Integrand size = 12, antiderivative size = 164 \[ \int \left (a+b \cos ^2(x)\right )^{5/2} \, dx=\frac {\left (23 a^2+23 a b+8 b^2\right ) \sqrt {a+b \cos ^2(x)} E\left (\frac {\pi }{2}+x|-\frac {b}{a}\right )}{15 \sqrt {\frac {a+b \cos ^2(x)}{a}}}-\frac {4 a (a+b) (2 a+b) \sqrt {\frac {a+b \cos ^2(x)}{a}} \operatorname {EllipticF}\left (\frac {\pi }{2}+x,-\frac {b}{a}\right )}{15 \sqrt {a+b \cos ^2(x)}}+\frac {4}{15} b (2 a+b) \cos (x) \sqrt {a+b \cos ^2(x)} \sin (x)+\frac {1}{5} b \cos (x) \left (a+b \cos ^2(x)\right )^{3/2} \sin (x) \] Output:
1/15*(23*a^2+23*a*b+8*b^2)*(a+b*cos(x)^2)^(1/2)*EllipticE(cos(x),(-b/a)^(1 /2))/((a+b*cos(x)^2)/a)^(1/2)-4/15*a*(a+b)*(2*a+b)*((a+b*cos(x)^2)/a)^(1/2 )*InverseJacobiAM(1/2*Pi+x,(-b/a)^(1/2))/(a+b*cos(x)^2)^(1/2)+4/15*b*(2*a+ b)*cos(x)*(a+b*cos(x)^2)^(1/2)*sin(x)+1/5*b*cos(x)*(a+b*cos(x)^2)^(3/2)*si n(x)
Time = 0.82 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.02 \[ \int \left (a+b \cos ^2(x)\right )^{5/2} \, dx=\frac {16 \left (23 a^3+46 a^2 b+31 a b^2+8 b^3\right ) \sqrt {\frac {2 a+b+b \cos (2 x)}{a+b}} E\left (x\left |\frac {b}{a+b}\right .\right )-64 a \left (2 a^2+3 a b+b^2\right ) \sqrt {\frac {2 a+b+b \cos (2 x)}{a+b}} \operatorname {EllipticF}\left (x,\frac {b}{a+b}\right )+\sqrt {2} b \left (88 a^2+88 a b+25 b^2+28 b (2 a+b) \cos (2 x)+3 b^2 \cos (4 x)\right ) \sin (2 x)}{240 \sqrt {2 a+b+b \cos (2 x)}} \] Input:
Integrate[(a + b*Cos[x]^2)^(5/2),x]
Output:
(16*(23*a^3 + 46*a^2*b + 31*a*b^2 + 8*b^3)*Sqrt[(2*a + b + b*Cos[2*x])/(a + b)]*EllipticE[x, b/(a + b)] - 64*a*(2*a^2 + 3*a*b + b^2)*Sqrt[(2*a + b + b*Cos[2*x])/(a + b)]*EllipticF[x, b/(a + b)] + Sqrt[2]*b*(88*a^2 + 88*a*b + 25*b^2 + 28*b*(2*a + b)*Cos[2*x] + 3*b^2*Cos[4*x])*Sin[2*x])/(240*Sqrt[ 2*a + b + b*Cos[2*x]])
Time = 0.98 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {3042, 3659, 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.
| \(\displaystyle \int \left (a+b \cos ^2(x)\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \sin \left (x+\frac {\pi }{2}\right )^2\right )^{5/2}dx\) |
| \(\Big \downarrow \) 3659 |
| \(\displaystyle \frac {1}{5} \int \sqrt {b \cos ^2(x)+a} \left (4 b (2 a+b) \cos ^2(x)+a (5 a+b)\right )dx+\frac {1}{5} b \sin (x) \cos (x) \left (a+b \cos ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \sqrt {b \sin \left (x+\frac {\pi }{2}\right )^2+a} \left (4 b (2 a+b) \sin \left (x+\frac {\pi }{2}\right )^2+a (5 a+b)\right )dx+\frac {1}{5} b \sin (x) \cos (x) \left (a+b \cos ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3649 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {b \left (23 a^2+23 b a+8 b^2\right ) \cos ^2(x)+a \left (15 a^2+11 b a+4 b^2\right )}{\sqrt {b \cos ^2(x)+a}}dx+\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \cos ^2(x)}\right )+\frac {1}{5} b \sin (x) \cos (x) \left (a+b \cos ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {b \left (23 a^2+23 b a+8 b^2\right ) \sin \left (x+\frac {\pi }{2}\right )^2+a \left (15 a^2+11 b a+4 b^2\right )}{\sqrt {b \sin \left (x+\frac {\pi }{2}\right )^2+a}}dx+\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \cos ^2(x)}\right )+\frac {1}{5} b \sin (x) \cos (x) \left (a+b \cos ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3651 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\left (23 a^2+23 a b+8 b^2\right ) \int \sqrt {b \cos ^2(x)+a}dx-4 a (a+b) (2 a+b) \int \frac {1}{\sqrt {b \cos ^2(x)+a}}dx\right )+\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \cos ^2(x)}\right )+\frac {1}{5} b \sin (x) \cos (x) \left (a+b \cos ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\left (23 a^2+23 a b+8 b^2\right ) \int \sqrt {b \sin \left (x+\frac {\pi }{2}\right )^2+a}dx-4 a (a+b) (2 a+b) \int \frac {1}{\sqrt {b \sin \left (x+\frac {\pi }{2}\right )^2+a}}dx\right )+\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \cos ^2(x)}\right )+\frac {1}{5} b \sin (x) \cos (x) \left (a+b \cos ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3657 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (23 a^2+23 a b+8 b^2\right ) \sqrt {a+b \cos ^2(x)} \int \sqrt {\frac {b \cos ^2(x)}{a}+1}dx}{\sqrt {\frac {b \cos ^2(x)}{a}+1}}-4 a (a+b) (2 a+b) \int \frac {1}{\sqrt {b \sin \left (x+\frac {\pi }{2}\right )^2+a}}dx\right )+\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \cos ^2(x)}\right )+\frac {1}{5} b \sin (x) \cos (x) \left (a+b \cos ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (23 a^2+23 a b+8 b^2\right ) \sqrt {a+b \cos ^2(x)} \int \sqrt {\frac {b \sin \left (x+\frac {\pi }{2}\right )^2}{a}+1}dx}{\sqrt {\frac {b \cos ^2(x)}{a}+1}}-4 a (a+b) (2 a+b) \int \frac {1}{\sqrt {b \sin \left (x+\frac {\pi }{2}\right )^2+a}}dx\right )+\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \cos ^2(x)}\right )+\frac {1}{5} b \sin (x) \cos (x) \left (a+b \cos ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3656 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (23 a^2+23 a b+8 b^2\right ) \sqrt {a+b \cos ^2(x)} E\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{\sqrt {\frac {b \cos ^2(x)}{a}+1}}-4 a (a+b) (2 a+b) \int \frac {1}{\sqrt {b \sin \left (x+\frac {\pi }{2}\right )^2+a}}dx\right )+\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \cos ^2(x)}\right )+\frac {1}{5} b \sin (x) \cos (x) \left (a+b \cos ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3662 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (23 a^2+23 a b+8 b^2\right ) \sqrt {a+b \cos ^2(x)} E\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{\sqrt {\frac {b \cos ^2(x)}{a}+1}}-\frac {4 a (a+b) (2 a+b) \sqrt {\frac {b \cos ^2(x)}{a}+1} \int \frac {1}{\sqrt {\frac {b \cos ^2(x)}{a}+1}}dx}{\sqrt {a+b \cos ^2(x)}}\right )+\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \cos ^2(x)}\right )+\frac {1}{5} b \sin (x) \cos (x) \left (a+b \cos ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (23 a^2+23 a b+8 b^2\right ) \sqrt {a+b \cos ^2(x)} E\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{\sqrt {\frac {b \cos ^2(x)}{a}+1}}-\frac {4 a (a+b) (2 a+b) \sqrt {\frac {b \cos ^2(x)}{a}+1} \int \frac {1}{\sqrt {\frac {b \sin \left (x+\frac {\pi }{2}\right )^2}{a}+1}}dx}{\sqrt {a+b \cos ^2(x)}}\right )+\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \cos ^2(x)}\right )+\frac {1}{5} b \sin (x) \cos (x) \left (a+b \cos ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3661 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (23 a^2+23 a b+8 b^2\right ) \sqrt {a+b \cos ^2(x)} E\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{\sqrt {\frac {b \cos ^2(x)}{a}+1}}-\frac {4 a (a+b) (2 a+b) \sqrt {\frac {b \cos ^2(x)}{a}+1} \operatorname {EllipticF}\left (x+\frac {\pi }{2},-\frac {b}{a}\right )}{\sqrt {a+b \cos ^2(x)}}\right )+\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \cos ^2(x)}\right )+\frac {1}{5} b \sin (x) \cos (x) \left (a+b \cos ^2(x)\right )^{3/2}\) |
Input:
Int[(a + b*Cos[x]^2)^(5/2),x]
Output:
(b*Cos[x]*(a + b*Cos[x]^2)^(3/2)*Sin[x])/5 + ((((23*a^2 + 23*a*b + 8*b^2)* Sqrt[a + b*Cos[x]^2]*EllipticE[Pi/2 + x, -(b/a)])/Sqrt[1 + (b*Cos[x]^2)/a] - (4*a*(a + b)*(2*a + b)*Sqrt[1 + (b*Cos[x]^2)/a]*EllipticF[Pi/2 + x, -(b /a)])/Sqrt[a + b*Cos[x]^2])/3 + (4*b*(2*a + b)*Cos[x]*Sqrt[a + b*Cos[x]^2] *Sin[x])/3)/5
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]
Leaf count of result is larger than twice the leaf count of optimal. \(330\) vs. \(2(141)=282\).
