Integrand size = 12, antiderivative size = 121 \[ \int \left (a+a \cos ^2(x)\right )^{5/2} \, dx=\frac {18 a^2 \sqrt {a+a \cos ^2(x)} E\left (\left .\frac {\pi }{2}+x\right |-1\right )}{5 \sqrt {1+\cos ^2(x)}}-\frac {8 a^3 \sqrt {1+\cos ^2(x)} \operatorname {EllipticF}\left (\frac {\pi }{2}+x,-1\right )}{5 \sqrt {a+a \cos ^2(x)}}+\frac {4}{5} a^2 \cos (x) \sqrt {a+a \cos ^2(x)} \sin (x)+\frac {1}{5} a \cos (x) \left (a+a \cos ^2(x)\right )^{3/2} \sin (x) \] Output:
18/5*a^2*(a+a*cos(x)^2)^(1/2)*EllipticE(cos(x),I)/(1+cos(x)^2)^(1/2)-8/5*a ^3*(1+cos(x)^2)^(1/2)*InverseJacobiAM(1/2*Pi+x,I)/(a+a*cos(x)^2)^(1/2)+4/5 *a^2*cos(x)*(a+a*cos(x)^2)^(1/2)*sin(x)+1/5*a*cos(x)*(a+a*cos(x)^2)^(3/2)* sin(x)
Time = 0.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.55 \[ \int \left (a+a \cos ^2(x)\right )^{5/2} \, dx=\frac {a^2 \sqrt {a (3+\cos (2 x))} \left (288 E\left (x\left |\frac {1}{2}\right .\right )-64 \operatorname {EllipticF}\left (x,\frac {1}{2}\right )+\sqrt {3+\cos (2 x)} (22 \sin (2 x)+\sin (4 x))\right )}{80 \sqrt {1+\cos ^2(x)}} \] Input:
Integrate[(a + a*Cos[x]^2)^(5/2),x]
Output:
(a^2*Sqrt[a*(3 + Cos[2*x])]*(288*EllipticE[x, 1/2] - 64*EllipticF[x, 1/2] + Sqrt[3 + Cos[2*x]]*(22*Sin[2*x] + Sin[4*x])))/(80*Sqrt[1 + Cos[x]^2])
Time = 0.79 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {3042, 3659, 27, 3042, 3649, 3042, 3651, 3042, 3657, 3042, 3656, 3662, 3042, 3661}
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.
| \(\displaystyle \int \left (a \cos ^2(x)+a\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin \left (x+\frac {\pi }{2}\right )^2+a\right )^{5/2}dx\) |
| \(\Big \downarrow \) 3659 |
| \(\displaystyle \frac {1}{5} \int 6 \sqrt {a \cos ^2(x)+a} \left (2 \cos ^2(x) a^2+a^2\right )dx+\frac {1}{5} a \sin (x) \cos (x) \left (a \cos ^2(x)+a\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{5} \int \sqrt {a \cos ^2(x)+a} \left (2 \cos ^2(x) a^2+a^2\right )dx+\frac {1}{5} a \sin (x) \cos (x) \left (a \cos ^2(x)+a\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6}{5} \int \sqrt {a \sin \left (x+\frac {\pi }{2}\right )^2+a} \left (2 \sin \left (x+\frac {\pi }{2}\right )^2 a^2+a^2\right )dx+\frac {1}{5} a \sin (x) \cos (x) \left (a \cos ^2(x)+a\right )^{3/2}\) |
| \(\Big \downarrow \) 3649 |
| \(\displaystyle \frac {6}{5} \left (\frac {1}{3} \int \frac {9 \cos ^2(x) a^3+5 a^3}{\sqrt {a \cos ^2(x)+a}}dx+\frac {2}{3} a^2 \sin (x) \cos (x) \sqrt {a \cos ^2(x)+a}\right )+\frac {1}{5} a \sin (x) \cos (x) \left (a \cos ^2(x)+a\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6}{5} \left (\frac {1}{3} \int \frac {9 \sin \left (x+\frac {\pi }{2}\right )^2 a^3+5 a^3}{\sqrt {a \sin \left (x+\frac {\pi }{2}\right )^2+a}}dx+\frac {2}{3} a^2 \sin (x) \cos (x) \sqrt {a \cos ^2(x)+a}\right )+\frac {1}{5} a \sin (x) \cos (x) \left (a \cos ^2(x)+a\right )^{3/2}\) |
