Integrand size = 25, antiderivative size = 671 \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+b \sin (c+d x))^4} \, dx=\frac {39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^{15/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 b^{15/2} \left (-a^2+b^2\right )^{3/4} d}+\frac {39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^{15/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 b^{15/2} \left (-a^2+b^2\right )^{3/4} d}+\frac {13 \left (231 a^4-203 a^2 b^2+20 b^4\right ) e^8 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{56 b^8 d \sqrt {e \cos (c+d x)}}-\frac {39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 b^8 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d x)}}-\frac {39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 b^8 \left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d x)}}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac {13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac {39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2-20 b^2+22 a b \sin (c+d x)\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac {13 e^7 \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d} \]
39/16*a*(11*a^4-17*a^2*b^2+6*b^4)*e^(15/2)*arctan(b^(1/2)*(e*cos(d*x+c))^( 1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/b^(15/2)/(-a^2+b^2)^(3/4)/d+39/16*a*(11*a^4 -17*a^2*b^2+6*b^4)*e^(15/2)*arctanh(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2 )^(1/4)/e^(1/2))/b^(15/2)/(-a^2+b^2)^(3/4)/d-1/3*e*(e*cos(d*x+c))^(13/2)/b /d/(a+b*sin(d*x+c))^3-13/84*e^3*(e*cos(d*x+c))^(9/2)*(11*a+4*b*sin(d*x+c)) /b^3/d/(a+b*sin(d*x+c))^2-39/280*e^5*(e*cos(d*x+c))^(5/2)*(77*a^2-20*b^2+2 2*a*b*sin(d*x+c))/b^5/d/(a+b*sin(d*x+c))+13/56*(231*a^4-203*a^2*b^2+20*b^4 )*e^8*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d* x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)/b^8/d/(e*cos(d*x+c))^(1/2)-39/16*a^2*(1 1*a^4-17*a^2*b^2+6*b^4)*e^8*(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^8/d/(a^2-b*(b-(-a^2+b^2)^(1/2)))/(e*cos(d*x+c))^(1/2)-39/16*a^2 *(11*a^4-17*a^2*b^2+6*b^4)*e^8*(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^8/d/(a^2-b*(b+(-a^2+b^2)^(1/2)))/(e*cos(d*x+c))^(1/2)+13/56* e^7*(21*a*(11*a^2-6*b^2)-b*(77*a^2-20*b^2)*sin(d*x+c))*(e*cos(d*x+c))^(1/2 )/b^7/d
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 24.57 (sec) , antiderivative size = 2102, normalized size of antiderivative = 3.13 \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+b \sin (c+d x))^4} \, dx=\text {Result too large to show} \]
((e*Cos[c + d*x])^(15/2)*((-2*(4410*a^3*b - 3418*a*b^3)*(a + b*Sqrt[1 - Co s[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*Co s[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)*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)^(3/4))*Sin[c + d*x])/(Sqr t[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) + ((5600*a^3*b - 3472*a*b^3)*( 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]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2) ^(3/4)) + (4*Sqrt[Cos[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(5/2))/(5*(a...
Time = 3.16 (sec) , antiderivative size = 633, normalized size of antiderivative = 0.94, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.040, Rules used = {3042, 3172, 3042, 3342, 27, 3042, 3342, 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.
| \(\displaystyle \int \frac {(e \cos (c+d x))^{15/2}}{(a+b \sin (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(e \cos (c+d x))^{15/2}}{(a+b \sin (c+d x))^4}dx\) |
| \(\Big \downarrow \) 3172 |
| \(\displaystyle -\frac {13 e^2 \int \frac {(e \cos (c+d x))^{11/2} \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {13 e^2 \int \frac {(e \cos (c+d x))^{11/2} \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3342 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}-\frac {9 e^2 \int -\frac {(e \cos (c+d x))^{7/2} (4 b+11 a \sin (c+d x))}{2 (a+b \sin (c+d x))^2}dx}{14 b^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \int \frac {(e \cos (c+d x))^{7/2} (4 b+11 a \sin (c+d x))}{(a+b \sin (c+d x))^2}dx}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \int \frac {(e \cos (c+d x))^{7/2} (4 b+11 a \sin (c+d x))}{(a+b \sin (c+d x))^2}dx}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3342 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}-\frac {e^2 \int -\frac {(e \cos (c+d x))^{3/2} \left (22 a b+\left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{2 (a+b \sin (c+d x))}dx}{b^2}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \int \frac {(e \cos (c+d x))^{3/2} \left (22 a b+\left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \int \frac {(e \cos (c+d x))^{3/2} \left (22 a b+\left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (\frac {2 e^2 \int -\frac {2 a b \left (77 a^2-53 b^2\right )+\left (231 a^4-203 b^2 a^2+20 b^4\right ) \sin (c+d x)}{2 \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (-\frac {e^2 \int \frac {2 a b \left (77 a^2-53 b^2\right )+\left (231 a^4-203 b^2 a^2+20 b^4\right ) \sin (c+d x)}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (-\frac {e^2 \int \frac {2 a b \left (77 a^2-53 b^2\right )+\left (231 a^4-203 b^2 a^2+20 b^4\right ) \sin (c+d x)}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3346 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (-\frac {e^2 \left (\frac {\left (231 a^4-203 a^2 b^2+20 b^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}}dx}{b}-\frac {21 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{b}\right )}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (-\frac {e^2 \left (\frac {\left (231 a^4-203 a^2 b^2+20 b^4\right ) \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {21 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{b}\right )}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (-\frac {e^2 \left (\frac {\left (231 a^4-203 a^2 b^2+20 b^4\right ) \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b \sqrt {e \cos (c+d x)}}-\frac {21 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{b}\right )}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (-\frac {e^2 \left (\frac {\left (231 a^4-203 a^2 b^2+20 b^4\right ) \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b \sqrt {e \cos (c+d x)}}-\frac {21 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{b}\right )}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (-\frac {e^2 \left (\frac {2 \left (231 a^4-203 a^2 b^2+20 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {21 