Integrand size = 22, antiderivative size = 490 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx=\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{15 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac {2 \left (23 a^2+9 \left (b^2+c^2\right )\right ) E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{15 \left (a^2-b^2-c^2\right )^3 e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {16 a \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}{15 \left (a^2-b^2-c^2\right )^2 e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}+\frac {2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{15 \left (a^2-b^2-c^2\right )^3 e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \] Output:
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*(23*a^2+9*b^2+9*c^2)*EllipticE(sin(1/2*d+ 1/2*e*x-1/2*arctan(c,b)),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*InverseJacobiAM(1/2*d+1/ 2*e*x-1/2*arctan(c,b),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))/(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)+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)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.56 (sec) , antiderivative size = 4116, normalized size of antiderivative = 8.40 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx=\text {Result too large to show} \] Input:
Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-7/2),x]
Output:
(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 = 2.31 (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 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {16 a \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{a^2-b^2-c^2}+\frac {2 \left (c \left (23 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (23 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{e \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}}{3 \left (a^2-b^2-c^2\right )}+\frac {16 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}}{5 \left (a^2-b^2-c^2\right )}+\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}\) |
Input:
Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-7/2),x]
Output:
(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))
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. \(5021\) vs. \(2(466)=932\).
Time = 3.76 (sec) , antiderivative size = 5022, normalized size of antiderivative = 10.25
Input:
int(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^(7/2),x,method=_RETURNVERBOSE)
Output:
result too large to display
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 4863, normalized size of antiderivative = 9.92 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^(7/2),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))**(7/2),x)
Output:
Timed out
\[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^(7/2),x, algorithm="maxima")
Output:
integrate((b*cos(e*x + d) + c*sin(e*x + d) + a)^(-7/2), x)
\[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^(7/2),x, algorithm="giac")
Output:
integrate((b*cos(e*x + d) + c*sin(e*x + d) + a)^(-7/2), x)
Timed out. \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx=\int \frac {1}{{\left (a+b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )\right )}^{7/2}} \,d x \] Input:
int(1/(a + b*cos(d + e*x) + c*sin(d + e*x))^(7/2),x)
Output:
int(1/(a + b*cos(d + e*x) + c*sin(d + e*x))^(7/2), x)
\[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx=\int \frac {\sqrt {a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )}}{\cos \left (e x +d \right )^{4} b^{4}+4 \cos \left (e x +d \right )^{3} \sin \left (e x +d \right ) b^{3} c +4 \cos \left (e x +d \right )^{3} a \,b^{3}+6 \cos \left (e x +d \right )^{2} \sin \left (e x +d \right )^{2} b^{2} c^{2}+12 \cos \left (e x +d \right )^{2} \sin \left (e x +d \right ) a \,b^{2} c +6 \cos \left (e x +d \right )^{2} a^{2} b^{2}+4 \cos \left (e x +d \right ) \sin \left (e x +d \right )^{3} b \,c^{3}+12 \cos \left (e x +d \right ) \sin \left (e x +d \right )^{2} a b \,c^{2}+12 \cos \left (e x +d \right ) \sin \left (e x +d \right ) a^{2} b c +4 \cos \left (e x +d \right ) a^{3} b +\sin \left (e x +d \right )^{4} c^{4}+4 \sin \left (e x +d \right )^{3} a \,c^{3}+6 \sin \left (e x +d \right )^{2} a^{2} c^{2}+4 \sin \left (e x +d \right ) a^{3} c +a^{4}}d x \] Input:
int(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^(7/2),x)
Output:
int(sqrt(cos(d + e*x)*b + sin(d + e*x)*c + a)/(cos(d + e*x)**4*b**4 + 4*co s(d + e*x)**3*sin(d + e*x)*b**3*c + 4*cos(d + e*x)**3*a*b**3 + 6*cos(d + e *x)**2*sin(d + e*x)**2*b**2*c**2 + 12*cos(d + e*x)**2*sin(d + e*x)*a*b**2* c + 6*cos(d + e*x)**2*a**2*b**2 + 4*cos(d + e*x)*sin(d + e*x)**3*b*c**3 + 12*cos(d + e*x)*sin(d + e*x)**2*a*b*c**2 + 12*cos(d + e*x)*sin(d + e*x)*a* *2*b*c + 4*cos(d + e*x)*a**3*b + sin(d + e*x)**4*c**4 + 4*sin(d + e*x)**3* a*c**3 + 6*sin(d + e*x)**2*a**2*c**2 + 4*sin(d + e*x)*a**3*c + a**4),x)