Integrand size = 25, antiderivative size = 591 \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx=\frac {7 a \left (5 a^2-6 b^2\right ) e^{9/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 b^{9/2} \left (-a^2+b^2\right )^{5/4} d}-\frac {7 a \left (5 a^2-6 b^2\right ) e^{9/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 b^{9/2} \left (-a^2+b^2\right )^{5/4} d}+\frac {7 \left (5 a^2-4 b^2\right ) e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^4 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}-\frac {7 a^2 \left (5 a^2-6 b^2\right ) e^5 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 b^5 \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {7 a^2 \left (5 a^2-6 b^2\right ) e^5 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 b^5 \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}+\frac {7 \left (5 a^2-4 b^2\right ) e^3 (e \cos (c+d x))^{3/2}}{8 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2} \]
7/16*a*(5*a^2-6*b^2)*e^(9/2)*arctan(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2 )^(1/4)/e^(1/2))/b^(9/2)/(-a^2+b^2)^(5/4)/d-7/16*a*(5*a^2-6*b^2)*e^(9/2)*a rctanh(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/b^(9/2)/(-a^ 2+b^2)^(5/4)/d-1/3*e*(e*cos(d*x+c))^(7/2)/b/d/(a+b*sin(d*x+c))^3+7/8*(5*a^ 2-4*b^2)*e^3*(e*cos(d*x+c))^(3/2)/b^3/(a^2-b^2)/d/(a+b*sin(d*x+c))-7/12*e^ 3*(e*cos(d*x+c))^(3/2)*(5*a+4*b*sin(d*x+c))/b^3/d/(a+b*sin(d*x+c))^2-7/16* a^2*(5*a^2-6*b^2)*e^5*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Elli pticPi(sin(1/2*d*x+1/2*c),2*b/(b-(-a^2+b^2)^(1/2)),2^(1/2))*cos(d*x+c)^(1/ 2)/b^5/(a^2-b^2)/d/(b-(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)-7/16*a^2*(5*a ^2-6*b^2)*e^5*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(s in(1/2*d*x+1/2*c),2*b/(b+(-a^2+b^2)^(1/2)),2^(1/2))*cos(d*x+c)^(1/2)/b^5/( a^2-b^2)/d/(b+(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)+7/8*(5*a^2-4*b^2)*e^4 *(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2 *c),2^(1/2))*(e*cos(d*x+c))^(1/2)/b^4/(a^2-b^2)/d/cos(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 20.76 (sec) , antiderivative size = 900, normalized size of antiderivative = 1.52 \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx=\frac {(e \cos (c+d x))^{9/2} \sec ^4(c+d x) \left (\frac {a^2 \cos (c+d x)-b^2 \cos (c+d x)}{3 b^3 (a+b \sin (c+d x))^3}-\frac {5 a \cos (c+d x)}{4 b^3 (a+b \sin (c+d x))^2}+\frac {-19 a^2 \cos (c+d x)+12 b^2 \cos (c+d x)}{8 b^3 \left (-a^2+b^2\right ) (a+b \sin (c+d x))}\right )}{d}+\frac {7 (e \cos (c+d x))^{9/2} \left (-\frac {4 a b \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (\frac {a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}\right ) \sin (c+d x)}{\sqrt {1-\cos ^2(c+d x)} (a+b \sin (c+d x))}-\frac {\left (5 a^2-4 b^2\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (8 b^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)+3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )\right )\right ) \sin ^2(c+d x)}{12 b^{3/2} \left (-a^2+b^2\right ) \left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}\right )}{16 (a-b) b^3 (a+b) d \cos ^{\frac {9}{2}}(c+d x)} \]
((e*Cos[c + d*x])^(9/2)*Sec[c + d*x]^4*((a^2*Cos[c + d*x] - b^2*Cos[c + d* x])/(3*b^3*(a + b*Sin[c + d*x])^3) - (5*a*Cos[c + d*x])/(4*b^3*(a + b*Sin[ c + d*x])^2) + (-19*a^2*Cos[c + d*x] + 12*b^2*Cos[c + d*x])/(8*b^3*(-a^2 + b^2)*(a + b*Sin[c + d*x]))))/d + (7*(e*Cos[c + d*x])^(9/2)*((-4*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(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[C os[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]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x ])) - ((5*a^2 - 4*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF 1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Co s[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sq rt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[ b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]* Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[ a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[ c + d*x]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^...