Time = 5.38 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.02
| method | result | size |
| default | \(-\frac {-\frac {b^{3} \cos \left (x \right ) \sin \left (x \right )^{6}}{5}+\frac {\left (14 a \,b^{2}+10 b^{3}\right ) \sin \left (x \right )^{4} \cos \left (x \right )}{15}+\frac {\left (-11 a^{2} b -18 a \,b^{2}-7 b^{3}\right ) \sin \left (x \right )^{2} \cos \left (x \right )}{15}-\frac {8 a^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\frac {b \sin \left (x \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticF}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right )}{15}-\frac {4 a^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\frac {b \sin \left (x \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticF}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) b}{5}-\frac {4 a \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\frac {b \sin \left (x \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticF}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}}{15}+\frac {23 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\frac {b \sin \left (x \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticE}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}}{15}+\frac {23 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\frac {b \sin \left (x \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticE}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b}{15}+\frac {8 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\frac {b \sin \left (x \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticE}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}}{15}}{\sin \left (x \right ) \sqrt {a +b \cos \left (x \right )^{2}}}\) | \(331\) |
Input:
int((a+b*cos(x)^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
-(-1/5*b^3*cos(x)*sin(x)^6+1/15*(14*a*b^2+10*b^3)*sin(x)^4*cos(x)+1/15*(-1 1*a^2*b-18*a*b^2-7*b^3)*sin(x)^2*cos(x)-8/15*a^3*(sin(x)^2)^(1/2)*(-b/a*si n(x)^2+(a+b)/a)^(1/2)*EllipticF(cos(x),(-b/a)^(1/2))-4/5*a^2*(sin(x)^2)^(1 /2)*(-b/a*sin(x)^2+(a+b)/a)^(1/2)*EllipticF(cos(x),(-b/a)^(1/2))*b-4/15*a* (sin(x)^2)^(1/2)*(-b/a*sin(x)^2+(a+b)/a)^(1/2)*EllipticF(cos(x),(-b/a)^(1/ 2))*b^2+23/15*(sin(x)^2)^(1/2)*(-b/a*sin(x)^2+(a+b)/a)^(1/2)*EllipticE(cos (x),(-b/a)^(1/2))*a^3+23/15*(sin(x)^2)^(1/2)*(-b/a*sin(x)^2+(a+b)/a)^(1/2) *EllipticE(cos(x),(-b/a)^(1/2))*a^2*b+8/15*(sin(x)^2)^(1/2)*(-b/a*sin(x)^2 +(a+b)/a)^(1/2)*EllipticE(cos(x),(-b/a)^(1/2))*a*b^2)/sin(x)/(a+b*cos(x)^2 )^(1/2)
\[ \int \left (a+b \cos ^2(x)\right )^{5/2} \, dx=\int { {\left (b \cos \left (x\right )^{2} + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*cos(x)^2)^(5/2),x, algorithm="fricas")
Output:
integral((b^2*cos(x)^4 + 2*a*b*cos(x)^2 + a^2)*sqrt(b*cos(x)^2 + a), x)
Timed out. \[ \int \left (a+b \cos ^2(x)\right )^{5/2} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(x)**2)**(5/2),x)
Output:
Timed out
\[ \int \left (a+b \cos ^2(x)\right )^{5/2} \, dx=\int { {\left (b \cos \left (x\right )^{2} + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*cos(x)^2)^(5/2),x, algorithm="maxima")
Output:
integrate((b*cos(x)^2 + a)^(5/2), x)
\[ \int \left (a+b \cos ^2(x)\right )^{5/2} \, dx=\int { {\left (b \cos \left (x\right )^{2} + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*cos(x)^2)^(5/2),x, algorithm="giac")
Output:
integrate((b*cos(x)^2 + a)^(5/2), x)
Timed out. \[ \int \left (a+b \cos ^2(x)\right )^{5/2} \, dx=\int {\left (b\,{\cos \left (x\right )}^2+a\right )}^{5/2} \,d x \] Input:
int((a + b*cos(x)^2)^(5/2),x)
Output:
int((a + b*cos(x)^2)^(5/2), x)
\[ \int \left (a+b \cos ^2(x)\right )^{5/2} \, dx=\left (\int \sqrt {\cos \left (x \right )^{2} b +a}d x \right ) a^{2}+\left (\int \sqrt {\cos \left (x \right )^{2} b +a}\, \cos \left (x \right )^{4}d x \right ) b^{2}+2 \left (\int \sqrt {\cos \left (x \right )^{2} b +a}\, \cos \left (x \right )^{2}d x \right ) a b \] Input:
int((a+b*cos(x)^2)^(5/2),x)
Output:
int(sqrt(cos(x)**2*b + a),x)*a**2 + int(sqrt(cos(x)**2*b + a)*cos(x)**4,x) *b**2 + 2*int(sqrt(cos(x)**2*b + a)*cos(x)**2,x)*a*b