Integrand size = 14, antiderivative size = 282 \[ \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx=\frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {16 a \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {2 b \sin (c+d x)}{5 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {16 a b \sin (c+d x)}{15 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{15 \left (a^2-b^2\right )^3 d \sqrt {a+b \cos (c+d x)}} \]
-2/5*b*sin(d*x+c)/(a^2-b^2)/d/(a+b*cos(d*x+c))^(5/2)-16/15*a*b*sin(d*x+c)/ (a^2-b^2)^2/d/(a+b*cos(d*x+c))^(3/2)-2/15*b*(23*a^2+9*b^2)*sin(d*x+c)/(a^2 -b^2)^3/d/(a+b*cos(d*x+c))^(1/2)+2/15*(23*a^2+9*b^2)*(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)*(b/(a+b))^ (1/2))*(a+b*cos(d*x+c))^(1/2)/(a^2-b^2)^3/d/((a+b*cos(d*x+c))/(a+b))^(1/2) -16/15*a*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2 *d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/(a^2-b ^2)^2/d/(a+b*cos(d*x+c))^(1/2)
Time = 0.92 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx=\frac {2 \left (\frac {\left (\frac {a+b \cos (c+d x)}{a+b}\right )^{5/2} \left (\left (23 a^2+9 b^2\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+8 a (-a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )}{(a-b)^3}+\frac {b \left (34 a^4-5 a^2 b^2+3 b^4+2 a b \left (27 a^2+5 b^2\right ) \cos (c+d x)+b^2 \left (23 a^2+9 b^2\right ) \cos ^2(c+d x)\right ) \sin (c+d x)}{\left (-a^2+b^2\right )^3}\right )}{15 d (a+b \cos (c+d x))^{5/2}} \]
(2*((((a + b*Cos[c + d*x])/(a + b))^(5/2)*((23*a^2 + 9*b^2)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 8*a*(-a + b)*EllipticF[(c + d*x)/2, (2*b)/(a + b )]))/(a - b)^3 + (b*(34*a^4 - 5*a^2*b^2 + 3*b^4 + 2*a*b*(27*a^2 + 5*b^2)*C os[c + d*x] + b^2*(23*a^2 + 9*b^2)*Cos[c + d*x]^2)*Sin[c + d*x])/(-a^2 + b ^2)^3))/(15*d*(a + b*Cos[c + d*x])^(5/2))
Time = 1.46 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.09, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.286, Rules used = {3042, 3143, 27, 3042, 3233, 27, 3042, 3233, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 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.
| \(\displaystyle \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 3143 |
| \(\displaystyle -\frac {2 \int -\frac {5 a-3 b \cos (c+d x)}{2 (a+b \cos (c+d x))^{5/2}}dx}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 a-3 b \cos (c+d x)}{(a+b \cos (c+d x))^{5/2}}dx}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {5 a-3 b \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {3 \left (5 a^2+3 b^2\right )-8 a b \cos (c+d x)}{2 (a+b \cos (c+d x))^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 \left (5 a^2+3 b^2\right )-8 a b \cos (c+d x)}{(a+b \cos (c+d x))^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {3 \left (5 a^2+3 b^2\right )-8 a b \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {-\frac {2 \int -\frac {a \left (15 a^2+17 b^2\right )+b \left (23 a^2+9 b^2\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a \left (15 a^2+17 b^2\right )+b \left (23 a^2+9 b^2\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a \left (15 a^2+17 b^2\right )+b \left (23 a^2+9 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {\left (23 a^2+9 b^2\right ) \int \sqrt {a+b \cos (c+d x)}dx-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\left (23 a^2+9 b^2\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {8 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {8 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {16 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
(-2*b*Sin[c + d*x])/(5*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(5/2)) + ((-16*a *b*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (((2*(23*a ^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] )/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (16*a*(a^2 - b^2)*Sqrt[(a + b*C os[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b *Cos[c + d*x]]))/(a^2 - b^2) - (2*b*(23*a^2 + 9*b^2)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]))/(3*(a^2 - b^2)))/(5*(a^2 - b^2))
3.6.46.3.1 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[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], 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[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 - b^2))), x] + Simp [1/((n + 1)*(a^2 - b^2)) Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Time = 7.51 (sec) , antiderivative size = 616, normalized size of antiderivative = 2.18
