Integrand size = 25, antiderivative size = 501 \[ \int \frac {1}{(a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}} \, dx=-\frac {b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{\left (-a^2+b^2\right )^{9/4} d e^{7/2}}+\frac {b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{\left (-a^2+b^2\right )^{9/4} d e^{7/2}}+\frac {2 (b-a \cos (c+d x))}{5 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{5/2}}-\frac {2 \left (5 b^3+a \left (3 a^2-8 b^2\right ) \cos (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \sin (c+d x)}}+\frac {a b^3 \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{\left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \sin (c+d x)}}+\frac {a b^3 \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{\left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \sin (c+d x)}}-\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 \left (a^2-b^2\right )^2 d e^4 \sqrt {\sin (c+d x)}} \]
-b^(7/2)*arctan(b^(1/2)*(e*sin(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/(-a ^2+b^2)^(9/4)/d/e^(7/2)+b^(7/2)*arctanh(b^(1/2)*(e*sin(d*x+c))^(1/2)/(-a^2 +b^2)^(1/4)/e^(1/2))/(-a^2+b^2)^(9/4)/d/e^(7/2)+2/5*(b-a*cos(d*x+c))/(a^2- b^2)/d/e/(e*sin(d*x+c))^(5/2)-2/5*(5*b^3+a*(3*a^2-8*b^2)*cos(d*x+c))/(a^2- b^2)^2/d/e^3/(e*sin(d*x+c))^(1/2)-a*b^3*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2 )/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*b/(b-(- a^2+b^2)^(1/2)),2^(1/2))*sin(d*x+c)^(1/2)/(a^2-b^2)^2/d/e^3/(b-(-a^2+b^2)^ (1/2))/(e*sin(d*x+c))^(1/2)-a*b^3*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin( 1/2*c+1/4*Pi+1/2*d*x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*b/(b+(-a^2+b^ 2)^(1/2)),2^(1/2))*sin(d*x+c)^(1/2)/(a^2-b^2)^2/d/e^3/(b+(-a^2+b^2)^(1/2)) /(e*sin(d*x+c))^(1/2)+2/5*a*(3*a^2-8*b^2)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1 /2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2)) *(e*sin(d*x+c))^(1/2)/(a^2-b^2)^2/d/e^4/sin(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.66 (sec) , antiderivative size = 811, normalized size of antiderivative = 1.62 \[ \int \frac {1}{(a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}} \, dx=\frac {\sin ^{\frac {7}{2}}(c+d x) \left (\frac {4 a^2 b-14 b^3+\left (-7 a^3+12 a b^2\right ) \cos (c+d x)+10 b^3 \cos (2 (c+d x))+3 a^3 \cos (3 (c+d x))-8 a b^2 \cos (3 (c+d x))}{2 \left (a^2-b^2\right )^2 \sin ^{\frac {5}{2}}(c+d x)}-\frac {\cos (c+d x) \left (a+b \sqrt {\cos ^2(c+d x)}\right ) \left (\frac {a \left (3 a^2-8 b^2\right ) \sec (c+d x) \left (3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )\right )+8 b^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)\right )}{\sqrt {b} \left (-a^2+b^2\right )}+\frac {24 \left (3 a^4-8 a^2 b^2-5 b^4\right ) \left (\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}+\frac {a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}\right )}{\sqrt {\cos ^2(c+d x)}}\right )}{12 (a-b)^2 (a+b)^2 (a+b \cos (c+d x))}\right )}{5 d (e \sin (c+d x))^{7/2}} \]
(Sin[c + d*x]^(7/2)*((4*a^2*b - 14*b^3 + (-7*a^3 + 12*a*b^2)*Cos[c + d*x] + 10*b^3*Cos[2*(c + d*x)] + 3*a^3*Cos[3*(c + d*x)] - 8*a*b^2*Cos[3*(c + d* x)])/(2*(a^2 - b^2)^2*Sin[c + d*x]^(5/2)) - (Cos[c + d*x]*(a + b*Sqrt[Cos[ c + d*x]^2])*((a*(3*a^2 - 8*b^2)*Sec[c + d*x]*(3*Sqrt[2]*a*(a^2 - b^2)^(3/ 4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Lo g[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4) *Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2) ))/(Sqrt[b]*(-a^2 + b^2)) + (24*(3*a^4 - 8*a^2*b^2 - 5*b^4)*(((1/8 + I/8)* (2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2 *ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[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[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^ (1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4) ) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a ^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2))))/Sqrt[Cos[c + d*x]^2]))/(1 2*(a - b)^2*(a + b)^2*(a + b*Cos[c + d*x]))))/(5*d*(e*Sin[c + d*x])^(7/2))
Time = 2.49 (sec) , antiderivative size = 494, normalized size of antiderivative = 0.99, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.840, Rules used = {3042, 3175, 27, 3042, 3345, 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.
