Integrand size = 22, antiderivative size = 490 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx=\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{15 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac {2 \left (23 a^2+9 \left (b^2+c^2\right )\right ) E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{15 \left (a^2-b^2-c^2\right )^3 e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {16 a \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}{15 \left (a^2-b^2-c^2\right )^2 e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}+\frac {2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{15 \left (a^2-b^2-c^2\right )^3 e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \]
2/5*(c*cos(e*x+d)-b*sin(e*x+d))/(a^2-b^2-c^2)/e/(a+b*cos(e*x+d)+c*sin(e*x+ d))^(5/2)+16/15*(a*c*cos(e*x+d)-a*b*sin(e*x+d))/(a^2-b^2-c^2)^2/e/(a+b*cos (e*x+d)+c*sin(e*x+d))^(3/2)+2/15*(c*(23*a^2+9*b^2+9*c^2)*cos(e*x+d)-b*(23* a^2+9*b^2+9*c^2)*sin(e*x+d))/(a^2-b^2-c^2)^3/e/(a+b*cos(e*x+d)+c*sin(e*x+d ))^(1/2)+2/15*(23*a^2+9*b^2+9*c^2)*(cos(1/2*d+1/2*e*x-1/2*arctan(b,c))^2)^ (1/2)/cos(1/2*d+1/2*e*x-1/2*arctan(b,c))*EllipticE(sin(1/2*d+1/2*e*x-1/2*a rctan(b,c)),2^(1/2)*((b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2))*(a+b*cos( e*x+d)+c*sin(e*x+d))^(1/2)/(a^2-b^2-c^2)^3/e/((a+b*cos(e*x+d)+c*sin(e*x+d) )/(a+(b^2+c^2)^(1/2)))^(1/2)-16/15*a*(cos(1/2*d+1/2*e*x-1/2*arctan(b,c))^2 )^(1/2)/cos(1/2*d+1/2*e*x-1/2*arctan(b,c))*EllipticF(sin(1/2*d+1/2*e*x-1/2 *arctan(b,c)),2^(1/2)*((b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2))*((a+b*c os(e*x+d)+c*sin(e*x+d))/(a+(b^2+c^2)^(1/2)))^(1/2)/(a^2-b^2-c^2)^2/e/(a+b* cos(e*x+d)+c*sin(e*x+d))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.57 (sec) , antiderivative size = 4116, normalized size of antiderivative = 8.40 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx=\text {Result too large to show} \]
(Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]*((-2*(b^2 + c^2)*(23*a^2 + 9*b^ 2 + 9*c^2))/(15*b*c*(-a^2 + b^2 + c^2)^3) + (2*(a*c + b^2*Sin[d + e*x] + c ^2*Sin[d + e*x]))/(5*b*(-a^2 + b^2 + c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^3) - (2*(5*a^2*c + 3*b^2*c + 3*c^3 + 8*a*b^2*Sin[d + e*x] + 8*a*c^2* Sin[d + e*x]))/(15*b*(-a^2 + b^2 + c^2)^2*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) + (2*(15*a^3*c + 17*a*b^2*c + 17*a*c^3 + 23*a^2*b^2*Sin[d + e*x] + 9*b^4*Sin[d + e*x] + 23*a^2*c^2*Sin[d + e*x] + 18*b^2*c^2*Sin[d + e*x] + 9*c^4*Sin[d + e*x]))/(15*b*(-a^2 + b^2 + c^2)^3*(a + b*Cos[d + e*x] + c*S in[d + e*x]))))/e - (2*a^3*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^ 2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b ^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(S qrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[ b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2) /c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[( b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2]) ])/(Sqrt[1 + b^2/c^2]*c*(-a^2 + b^2 + c^2)^3*e) - (34*a*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqr t[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2] *c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2...