| method | result | size |
| default | \(-\frac {\sqrt {-\left (-2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{10 b^{2} \left (a -b \right ) \left (a +b \right ) \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {a -b}{2 b}\right )^{3}}+\frac {8 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{15 b \left (a -b \right )^{2} \left (a +b \right )^{2} \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {a -b}{2 b}\right )^{2}}+\frac {4 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (23 a^{2}+9 b^{2}\right )}{15 \left (a -b \right )^{3} \left (a +b \right )^{3} \sqrt {-\left (-2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}+\frac {2 \left (15 a^{2}-8 a b +9 b^{2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b}{a -b}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )}{\left (15 a^{5}+15 a^{4} b -30 a^{3} b^{2}-30 a^{2} b^{3}+15 a \,b^{4}+15 b^{5}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}-\frac {2 \left (23 a^{2}+9 b^{2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b}{a -b}}\, \left (F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )-E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )\right )}{15 \left (a -b \right )^{2} \left (a +b \right )^{3} \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}\, d}\) | \(616\) |
-(-(-2*b*cos(1/2*d*x+1/2*c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(1/10/b^2/( a-b)/(a+b)*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x +1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2+1/2*(a-b)/b)^3+8/15*a/b/(a-b)^2/(a+ b)^2*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c )^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2+1/2*(a-b)/b)^2+4/15*sin(1/2*d*x+1/2*c)^2* b/(a-b)^3/(a+b)^3*cos(1/2*d*x+1/2*c)*(23*a^2+9*b^2)/(-(-2*b*cos(1/2*d*x+1/ 2*c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*(15*a^2-8*a*b+9*b^2)/(15*a^5+15* a^4*b-30*a^3*b^2-30*a^2*b^3+15*a*b^4+15*b^5)*(sin(1/2*d*x+1/2*c)^2)^(1/2)* ((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a +b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^ (1/2))-2/15*(23*a^2+9*b^2)/(a-b)^2/(a+b)^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(( 2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b )*sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^( 1/2))-EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))))/sin(1/2*d*x+1/2*c )/(-2*b*sin(1/2*d*x+1/2*c)^2+a+b)^(1/2)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 985, normalized size of antiderivative = 3.49 \[ \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx=\text {Too large to display} \]
-1/45*(6*(34*a^4*b^2 - 5*a^2*b^4 + 3*b^6 + (23*a^2*b^4 + 9*b^6)*cos(d*x + c)^2 + 2*(27*a^3*b^3 + 5*a*b^5)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)*sin (d*x + c) + (sqrt(2)*(-I*a^3*b^3 + 33*I*a*b^5)*cos(d*x + c)^3 - 3*sqrt(2)* (I*a^4*b^2 - 33*I*a^2*b^4)*cos(d*x + c)^2 - 3*sqrt(2)*(I*a^5*b - 33*I*a^3* b^3)*cos(d*x + c) + sqrt(2)*(-I*a^6 + 33*I*a^4*b^2))*sqrt(b)*weierstrassPI nverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos( d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b) + (sqrt(2)*(I*a^3*b^3 - 33*I*a*b^5 )*cos(d*x + c)^3 - 3*sqrt(2)*(-I*a^4*b^2 + 33*I*a^2*b^4)*cos(d*x + c)^2 - 3*sqrt(2)*(-I*a^5*b + 33*I*a^3*b^3)*cos(d*x + c) + sqrt(2)*(I*a^6 - 33*I*a ^4*b^2))*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) - 3* (sqrt(2)*(23*I*a^2*b^4 + 9*I*b^6)*cos(d*x + c)^3 + 3*sqrt(2)*(23*I*a^3*b^3 + 9*I*a*b^5)*cos(d*x + c)^2 + 3*sqrt(2)*(23*I*a^4*b^2 + 9*I*a^2*b^4)*cos( d*x + c) + sqrt(2)*(23*I*a^5*b + 9*I*a^3*b^3))*sqrt(b)*weierstrassZeta(4/3 *(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3 *(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b)) - 3*(sqrt(2)*(-23*I*a^2*b^4 - 9*I*b^6)*cos( d*x + c)^3 + 3*sqrt(2)*(-23*I*a^3*b^3 - 9*I*a*b^5)*cos(d*x + c)^2 + 3*sqrt (2)*(-23*I*a^4*b^2 - 9*I*a^2*b^4)*cos(d*x + c) + sqrt(2)*(-23*I*a^5*b - 9* I*a^3*b^3))*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a...
Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx=\int \frac {1}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]