2 + b^2)^(1/4)]/(Sqr t[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*Sqrt[b]*(-a^2 + b^2)^(3/4)*e^(3/2))))/d + (a*Ellipt icPi[(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]]) ))/b))/(3*b^2)))/(7*b^2)))/b
3.1.58.3.1 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[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[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*(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]
Time = 6.22 (sec) , antiderivative size = 930, normalized size of antiderivative = 1.71
(-2*e*b*(1/45/b^6*(e*sin(d*x+c))^(1/2)*e^4*(5*b^4*cos(d*x+c)^4+9*a^2*b^2*c os(d*x+c)^2-19*b^4*cos(d*x+c)^2+45*a^4-99*a^2*b^2+59*b^4)-1/8*e^6*(a^6-3*a ^4*b^2+3*a^2*b^4-b^6)/b^6*(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*s in(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^6*a*(-1/21/b^6/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)*(-6*b^4*cos(d*x+c)^4*s in(d*x+c)+21*a^4*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1 /2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))-49*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)*EllipticF((1-sin(d*x+c))^(1/ 2),1/2*2^(1/2))+33*b^4*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x +c)^(1/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))-14*a^2*b^2*cos(d*x+c )^2*sin(d*x+c)+30*b^4*cos(d*x+c)^2*sin(d*x+c))+(-a^6+3*a^4*b^2-3*a^2*b^4+b ^6)/b^6*(-1/2/(-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 /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)*Ellip...
Timed out. \[ \int \frac {(e \sin (c+d x))^{11/2}}{a+b \cos (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e \sin (c+d x))^{11/2}}{a+b \cos (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e \sin (c+d x))^{11/2}}{a+b \cos (c+d x)} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {11}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {(e \sin (c+d x))^{11/2}}{a+b \cos (c+d x)} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {11}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e \sin (c+d x))^{11/2}}{a+b \cos (c+d x)} \, dx=\int \frac {{\left (e\,\sin \left (c+d\,x\right )\right )}^{11/2}}{a+b\,\cos \left (c+d\,x\right )} \,d x \]