| \(\Big \downarrow \) 3651 |
| \(\displaystyle \frac {6}{5} \left (\frac {1}{3} \left (9 a^2 \int \sqrt {a \cos ^2(x)+a}dx-4 a^3 \int \frac {1}{\sqrt {a \cos ^2(x)+a}}dx\right )+\frac {2}{3} a^2 \sin (x) \cos (x) \sqrt {a \cos ^2(x)+a}\right )+\frac {1}{5} a \sin (x) \cos (x) \left (a \cos ^2(x)+a\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6}{5} \left (\frac {1}{3} \left (9 a^2 \int \sqrt {a \sin \left (x+\frac {\pi }{2}\right )^2+a}dx-4 a^3 \int \frac {1}{\sqrt {a \sin \left (x+\frac {\pi }{2}\right )^2+a}}dx\right )+\frac {2}{3} a^2 \sin (x) \cos (x) \sqrt {a \cos ^2(x)+a}\right )+\frac {1}{5} a \sin (x) \cos (x) \left (a \cos ^2(x)+a\right )^{3/2}\) |
| \(\Big \downarrow \) 3657 |
| \(\displaystyle \frac {6}{5} \left (\frac {1}{3} \left (\frac {9 a^2 \sqrt {a \cos ^2(x)+a} \int \sqrt {\cos ^2(x)+1}dx}{\sqrt {\cos ^2(x)+1}}-4 a^3 \int \frac {1}{\sqrt {a \sin \left (x+\frac {\pi }{2}\right )^2+a}}dx\right )+\frac {2}{3} a^2 \sin (x) \cos (x) \sqrt {a \cos ^2(x)+a}\right )+\frac {1}{5} a \sin (x) \cos (x) \left (a \cos ^2(x)+a\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6}{5} \left (\frac {1}{3} \left (\frac {9 a^2 \sqrt {a \cos ^2(x)+a} \int \sqrt {\sin \left (x+\frac {\pi }{2}\right )^2+1}dx}{\sqrt {\cos ^2(x)+1}}-4 a^3 \int \frac {1}{\sqrt {a \sin \left (x+\frac {\pi }{2}\right )^2+a}}dx\right )+\frac {2}{3} a^2 \sin (x) \cos (x) \sqrt {a \cos ^2(x)+a}\right )+\frac {1}{5} a \sin (x) \cos (x) \left (a \cos ^2(x)+a\right )^{3/2}\) |
| \(\Big \downarrow \) 3656 |
| \(\displaystyle \frac {6}{5} \left (\frac {1}{3} \left (\frac {9 a^2 E\left (\left .x+\frac {\pi }{2}\right |-1\right ) \sqrt {a \cos ^2(x)+a}}{\sqrt {\cos ^2(x)+1}}-4 a^3 \int \frac {1}{\sqrt {a \sin \left (x+\frac {\pi }{2}\right )^2+a}}dx\right )+\frac {2}{3} a^2 \sin (x) \cos (x) \sqrt {a \cos ^2(x)+a}\right )+\frac {1}{5} a \sin (x) \cos (x) \left (a \cos ^2(x)+a\right )^{3/2}\) |
| \(\Big \downarrow \) 3662 |
| \(\displaystyle \frac {6}{5} \left (\frac {1}{3} \left (\frac {9 a^2 E\left (\left .x+\frac {\pi }{2}\right |-1\right ) \sqrt {a \cos ^2(x)+a}}{\sqrt {\cos ^2(x)+1}}-\frac {4 a^3 \sqrt {\cos ^2(x)+1} \int \frac {1}{\sqrt {\cos ^2(x)+1}}dx}{\sqrt {a \cos ^2(x)+a}}\right )+\frac {2}{3} a^2 \sin (x) \cos (x) \sqrt {a \cos ^2(x)+a}\right )+\frac {1}{5} a \sin (x) \cos (x) \left (a \cos ^2(x)+a\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6}{5} \left (\frac {1}{3} \left (\frac {9 a^2 E\left (\left .x+\frac {\pi }{2}\right |-1\right ) \sqrt {a \cos ^2(x)+a}}{\sqrt {\cos ^2(x)+1}}-\frac {4 a^3 \sqrt {\cos ^2(x)+1} \int \frac {1}{\sqrt {\sin \left (x+\frac {\pi }{2}\right )^2+1}}dx}{\sqrt {a \cos ^2(x)+a}}\right )+\frac {2}{3} a^2 \sin (x) \cos (x) \sqrt {a \cos ^2(x)+a}\right )+\frac {1}{5} a \sin (x) \cos (x) \left (a \cos ^2(x)+a\right )^{3/2}\) |
| \(\Big \downarrow \) 3661 |
| \(\displaystyle \frac {6}{5} \left (\frac {2}{3} a^2 \sin (x) \cos (x) \sqrt {a \cos ^2(x)+a}+\frac {1}{3} \left (\frac {9 a^2 E\left (\left .x+\frac {\pi }{2}\right |-1\right ) \sqrt {a \cos ^2(x)+a}}{\sqrt {\cos ^2(x)+1}}-\frac {4 a^3 \sqrt {\cos ^2(x)+1} \operatorname {EllipticF}\left (x+\frac {\pi }{2},-1\right )}{\sqrt {a \cos ^2(x)+a}}\right )\right )+\frac {1}{5} a \sin (x) \cos (x) \left (a \cos ^2(x)+a\right )^{3/2}\) |
Input:
Int[(a + a*Cos[x]^2)^(5/2),x]
Output:
(a*Cos[x]*(a + a*Cos[x]^2)^(3/2)*Sin[x])/5 + (6*(((9*a^2*Sqrt[a + a*Cos[x] ^2]*EllipticE[Pi/2 + x, -1])/Sqrt[1 + Cos[x]^2] - (4*a^3*Sqrt[1 + Cos[x]^2 ]*EllipticF[Pi/2 + x, -1])/Sqrt[a + a*Cos[x]^2])/3 + (2*a^2*Cos[x]*Sqrt[a + a*Cos[x]^2]*Sin[x])/3))/5
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_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-B)*Cos[e + f*x]*Sin[e + f*x]*((a + b* Sin[e + f*x]^2)^p/(2*f*(p + 1))), x] + Simp[1/(2*(p + 1)) Int[(a + b*Sin[ e + f*x]^2)^(p - 1)*Simp[a*B + 2*a*A*(p + 1) + (2*A*b*(p + 1) + B*(b + 2*a* p + 2*b*p))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && G tQ[p, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[B/b Int[Sqrt[a + b*Sin[e + f*x]^2], x] , x] + Simp[(A*b - a*B)/b Int[1/Sqrt[a + b*Sin[e + f*x]^2], x], x] /; Fre eQ[{a, b, e, f, A, B}, x]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a ]/f)*EllipticE[e + f*x, -b/a], x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[1 + b*(Sin[e + f*x]^2/a)] Int[Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p - 1)/(2*f*p)), x] + Sim p[1/(2*p) Int[(a + b*Sin[e + f*x]^2)^(p - 2)*Simp[a*(b + 2*a*p) + b*(2*a + b)*(2*p - 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[ a + b, 0] && GtQ[p, 1]
Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(1/(S qrt[a]*f))*EllipticF[e + f*x, -b/a], x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[Sqrt[ 1 + b*(Sin[e + f*x]^2/a)]/Sqrt[a + b*Sin[e + f*x]^2] Int[1/Sqrt[1 + (b*Si n[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] && !GtQ[a, 0]
