Integrand size = 12, antiderivative size = 111 \[ \int \left (a+b \sin ^2(x)\right )^{3/2} \, dx=-\frac {1}{3} b \cos (x) \sin (x) \sqrt {a+b \sin ^2(x)}+\frac {2 (2 a+b) E\left (x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(x)}}{3 \sqrt {\frac {a+b \sin ^2(x)}{a}}}-\frac {a (a+b) \operatorname {EllipticF}\left (x,-\frac {b}{a}\right ) \sqrt {\frac {a+b \sin ^2(x)}{a}}}{3 \sqrt {a+b \sin ^2(x)}} \] Output:
-1/3*b*cos(x)*sin(x)*(a+b*sin(x)^2)^(1/2)+2/3*(2*a+b)*EllipticE(sin(x),(-b /a)^(1/2))*(a+b*sin(x)^2)^(1/2)/((a+b*sin(x)^2)/a)^(1/2)-1/3*a*(a+b)*Inver seJacobiAM(x,(-b/a)^(1/2))*((a+b*sin(x)^2)/a)^(1/2)/(a+b*sin(x)^2)^(1/2)
Time = 0.44 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.04 \[ \int \left (a+b \sin ^2(x)\right )^{3/2} \, dx=\frac {8 a (2 a+b) \sqrt {\frac {2 a+b-b \cos (2 x)}{a}} E\left (x\left |-\frac {b}{a}\right .\right )-4 a (a+b) \sqrt {\frac {2 a+b-b \cos (2 x)}{a}} \operatorname {EllipticF}\left (x,-\frac {b}{a}\right )+\sqrt {2} b (-2 a-b+b \cos (2 x)) \sin (2 x)}{12 \sqrt {2 a+b-b \cos (2 x)}} \] Input:
Integrate[(a + b*Sin[x]^2)^(3/2),x]
Output:
(8*a*(2*a + b)*Sqrt[(2*a + b - b*Cos[2*x])/a]*EllipticE[x, -(b/a)] - 4*a*( a + b)*Sqrt[(2*a + b - b*Cos[2*x])/a]*EllipticF[x, -(b/a)] + Sqrt[2]*b*(-2 *a - b + b*Cos[2*x])*Sin[2*x])/(12*Sqrt[2*a + b - b*Cos[2*x]])
Time = 0.69 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {3042, 3659, 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.
\(\displaystyle \int \left (a+b \sin ^2(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a+b \sin (x)^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 3659 |
\(\displaystyle \frac {1}{3} \int \frac {2 b (2 a+b) \sin ^2(x)+a (3 a+b)}{\sqrt {b \sin ^2(x)+a}}dx-\frac {1}{3} b \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \int \frac {2 b (2 a+b) \sin (x)^2+a (3 a+b)}{\sqrt {b \sin (x)^2+a}}dx-\frac {1}{3} b \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\) |
\(\Big \downarrow \) 3651 |
\(\displaystyle \frac {1}{3} \left (2 (2 a+b) \int \sqrt {b \sin ^2(x)+a}dx-a (a+b) \int \frac {1}{\sqrt {b \sin ^2(x)+a}}dx\right )-\frac {1}{3} b \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (2 (2 a+b) \int \sqrt {b \sin (x)^2+a}dx-a (a+b) \int \frac {1}{\sqrt {b \sin (x)^2+a}}dx\right )-\frac {1}{3} b \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\) |
\(\Big \downarrow \) 3657 |
\(\displaystyle \frac {1}{3} \left (\frac {2 (2 a+b) \sqrt {a+b \sin ^2(x)} \int \sqrt {\frac {b \sin ^2(x)}{a}+1}dx}{\sqrt {\frac {b \sin ^2(x)}{a}+1}}-a (a+b) \int \frac {1}{\sqrt {b \sin (x)^2+a}}dx\right )-\frac {1}{3} b \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (\frac {2 (2 a+b) \sqrt {a+b \sin ^2(x)} \int \sqrt {\frac {b \sin (x)^2}{a}+1}dx}{\sqrt {\frac {b \sin ^2(x)}{a}+1}}-a (a+b) \int \frac {1}{\sqrt {b \sin (x)^2+a}}dx\right )-\frac {1}{3} b \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\) |
\(\Big \downarrow \) 3656 |
\(\displaystyle \frac {1}{3} \left (\frac {2 (2 a+b) \sqrt {a+b \sin ^2(x)} E\left (x\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(x)}{a}+1}}-a (a+b) \int \frac {1}{\sqrt {b \sin (x)^2+a}}dx\right )-\frac {1}{3} b \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\) |
\(\Big \downarrow \) 3662 |
\(\displaystyle \frac {1}{3} \left (\frac {2 (2 a+b) \sqrt {a+b \sin ^2(x)} E\left (x\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(x)}{a}+1}}-\frac {a (a+b) \sqrt {\frac {b \sin ^2(x)}{a}+1} \int \frac {1}{\sqrt {\frac {b \sin ^2(x)}{a}+1}}dx}{\sqrt {a+b \sin ^2(x)}}\right )-\frac {1}{3} b \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (\frac {2 (2 a+b) \sqrt {a+b \sin ^2(x)} E\left (x\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(x)}{a}+1}}-\frac {a (a+b) \sqrt {\frac {b \sin ^2(x)}{a}+1} \int \frac {1}{\sqrt {\frac {b \sin (x)^2}{a}+1}}dx}{\sqrt {a+b \sin ^2(x)}}\right )-\frac {1}{3} b \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\) |
