Integrand size = 25, antiderivative size = 571 \[ \int \frac {(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^4} \, dx=-\frac {15 a \left (7 a^2-6 b^2\right ) e^{11/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 b^{11/2} \left (-a^2+b^2\right )^{3/4} d}-\frac {15 a \left (7 a^2-6 b^2\right ) e^{11/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 b^{11/2} \left (-a^2+b^2\right )^{3/4} d}-\frac {5 \left (21 a^2-4 b^2\right ) e^6 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{8 b^6 d \sqrt {e \cos (c+d x)}}+\frac {15 a^2 \left (7 a^2-6 b^2\right ) e^6 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 b^6 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d x)}}+\frac {15 a^2 \left (7 a^2-6 b^2\right ) e^6 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 b^6 \left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d x)}}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}-\frac {e^3 (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{4 b^3 d (a+b \sin (c+d x))^2}-\frac {5 e^5 \sqrt {e \cos (c+d x)} \left (21 a^2-4 b^2+14 a b \sin (c+d x)\right )}{8 b^5 d (a+b \sin (c+d x))} \]
-15/16*a*(7*a^2-6*b^2)*e^(11/2)*arctan(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+ b^2)^(1/4)/e^(1/2))/b^(11/2)/(-a^2+b^2)^(3/4)/d-15/16*a*(7*a^2-6*b^2)*e^(1 1/2)*arctanh(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/b^(11/ 2)/(-a^2+b^2)^(3/4)/d-1/3*e*(e*cos(d*x+c))^(9/2)/b/d/(a+b*sin(d*x+c))^3-1/ 4*e^3*(e*cos(d*x+c))^(5/2)*(7*a+4*b*sin(d*x+c))/b^3/d/(a+b*sin(d*x+c))^2-5 /8*(21*a^2-4*b^2)*e^6*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Elli pticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)/b^6/d/(e*cos(d*x+c))^(1 /2)+15/16*a^2*(7*a^2-6*b^2)*e^6*(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^6/d/(a^2-b*(b-(-a^2+b^2)^(1/2)))/(e*cos(d*x+c))^(1/2)+15/16 *a^2*(7*a^2-6*b^2)*e^6*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ell ipticPi(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^6/d/(a^2-b*(b+(-a^2+b^2)^(1/2)))/(e*cos(d*x+c))^(1/2)-5/8*e^5*(21*a^ 2-4*b^2+14*a*b*sin(d*x+c))*(e*cos(d*x+c))^(1/2)/b^5/d/(a+b*sin(d*x+c))
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 20.51 (sec) , antiderivative size = 2020, normalized size of antiderivative = 3.54 \[ \int \frac {(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^4} \, dx=\text {Result too large to show} \]
((e*Cos[c + d*x])^(11/2)*Sec[c + d*x]^5*((2*Sin[c + d*x])/(3*b^4) - (-a^2 + b^2)^2/(3*b^5*(a + b*Sin[c + d*x])^3) + (25*a*(a^2 - b^2))/(12*b^5*(a + b*Sin[c + d*x])^2) + (-165*a^2 + 52*b^2)/(24*b^5*(a + b*Sin[c + d*x]))))/d - ((e*Cos[c + d*x])^(11/2)*((-76*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5 *a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x ]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^ 2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(- 1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(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)*S qrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqr t[-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)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x] ^2]*(a + b*Sin[c + d*x])) + (32*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*Cos[2 *(c + d*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[ Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-...
Time = 2.52 (sec) , antiderivative size = 545, normalized size of antiderivative = 0.95, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.920, Rules used = {3042, 3172, 3042, 3342, 27, 3042, 3342, 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.
| \(\displaystyle \int \frac {(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^4}dx\) |
| \(\Big \downarrow \) 3172 |
| \(\displaystyle -\frac {3 e^2 \int \frac {(e \cos (c+d x))^{7/2} \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 e^2 \int \frac {(e \cos (c+d x))^{7/2} \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3342 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}-\frac {5 e^2 \int -\frac {(e \cos (c+d x))^{3/2} (4 b+7 a \sin (c+d x))}{2 (a+b \sin (c+d x))^2}dx}{6 b^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \int \frac {(e \cos (c+d x))^{3/2} (4 b+7 a \sin (c+d x))}{(a+b \sin (c+d x))^2}dx}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \int \frac {(e \cos (c+d x))^{3/2} (4 b+7 a \sin (c+d x))}{(a+b \sin (c+d x))^2}dx}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3342 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}-\frac {e^2 \int -\frac {14 a b+\left (21 a^2-4 b^2\right ) \sin (c+d x)}{2 \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{b^2}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \int \frac {14 a b+\left (21 a^2-4 b^2\right ) \sin (c+d x)}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \int \frac {14 a b+\left (21 a^2-4 b^2\right ) \sin (c+d x)}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3346 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \left (\frac {\left (21 a^2-4 b^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}}dx}{b}-\frac {3 a \left (7 a^2-6 b^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{b}\right )}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \left (\frac {\left (21 a^2-4 b^2\right ) \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {3 a \left (7 a^2-6 b^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{b}\right )}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \left (\frac {\left (21 a^2-4 b^2\right ) \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b \sqrt {e \cos (c+d x)}}-\frac {3 a \left (7 a^2-6 b^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{b}\right )}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \left (\frac {\left (21 a^2-4 b^2\right ) \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b \sqrt {e \cos (c+d x)}}-\frac {3 a \left (7 a^2-6 b^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{b}\right )}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \left (21 a^2-4 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {3 a \left (7 a^2-6 b^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{b}\right )}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3181 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \left (21 