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}dx}{b}\right )}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3181 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (-\frac {e^2 \left (\frac {2 \left (231 a^4-203 a^2 b^2+20 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {21 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) \left (\frac {b e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b^2 \cos ^2(c+d x) e^2+\left (a^2-b^2\right ) e^2\right )}d(e \cos (c+d x))}{d}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (-\frac {e^2 \left (\frac {2 \left (231 a^4-203 a^2 b^2+20 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {21 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) \left (\frac {2 b e \int \frac {1}{b^2 e^4 \cos ^4(c+d x)+\left (a^2-b^2\right ) e^2}d\sqrt {e \cos (c+d x)}}{d}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (-\frac {e^2 \left (\frac {2 \left (231 a^4-203 a^2 b^2+20 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {21 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) \left (\frac {2 b e \left (-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \cos ^2(c+d x)}d\sqrt {e \cos (c+d x)}}{2 e \sqrt {b^2-a^2}}-\frac {\int \frac {1}{b e^2 \cos ^2(c+d x)+\sqrt {b^2-a^2} e}d\sqrt {e \cos (c+d x)}}{2 e \sqrt {b^2-a^2}}\right )}{d}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (-\frac {e^2 \left (\frac {2 \left (231 a^4-203 a^2 b^2+20 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {21 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) \left (\frac {2 b e \left (-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \cos ^2(c+d x)}d\sqrt {e \cos (c+d x)}}{2 e \sqrt {b^2-a^2}}-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}\right )}{b}\right )}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (-\frac {e^2 \left (\frac {2 \left (231 a^4-203 a^2 b^2+20 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {21 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) \left (-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}+\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (-\frac {e^2 \left (\frac {2 \left (231 a^4-203 a^2 b^2+20 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {21 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) \left (-\frac {a \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 \sqrt {b^2-a^2}}-\frac {a \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \sin \left (c+d x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2}}+\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (-\frac {e^2 \left (\frac {2 \left (231 a^4-203 a^2 b^2+20 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {21 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) \left (-\frac {a \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \cos (c+d x)}}-\frac {a \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \cos (c+d x)}}+\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e^2 \left (-\frac {e^2 \left (\frac {2 \left (231 a^4-203 a^2 b^2+20 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {21 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) \left (-\frac {a \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \cos (c+d x)}}-\frac {a \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \sin \left (c+d x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 \sqrt {b^2-a^2} \sqrt {e \cos (c+d x)}}+\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}\right )}{b}\right )}{3 b^2}-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}\right )}{2 b^2}+\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle -\frac {13 e^2 \left (\frac {9 e^2 \left (\frac {e (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{5 b^2 d (a+b \sin (c+d x))}+\frac {e^2 \left (-\frac {2 e \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{3 b^2 d}-\frac {e^2 \left (\frac {2 \left (231 a^4-203 a^2 b^2+20 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d \sqrt {e \cos (c+d x)}}-\frac {21 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) \left (\frac {2 b e \left (-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} e^{3/2} \left (b^2-a^2\right )^{3/4}}\right )}{d}+\frac {a \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{d \sqrt {b^2-a^2} \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}-\frac {a \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{d \sqrt {b^2-a^2} \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}\right )}{b}\right )}{3 b^2}\right )}{2 b^2}\right )}{28 b^2}+\frac {e (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{14 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}\) |
-1/3*(e*(e*Cos[c + d*x])^(13/2))/(b*d*(a + b*Sin[c + d*x])^3) - (13*e^2*(( e*(e*Cos[c + d*x])^(9/2)*(11*a + 4*b*Sin[c + d*x]))/(14*b^2*d*(a + b*Sin[c + d*x])^2) + (9*e^2*((e*(e*Cos[c + d*x])^(5/2)*(77*a^2 - 20*b^2 + 22*a*b* Sin[c + d*x]))/(5*b^2*d*(a + b*Sin[c + d*x])) + (e^2*(-1/3*(e^2*((2*(231*a ^4 - 203*a^2*b^2 + 20*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/( b*d*Sqrt[e*Cos[c + d*x]]) - (21*a*(11*a^4 - 17*a^2*b^2 + 6*b^4)*((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))/b^2 - (2*e*Sqrt [e*Cos[c + d*x]]*(21*a*(11*a^2 - 6*b^2) - b*(77*a^2 - 20*b^2)*Sin[c + d*x] ))/(3*b^2*d)))/(2*b^2)))/(28*b^2)))/(6*b)
3.7.5.3.1 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_)*((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 = 372.50 (sec) , antiderivative size = 4938, normalized size of antiderivative = 7.36
(-16*e^8*a*b*(1/10/b^8/e*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*(4*sin(1/2*d* x+1/2*c)^4*b^2-4*sin(1/2*d*x+1/2*c)^2*b^2-25*a^2+16*b^2)+2*(7*a^4-10*a^2*b ^2+3*b^4)/b^8*(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*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)))/(1 6*a^2-16*b^2)/e-1/16*(4*a^6-9*a^4*b^2+6*a^2*b^4-b^6)/b^8*(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*c os(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*(9*a^8-28*a^6*b^2+30*a^4*b^4-12*a^2*b^6+b^8)/b^8*(21*(e...
Timed out. \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+b \sin (c+d x))^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+b \sin (c+d x))^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+b \sin (c+d x))^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+b \sin (c+d x))^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+b \sin (c+d x))^4} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{15/2}}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]