\(\Big \downarrow \) 3661 |
\(\displaystyle \frac {1}{3} \left (\frac {2 (2 a+b) \sqrt {a+b \sin ^2(x)} E\left (x\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(x)}{a}+1}}-\frac {a (a+b) \sqrt {\frac {b \sin ^2(x)}{a}+1} \operatorname {EllipticF}\left (x,-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(x)}}\right )-\frac {1}{3} b \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\) |
Input:
Int[(a + b*Sin[x]^2)^(3/2),x]
Output:
-1/3*(b*Cos[x]*Sin[x]*Sqrt[a + b*Sin[x]^2]) + ((2*(2*a + b)*EllipticE[x, - (b/a)]*Sqrt[a + b*Sin[x]^2])/Sqrt[1 + (b*Sin[x]^2)/a] - (a*(a + b)*Ellipti cF[x, -(b/a)]*Sqrt[1 + (b*Sin[x]^2)/a])/Sqrt[a + b*Sin[x]^2])/3
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 = 2.60 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.72
method | result | size |
default | \(\frac {-\frac {a^{2} \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (x \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\sin \left (x \right ), \sqrt {-\frac {b}{a}}\right )}{3}-\frac {a \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (x \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\sin \left (x \right ), \sqrt {-\frac {b}{a}}\right ) b}{3}+\frac {4 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (x \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\sin \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}}{3}+\frac {2 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \sin \left (x \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\sin \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a b}{3}+\frac {b^{2} \sin \left (x \right )^{5}}{3}+\frac {a b \sin \left (x \right )^{3}}{3}-\frac {b^{2} \sin \left (x \right )^{3}}{3}-\frac {a b \sin \left (x \right )}{3}}{\cos \left (x \right ) \sqrt {a +b \sin \left (x \right )^{2}}}\) | \(191\) |
Input:
int((a+b*sin(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
(-1/3*a^2*(cos(x)^2)^(1/2)*((a+b*sin(x)^2)/a)^(1/2)*EllipticF(sin(x),(-b/a )^(1/2))-1/3*a*(cos(x)^2)^(1/2)*((a+b*sin(x)^2)/a)^(1/2)*EllipticF(sin(x), (-b/a)^(1/2))*b+4/3*(cos(x)^2)^(1/2)*((a+b*sin(x)^2)/a)^(1/2)*EllipticE(si n(x),(-b/a)^(1/2))*a^2+2/3*(cos(x)^2)^(1/2)*((a+b*sin(x)^2)/a)^(1/2)*Ellip ticE(sin(x),(-b/a)^(1/2))*a*b+1/3*b^2*sin(x)^5+1/3*a*b*sin(x)^3-1/3*b^2*si n(x)^3-1/3*a*b*sin(x))/cos(x)/(a+b*sin(x)^2)^(1/2)
\[ \int \left (a+b \sin ^2(x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (x\right )^{2} + a\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b*sin(x)^2)^(3/2),x, algorithm="fricas")
Output:
integral((-b*cos(x)^2 + a + b)^(3/2), x)
\[ \int \left (a+b \sin ^2(x)\right )^{3/2} \, dx=\int \left (a + b \sin ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((a+b*sin(x)**2)**(3/2),x)
Output:
Integral((a + b*sin(x)**2)**(3/2), x)
\[ \int \left (a+b \sin ^2(x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (x\right )^{2} + a\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b*sin(x)^2)^(3/2),x, algorithm="maxima")
Output:
integrate((b*sin(x)^2 + a)^(3/2), x)
\[ \int \left (a+b \sin ^2(x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (x\right )^{2} + a\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b*sin(x)^2)^(3/2),x, algorithm="giac")
Output:
integrate((b*sin(x)^2 + a)^(3/2), x)
Timed out. \[ \int \left (a+b \sin ^2(x)\right )^{3/2} \, dx=\int {\left (b\,{\sin \left (x\right )}^2+a\right )}^{3/2} \,d x \] Input:
int((a + b*sin(x)^2)^(3/2),x)
Output:
int((a + b*sin(x)^2)^(3/2), x)
\[ \int \left (a+b \sin ^2(x)\right )^{3/2} \, dx=\left (\int \sqrt {\sin \left (x \right )^{2} b +a}d x \right ) a +\left (\int \sqrt {\sin \left (x \right )^{2} b +a}\, \sin \left (x \right )^{2}d x \right ) b \] Input:
int((a+b*sin(x)^2)^(3/2),x)
Output:
int(sqrt(sin(x)**2*b + a),x)*a + int(sqrt(sin(x)**2*b + a)*sin(x)**2,x)*b