| \(\displaystyle \int \frac {1}{(e \sin (c+d x))^{7/2} (a+b \cos (c+d x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{7/2} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 3175 |
| \(\displaystyle \frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}-\frac {2 \int -\frac {3 a^2+3 b \cos (c+d x) a-5 b^2}{2 (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}dx}{5 e^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 a^2+3 b \cos (c+d x) a-5 b^2}{(a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}dx}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 a^2-3 b \sin \left (c+d x-\frac {\pi }{2}\right ) a-5 b^2}{\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {-\frac {2 \int \frac {\left (3 a^4-8 b^2 a^2+b \left (3 a^2-8 b^2\right ) \cos (c+d x) a-5 b^4\right ) \sqrt {e \sin (c+d x)}}{2 (a+b \cos (c+d x))}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\left (3 a^4-8 b^2 a^2+b \left (3 a^2-8 b^2\right ) \cos (c+d x) a-5 b^4\right ) \sqrt {e \sin (c+d x)}}{a+b \cos (c+d x)}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {\sqrt {-e \cos \left (c+d x+\frac {\pi }{2}\right )} \left (3 a^4-8 b^2 a^2+b \left (3 a^2-8 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a-5 b^4\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3346 |
| \(\displaystyle \frac {-\frac {a \left (3 a^2-8 b^2\right ) \int \sqrt {e \sin (c+d x)}dx-5 b^4 \int \frac {\sqrt {e \sin (c+d x)}}{a+b \cos (c+d x)}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {a \left (3 a^2-8 b^2\right ) \int \sqrt {e \sin (c+d x)}dx-5 b^4 \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {-\frac {\frac {a \left (3 a^2-8 b^2\right ) \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}-5 b^4 \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {a \left (3 a^2-8 b^2\right ) \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}-5 b^4 \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-5 b^4 \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3180 |
| \(\displaystyle \frac {-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-5 b^4 \left (-\frac {b e \int \frac {\sqrt {e \sin (c+d x)}}{b^2 \sin ^2(c+d x) e^2+\left (a^2-b^2\right ) e^2}d(e \sin (c+d x))}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-5 b^4 \left (-\frac {2 b e \int \frac {e^2 \sin ^2(c+d x)}{b^2 e^4 \sin ^4(c+d x)+\left (a^2-b^2\right ) e^2}d\sqrt {e \sin (c+d x)}}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-5 b^4 \left (-\frac {2 b e \left (\frac {\int \frac {1}{b e^2 \sin ^2(c+d x)+\sqrt {b^2-a^2} e}d\sqrt {e \sin (c+d x)}}{2 b}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-5 b^4 \left (-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-5 b^4 \left (-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-5 b^4 \left (-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-5 b^4 \left (-\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b \sqrt {e \sin (c+d x)}}+\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \sin (c+d x)}}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-5 b^4 \left (-\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b \sqrt {e \sin (c+d x)}}+\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \sin (c+d x)}}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {-\frac {\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}-5 b^4 \left (-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}+\frac {a e \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \sin (c+d x)}}+\frac {a e \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \sin (c+d x)}}\right )}{e^2 \left (a^2-b^2\right )}-\frac {2 \left (a \left (3 a^2-8 b^2\right ) \cos (c+d x)+5 b^3\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}}{5 e^2 \left (a^2-b^2\right )}+\frac {2 (b-a \cos (c+d x))}{5 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{5/2}}\) |