Time = 2.25 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.07, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {3042, 3608, 27, 3042, 3635, 27, 3042, 3635, 27, 3042, 3628, 3042, 3598, 3042, 3132, 3606, 3042, 3140}
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.
| \(\displaystyle \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3608 |
| \(\displaystyle \frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}-\frac {2 \int -\frac {5 a-3 b \cos (d+e x)-3 c \sin (d+e x)}{2 (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}dx}{5 \left (a^2-b^2-c^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 a-3 b \cos (d+e x)-3 c \sin (d+e x)}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}dx}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {5 a-3 b \cos (d+e x)-3 c \sin (d+e x)}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}dx}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3635 |
| \(\displaystyle \frac {\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}-\frac {2 \int -\frac {3 \left (5 a^2+3 \left (b^2+c^2\right )\right )-8 a b \cos (d+e x)-8 a c \sin (d+e x)}{2 (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}dx}{3 \left (a^2-b^2-c^2\right )}}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 \left (5 a^2+3 \left (b^2+c^2\right )\right )-8 a b \cos (d+e x)-8 a c \sin (d+e x)}{(a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}dx}{3 \left (a^2-b^2-c^2\right )}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {3 \left (5 a^2+3 \left (b^2+c^2\right )\right )-8 a b \cos (d+e x)-8 a c \sin (d+e x)}{(a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}dx}{3 \left (a^2-b^2-c^2\right )}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3635 |
| \(\displaystyle \frac {\frac {\frac {2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}-\frac {2 \int -\frac {a \left (15 a^2+17 \left (b^2+c^2\right )\right )+b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)+c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)}{2 \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{a^2-b^2-c^2}}{3 \left (a^2-b^2-c^2\right )}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a \left (15 a^2+17 \left (b^2+c^2\right )\right )+b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)+c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{a^2-b^2-c^2}+\frac {2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a \left (15 a^2+17 \left (b^2+c^2\right )\right )+b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)+c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{a^2-b^2-c^2}+\frac {2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3628 |
| \(\displaystyle \frac {\frac {\frac {\left (23 a^2+9 \left (b^2+c^2\right )\right ) \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}dx-8 a \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{a^2-b^2-c^2}+\frac {2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\left (23 a^2+9 \left (b^2+c^2\right )\right ) \int \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}dx-8 a \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{a^2-b^2-c^2}+\frac {2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3598 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-8 a \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{a^2-b^2-c^2}+\frac {2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-8 a \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{a^2-b^2-c^2}+\frac {2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-8 a \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx}{a^2-b^2-c^2}+\frac {2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3606 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {8 a \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{a^2-b^2-c^2}+\frac {2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {8 a \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{a^2-b^2-c^2}+\frac {2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {16 a \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{a^2-b^2-c^2}+\frac {2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
(2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(5*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)) + ((16*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x ]))/(3*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)) + ((2*(c*(23*a^2 + 9*(b^2 + c^2))*Cos[d + e*x] - b*(23*a^2 + 9*(b^2 + c^2))* Sin[d + e*x]))/((a^2 - b^2 - c^2)*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e* x]]) + ((2*(23*a^2 + 9*(b^2 + c^2))*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[d + e*x] + c*Sin [d + e*x]])/(e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]) - (16*a*(a^2 - b^2 - c^2)*EllipticF[(d + e*x - ArcTan[b, c])/2, (2 *Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[d + e*x] + c*Sin[ d + e*x])/(a + Sqrt[b^2 + c^2])])/(e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e *x]]))/(a^2 - b^2 - c^2))/(3*(a^2 - b^2 - c^2)))/(5*(a^2 - b^2 - c^2))
3.5.16.3.1 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[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ )]], x_Symbol] :> Simp[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])] Int[Sqrt[a/(a + S qrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - Arc Tan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*( x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sq rt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] Int[1/Sqrt[a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2 , 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (n_), x_Symbol] :> Simp[((-c)*Cos[d + e*x] + b*Sin[d + e*x])*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(a^2 - b^2 - c^2))), x] + Simp[ 1/((n + 1)*(a^2 - b^2 - c^2)) Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c *(n + 2)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x ] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1] && NeQ[n, -3/2]
Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) /Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]] , x_Symbol] :> Simp[B/b Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] , x] + Simp[(A*b - a*B)/b Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] , x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[ A*b - a*B, 0]
Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]) ^(n_)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) ]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A) *Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*( a^2 - b^2 - c^2))), x] + Simp[1/((n + 1)*(a^2 - b^2 - c^2)) Int[(a + b*Co s[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C) + (n + 2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]
Leaf count of result is larger than twice the leaf count of optimal. \(5027\) vs. \(2(535)=1070\).