Time = 2.56 (sec) , antiderivative size = 532, normalized size of antiderivative = 0.90, 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, 3343, 27, 3042, 3346, 3042, 3121, 3042, 3119, 3180, 266, 827, 218, 221, 3042, 3286, 3042, 3284}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4}dx\) |
| \(\Big \downarrow \) 3172 |
| \(\displaystyle -\frac {7 e^2 \int \frac {(e \cos (c+d x))^{5/2} \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 e^2 \int \frac {(e \cos (c+d x))^{5/2} \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3342 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {3 e^2 \int -\frac {\sqrt {e \cos (c+d x)} (4 b+5 a \sin (c+d x))}{2 (a+b \sin (c+d x))^2}dx}{2 b^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \int \frac {\sqrt {e \cos (c+d x)} (4 b+5 a \sin (c+d x))}{(a+b \sin (c+d x))^2}dx}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \int \frac {\sqrt {e \cos (c+d x)} (4 b+5 a \sin (c+d x))}{(a+b \sin (c+d x))^2}dx}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3343 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (2 a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{2 (a+b \sin (c+d x))}dx}{a^2-b^2}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (2 a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (2 a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3346 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\frac {\left (5 a^2-4 b^2\right ) \int \sqrt {e \cos (c+d x)}dx}{b}-\frac {a \left (5 a^2-6 b^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{b}}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\frac {\left (5 a^2-4 b^2\right ) \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {a \left (5 a^2-6 b^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{b}}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\frac {\left (5 a^2-4 b^2\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{b \sqrt {\cos (c+d x)}}-\frac {a \left (5 a^2-6 b^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{b}}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\frac {\left (5 a^2-4 b^2\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b \sqrt {\cos (c+d x)}}-\frac {a \left (5 a^2-6 b^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{b}}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\frac {2 \left (5 a^2-4 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (5 a^2-6 b^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)}dx}{b}}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3180 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\frac {2 \left (5 a^2-4 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (5 a^2-6 b^2\right ) \left (\frac {b e \int \frac {\sqrt {e \cos (c+d x)}}{b^2 \cos ^2(c+d x) e^2+\left (a^2-b^2\right ) e^2}d(e \cos (c+d x))}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{b}}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\frac {2 \left (5 a^2-4 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (5 a^2-6 b^2\right ) \left (\frac {2 b e \int \frac {e^2 \cos ^2(c+d x)}{b^2 e^4 \cos ^4(c+d x)+\left (a^2-b^2\right ) e^2}d\sqrt {e \cos (c+d x)}}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{b}}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\frac {2 \left (5 a^2-4 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (5 a^2-6 b^2\right ) \left (\frac {2 b e \left (\frac {\int \frac {1}{b e^2 \cos ^2(c+d x)+\sqrt {b^2-a^2} e}d\sqrt {e \cos (c+d x)}}{2 b}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \cos ^2(c+d x)}d\sqrt {e \cos (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{b}}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\frac {2 \left (5 a^2-4 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (5 a^2-6 b^2\right ) \left (\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \cos ^2(c+d x)}d\sqrt {e \cos (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{b}}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\frac {2 \left (5 a^2-4 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (5 a^2-6 b^2\right ) \left (-\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{b}}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\frac {2 \left (5 a^2-4 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (5 a^2-6 b^2\right ) \left (-\frac {a e \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \sin \left (c+d x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 b}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{b}}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\frac {2 \left (5 a^2-4 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (5 a^2-6 b^2\right ) \left (-\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {b^2-a^2}-b \cos (c+d x)\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \left (b \cos (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{b}}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\frac {2 \left (5 a^2-4 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (5 a^2-6 b^2\right ) \left (-\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sqrt {b^2-a^2}-b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {a e \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \sin \left (c+d x+\frac {\pi }{2}\right )+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \cos (c+d x)}}+\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{b}}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle -\frac {7 e^2 \left (\frac {3 e^2 \left (-\frac {\frac {2 \left (5 a^2-4 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (5 a^2-6 b^2\right ) \left (\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \cos (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}+\frac {a e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{b d \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {a e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{b d \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}\right )}{b}}{2 \left (a^2-b^2\right )}-\frac {\left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{4 b^2}+\frac {e (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}\right )}{6 b}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}\) |
-1/3*(e*(e*Cos[c + d*x])^(7/2))/(b*d*(a + b*Sin[c + d*x])^3) - (7*e^2*((e* (e*Cos[c + d*x])^(3/2)*(5*a + 4*b*Sin[c + d*x]))/(2*b^2*d*(a + b*Sin[c + d *x])^2) + (3*e^2*(-1/2*((2*(5*a^2 - 4*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[ (c + d*x)/2, 2])/(b*d*Sqrt[Cos[c + d*x]]) - (a*(5*a^2 - 6*b^2)*((2*b*e*(Ar cTan[(Sqrt[b]*Sqrt[e]*Cos[c + d*x])/(-a^2 + b^2)^(1/4)]/(2*b^(3/2)*(-a^2 + b^2)^(1/4)*Sqrt[e]) - ArcTanh[(Sqrt[b]*Sqrt[e]*Cos[c + d*x])/(-a^2 + b^2) ^(1/4)]/(2*b^(3/2)*(-a^2 + b^2)^(1/4)*Sqrt[e])))/d + (a*e*Sqrt[Cos[c + d*x ]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b*(b - Sqrt[ -a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (a*e*Sqrt[Cos[c + d*x]]*EllipticPi[ (2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b*(b + Sqrt[-a^2 + b^2])*d *Sqrt[e*Cos[c + d*x]])))/b)/(a^2 - b^2) - ((5*a^2 - 4*b^2)*(e*Cos[c + d*x] )^(3/2))/((a^2 - b^2)*d*e*(a + b*Sin[c + d*x]))))/(4*b^2)))/(6*b)
3.7.8.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + 1))), x] + Simp[g^2*((p - 1)/(b*(m + 1))) Int[(g*Cos [e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; Fre eQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && I ntegersQ[2*m, 2*p]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_ )]), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Simp[a*(g/(2*b)) Int[1/(S qrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (-Simp[a*(g/(2*b)) In t[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x], x] + Simp[b*(g/f) Su bst[Int[Sqrt[x]/(g^2*(a^2 - b^2) + b^2*x^2), x], x, g*Cos[e + f*x]], x])] / ; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[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[(-(b*c - a*d))*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) Int[(g*Cos[e + f*x])^p *(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ [a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((c_.) + (d_.)*sin[(e_.) + (f_.)* (x_)]))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d/b Int [(g*Cos[e + f*x])^p, x], x] + Simp[(b*c - a*d)/b Int[(g*Cos[e + f*x])^p/( a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 341.36 (sec) , antiderivative size = 5039, normalized size of antiderivative = 8.53
Timed out. \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx=\text {Timed out} \]
\[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]
\[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{9/2}}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]