(2*(b - a*Cos[c + d*x]))/(5*(a^2 - b^2)*d*e*(e*Sin[c + d*x])^(5/2)) + ((-2 *(5*b^3 + a*(3*a^2 - 8*b^2)*Cos[c + d*x]))/((a^2 - b^2)*d*e*Sqrt[e*Sin[c + d*x]]) - ((2*a*(3*a^2 - 8*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Si n[c + d*x]])/(d*Sqrt[Sin[c + d*x]]) - 5*b^4*((-2*b*e*(ArcTan[(Sqrt[b]*Sqrt [e]*Sin[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]*Sin[c + d*x])/(-a^2 + b^2)^(1/4)]/(2*b^(3/2) *(-a^2 + b^2)^(1/4)*Sqrt[e])))/d + (a*e*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b*(b - Sqrt[-a^2 + b^2] )*d*Sqrt[e*Sin[c + d*x]]) + (a*e*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b*(b + Sqrt[-a^2 + b^2])*d*Sqr t[e*Sin[c + d*x]])))/((a^2 - b^2)*e^2))/(5*(a^2 - b^2)*e^2)
3.1.67.3.1 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*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ (m + 1)*((b - a*Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2* (a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m* (a^2*(p + 2) - b^2*(m + p + 2) + a*b*(m + p + 3)*Sin[e + f*x]), x], x] /; F reeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegersQ [2*m, 2*p]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_ )]), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Simp[a*(g/(2*b)) Int[1/(S qrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (-Simp[a*(g/(2*b)) In t[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x], x] + Simp[b*(g/f) Su bst[Int[Sqrt[x]/(g^2*(a^2 - b^2) + b^2*x^2), x], x, g*Cos[e + f*x]], x])] / ; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt Q[p, -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]
Time = 3.46 (sec) , antiderivative size = 1007, normalized size of antiderivative = 2.01
(-2*e*b*(1/8*b^2/e^4/(a-b)^2/(a+b)^2/(e^2*(a^2-b^2)/b^2)^(1/4)*2^(1/2)*(ln ((e*sin(d*x+c)-(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)*2^(1/2)+(e^2 *(a^2-b^2)/b^2)^(1/2))/(e*sin(d*x+c)+(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+ c))^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2)))+2*arctan(2^(1/2)/(e^2*(a^2-b ^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)+1)+2*arctan(2^(1/2)/(e^2*(a^2-b^2)/b^2 )^(1/4)*(e*sin(d*x+c))^(1/2)-1))-1/5/e^2/(a+b)/(a-b)/(e*sin(d*x+c))^(5/2)+ 1/e^4/(a-b)^2/(a+b)^2*b^2/(e*sin(d*x+c))^(1/2))-1/10/e^3*(12*(1-sin(d*x+c) )^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(7/2)*EllipticE((1-sin(d*x+c))^( 1/2),1/2*2^(1/2))*a^4-32*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d *x+c)^(7/2)*EllipticE((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*a^2*b^2-6*(1-sin(d *x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(7/2)*EllipticF((1-sin(d*x+ c))^(1/2),1/2*2^(1/2))*a^4+16*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)* sin(d*x+c)^(7/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*a^2*b^2-5*(1- sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(7/2)*EllipticPi((1-si n(d*x+c))^(1/2),-b/(-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^3 -5*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(7/2)*EllipticPi ((1-sin(d*x+c))^(1/2),-b/(-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*b^4+5*(1-sin(d *x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(7/2)*EllipticPi((1-sin(d*x +c))^(1/2),1/(b+(-a^2+b^2)^(1/2))*b,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^3-5*(1 -sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(7/2)*EllipticPi((...
Timed out. \[ \int \frac {1}{(a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}} \, dx=\int \frac {1}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}\,\left (a+b\,\cos \left (c+d\,x\right )\right )} \,d x \]