Time = 1.58 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(\frac {\sqrt {a \left (1+\cos \left (x \right )^{2}\right ) \sin \left (x \right )^{2}}\, a^{3} \left (\sin \left (x \right )^{6} \cos \left (x \right )-8 \sin \left (x \right )^{4} \cos \left (x \right )+8 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\sin \left (x \right )^{2}+2}\, \operatorname {EllipticF}\left (\cos \left (x \right ), i\right )-18 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\sin \left (x \right )^{2}+2}\, \operatorname {EllipticE}\left (\cos \left (x \right ), i\right )+12 \sin \left (x \right )^{2} \cos \left (x \right )\right )}{5 \sqrt {-a \left (\cos \left (x \right )^{4}-1\right )}\, \sin \left (x \right ) \sqrt {a \left (1+\cos \left (x \right )^{2}\right )}}\) | \(115\) |
Input:
int((a+a*cos(x)^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/5*(a*(1+cos(x)^2)*sin(x)^2)^(1/2)*a^3*(sin(x)^6*cos(x)-8*sin(x)^4*cos(x) +8*(sin(x)^2)^(1/2)*(-sin(x)^2+2)^(1/2)*EllipticF(cos(x),I)-18*(sin(x)^2)^ (1/2)*(-sin(x)^2+2)^(1/2)*EllipticE(cos(x),I)+12*sin(x)^2*cos(x))/(-a*(cos (x)^4-1))^(1/2)/sin(x)/(a*(1+cos(x)^2))^(1/2)
\[ \int \left (a+a \cos ^2(x)\right )^{5/2} \, dx=\int { {\left (a \cos \left (x\right )^{2} + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+a*cos(x)^2)^(5/2),x, algorithm="fricas")
Output:
integral((a^2*cos(x)^4 + 2*a^2*cos(x)^2 + a^2)*sqrt(a*cos(x)^2 + a), x)
Timed out. \[ \int \left (a+a \cos ^2(x)\right )^{5/2} \, dx=\text {Timed out} \] Input:
integrate((a+a*cos(x)**2)**(5/2),x)
Output:
Timed out
\[ \int \left (a+a \cos ^2(x)\right )^{5/2} \, dx=\int { {\left (a \cos \left (x\right )^{2} + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+a*cos(x)^2)^(5/2),x, algorithm="maxima")
Output:
integrate((a*cos(x)^2 + a)^(5/2), x)
\[ \int \left (a+a \cos ^2(x)\right )^{5/2} \, dx=\int { {\left (a \cos \left (x\right )^{2} + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+a*cos(x)^2)^(5/2),x, algorithm="giac")
Output:
integrate((a*cos(x)^2 + a)^(5/2), x)
Timed out. \[ \int \left (a+a \cos ^2(x)\right )^{5/2} \, dx=\int {\left (a\,{\cos \left (x\right )}^2+a\right )}^{5/2} \,d x \] Input:
int((a + a*cos(x)^2)^(5/2),x)
Output:
int((a + a*cos(x)^2)^(5/2), x)
\[ \int \left (a+a \cos ^2(x)\right )^{5/2} \, dx=\sqrt {a}\, a^{2} \left (\int \sqrt {\cos \left (x \right )^{2}+1}d x +\int \sqrt {\cos \left (x \right )^{2}+1}\, \cos \left (x \right )^{4}d x +2 \left (\int \sqrt {\cos \left (x \right )^{2}+1}\, \cos \left (x \right )^{2}d x \right )\right ) \] Input:
int((a+a*cos(x)^2)^(5/2),x)
Output:
sqrt(a)*a**2*(int(sqrt(cos(x)**2 + 1),x) + int(sqrt(cos(x)**2 + 1)*cos(x)* *4,x) + 2*int(sqrt(cos(x)**2 + 1)*cos(x)**2,x))