a^2-4 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {3 a \left (7 a^2-6 b^2\right ) \left (\frac {b e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b^2 \cos ^2(c+d x) e^2+\left (a^2-b^2\right ) e^2\right )}d(e \cos (c+d x))}{d}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \left (21 a^2-4 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {3 a \left (7 a^2-6 b^2\right ) \left (\frac {2 b e \int \frac {1}{b^2 e^4 \cos ^4(c+d x)+\left (a^2-b^2\right ) e^2}d\sqrt {e \cos (c+d x)}}{d}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \left (21 a^2-4 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {3 a \left (7 a^2-6 b^2\right ) \left (\frac {2 b e \left (-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \cos ^2(c+d x)}d\sqrt {e \cos (c+d x)}}{2 e \sqrt {b^2-a^2}}-\frac {\int \frac {1}{b e^2 \cos ^2(c+d x)+\sqrt {b^2-a^2} e}d\sqrt {e \cos (c+d x)}}{2 e \sqrt {b^2-a^2}}\right )}{d}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \left (21 a^2-4 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {3 a \left (7 a^2-6 b^2\right ) \left (\frac {2 b e \left (-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \cos ^2(c+d x)}d\sqrt {e \cos (c+d x)}}{2 e \sqrt {b^2-a^2}}-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \left (21 a^2-4 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {3 a \left (7 a^2-6 b^2\right ) \left (-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}+\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \left (21 a^2-4 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {3 a \left (7 a^2-6 b^2\right ) \left (-\frac {a \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \sin \left (c+d x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}+\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \left (21 a^2-4 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {3 a \left (7 a^2-6 b^2\right ) \left (-\frac {a \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \cos (c+d x)}}-\frac {a \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \cos (c+d x)}}+\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \left (21 a^2-4 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {3 a \left (7 a^2-6 b^2\right ) \left (-\frac {a \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \cos (c+d x)}}-\frac {a \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \sin \left (c+d x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \cos (c+d x)}}+\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle -\frac {3 e^2 \left (\frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \left (21 a^2-4 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {3 a \left (7 a^2-6 b^2\right ) \left (\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}+\frac {a \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{d \sqrt {b^2-a^2} \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}-\frac {a \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{d \sqrt {b^2-a^2} \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}\right )}{b}\right )}{2 b^2}+\frac {e \sqrt {e \cos (c+d x)} \left (21 a^2+14 a b \sin (c+d x)-4 b^2\right )}{b^2 d (a+b \sin (c+d x))}\right )}{12 b^2}+\frac {e (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^2}\right )}{2 b}-\frac {e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}\) |
-1/3*(e*(e*Cos[c + d*x])^(9/2))/(b*d*(a + b*Sin[c + d*x])^3) - (3*e^2*((e* (e*Cos[c + d*x])^(5/2)*(7*a + 4*b*Sin[c + d*x]))/(6*b^2*d*(a + b*Sin[c + d *x])^2) + (5*e^2*((e^2*((2*(21*a^2 - 4*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[( c + d*x)/2, 2])/(b*d*Sqrt[e*Cos[c + d*x]]) - (3*a*(7*a^2 - 6*b^2)*((2*b*e* (-1/2*ArcTan[(Sqrt[b]*Sqrt[e]*Cos[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]*Cos[c + d*x])/(-a^2 + b^2)^(1/4)]/(2*Sqrt[b]*(-a^2 + b^2)^(3/4)*e^(3/2))))/d + (a*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(Sqrt[-a^ 2 + b^2]*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (a*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(Sqrt[-a^2 + b^2]*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]])))/b))/(2*b^2) + (e* Sqrt[e*Cos[c + d*x]]*(21*a^2 - 4*b^2 + 14*a*b*Sin[c + d*x]))/(b^2*d*(a + b *Sin[c + d*x]))))/(12*b^2)))/(2*b)
3.7.7.3.1 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 + 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[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*C os[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d*p + b*d*(m + 1)*Sin[e + f*x])/(b^2*f*(m + 1)*(m + p + 1))), x] + Simp[g^2*(( p - 1)/(b^2*(m + 1)*(m + p + 1))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin [e + f*x])^(m + 1)*Simp[b*d*(m + 1) + (b*c*(m + p + 1) - a*d*p)*Sin[e + f*x ], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && Lt Q[m, -1] && GtQ[p, 1] && 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 = 336.32 (sec) , antiderivative size = 4681, normalized size of antiderivative = 8.20
(-16*e^6*a*b*(1/2/b^6/e*(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)-(5*a^2-3*b^2) /b^6*(e^2*(a^2-b^2)/b^2)^(1/4)*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*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*ar ctan((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)))/(16*a^2-16* b^2)/e+3/64*(3*a^4-4*a^2*b^2+b^4)/b^6*(3*(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*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* 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)))*(4*cos(1/ 2*d*x+1/2*c)^4*b^2-4*cos(1/2*d*x+1/2*c)^2*b^2+a^2)*2^(1/2)*(e^2*(a^2-b^2)/ b^2)^(1/4)+(8*a^2-8*b^2)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2))/e/(a-b)^2/(a+ b)^2/(4*cos(1/2*d*x+1/2*c)^4*b^2-4*cos(1/2*d*x+1/2*c)^2*b^2+a^2)-1/512*(7* a^6-15*a^4*b^2+9*a^2*b^4-b^6)/b^6*(21*(e^2*(a^2-b^2)/b^2)^(1/4)*(4*sin(1/2 *d*x+1/2*c)^4*b^2-4*sin(1/2*d*x+1/2*c)^2*b^2+a^2)^2*2^(1/2)*ln((2*e*cos...
Exception generated. \[ \int \frac {(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^4} \, dx=\text {Exception raised: TypeError} \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^4} \, dx=\text {Timed out} \]
\[ \int \frac {(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^4} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {11}{2}}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^4} \, dx=\text {Hanged} \]