Time = 30.19 (sec) , antiderivative size = 5028, normalized size of antiderivative = 10.26
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.51 (sec) , antiderivative size = 4955, normalized size of antiderivative = 10.11 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx=\text {Too large to display} \]
1/45*((sqrt(2)*(-I*a^3*b^4 + 33*I*a*b^6 - 99*I*a*b^2*c^4 - 99*a*b*c^5 + 3* (a^3*b - 22*a*b^3)*c^3 + 3*I*(a^3*b^2 - 22*a*b^4)*c^2 - (a^3*b^3 - 33*a*b^ 5)*c)*cos(e*x + d)^3 - 3*sqrt(2)*(I*a^4*b^3 - 33*I*a^2*b^5 - I*a^4*b*c^2 - a^4*c^3 + 33*I*a^2*b*c^4 + 33*a^2*c^5 + (a^4*b^2 - 33*a^2*b^4)*c)*cos(e*x + d)^2 - 3*sqrt(2)*(I*a^5*b^2 - 33*I*a^3*b^4 - 33*I*a*b^2*c^4 - 33*a*b*c^ 5 - (32*a^3*b + 33*a*b^3)*c^3 - I*(32*a^3*b^2 + 33*a*b^4)*c^2 + (a^5*b - 3 3*a^3*b^3)*c)*cos(e*x + d) + (sqrt(2)*(-33*I*a*b*c^5 - 33*a*c^6 + (a^3 + 6 6*a*b^2)*c^4 + I*(a^3*b + 66*a*b^3)*c^3 - 3*(a^3*b^2 - 33*a*b^4)*c^2 - 3*I *(a^3*b^3 - 33*a*b^5)*c)*cos(e*x + d)^2 - 6*sqrt(2)*(-33*I*a^2*b^2*c^3 - 3 3*a^2*b*c^4 + (a^4*b - 33*a^2*b^3)*c^2 + I*(a^4*b^2 - 33*a^2*b^4)*c)*cos(e *x + d) + sqrt(2)*(33*I*a*b*c^5 + 33*a*c^6 + (98*a^3 + 33*a*b^2)*c^4 + I*( 98*a^3*b + 33*a*b^3)*c^3 - 3*(a^5 - 33*a^3*b^2)*c^2 - 3*I*(a^5*b - 33*a^3* b^3)*c))*sin(e*x + d) + sqrt(2)*(-I*a^6*b + 33*I*a^4*b^3 + 99*I*a^2*b*c^4 + 99*a^2*c^5 + 3*(10*a^4 + 33*a^2*b^2)*c^3 + 3*I*(10*a^4*b + 33*a^2*b^3)*c ^2 - (a^6 - 33*a^4*b^2)*c))*sqrt(b + I*c)*weierstrassPInverse(4/3*(4*a^2*b ^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 + 3*c^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5 + 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), 1/3*(2*a*b - 2*I*a*c + 3* (b^2 + c^2)*cos(e*x + d) - 3*(I*b^2 + I*c^2)*sin(e*x + d))/(b^2 + c^2))...
Timed out. \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx=\int \frac {1}{{\left (a+b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )\right )}^{7/2}} \,d x \]