Integrand size = 13, antiderivative size = 34 \[ \int \left (a-a \cos ^2(x)\right )^{3/2} \, dx=-\frac {2}{3} a \cot (x) \sqrt {a \sin ^2(x)}-\frac {1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2} \] Output:
-2/3*a*cot(x)*(a*sin(x)^2)^(1/2)-1/3*cot(x)*(a*sin(x)^2)^(3/2)
Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \left (a-a \cos ^2(x)\right )^{3/2} \, dx=\frac {1}{12} a (-9 \cos (x)+\cos (3 x)) \csc (x) \sqrt {a \sin ^2(x)} \] Input:
Integrate[(a - a*Cos[x]^2)^(3/2),x]
Output:
(a*(-9*Cos[x] + Cos[3*x])*Csc[x]*Sqrt[a*Sin[x]^2])/12
Time = 0.32 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3042, 3655, 3042, 3682, 3042, 3686, 3042, 3118}
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.
\(\displaystyle \int \left (a-a \cos ^2(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a-a \sin \left (x+\frac {\pi }{2}\right )^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 3655 |
\(\displaystyle \int \left (a \sin ^2(x)\right )^{3/2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \sin (x)^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 3682 |
\(\displaystyle \frac {2}{3} a \int \sqrt {a \sin ^2(x)}dx-\frac {1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} a \int \sqrt {a \sin (x)^2}dx-\frac {1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2}\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle \frac {2}{3} a \csc (x) \sqrt {a \sin ^2(x)} \int \sin (x)dx-\frac {1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} a \csc (x) \sqrt {a \sin ^2(x)} \int \sin (x)dx-\frac {1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle -\frac {1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2}-\frac {2}{3} a \cot (x) \sqrt {a \sin ^2(x)}\) |
Input:
Int[(a - a*Cos[x]^2)^(3/2),x]
Output:
(-2*a*Cot[x]*Sqrt[a*Sin[x]^2])/3 - (Cot[x]*(a*Sin[x]^2)^(3/2))/3
Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(a*cos[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a + b, 0]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-Cot[e + f*x ])*((b*Sin[e + f*x]^2)^p/(2*f*p)), x] + Simp[b*((2*p - 1)/(2*p)) Int[(b*S in[e + f*x]^2)^(p - 1), x], x] /; FreeQ[{b, e, f}, x] && !IntegerQ[p] && G tQ[p, 1]
Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Si n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {\sin \left (x \right ) a^{2} \cos \left (x \right ) \left (\cos \left (x \right )^{2}-3\right )}{3 \sqrt {a \sin \left (x \right )^{2}}}\) | \(24\) |
risch | \(\frac {i a \,{\mathrm e}^{4 i x} \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{24 \,{\mathrm e}^{2 i x}-24}-\frac {3 i a \,{\mathrm e}^{2 i x} \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{8 \left ({\mathrm e}^{2 i x}-1\right )}-\frac {3 i \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}\, a}{8 \left ({\mathrm e}^{2 i x}-1\right )}+\frac {i a \,{\mathrm e}^{-2 i x} \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{24 \,{\mathrm e}^{2 i x}-24}\) | \(145\) |
Input:
int((a-a*cos(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/3*sin(x)*a^2*cos(x)*(cos(x)^2-3)/(a*sin(x)^2)^(1/2)
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \left (a-a \cos ^2(x)\right )^{3/2} \, dx=\frac {{\left (a \cos \left (x\right )^{3} - 3 \, a \cos \left (x\right )\right )} \sqrt {-a \cos \left (x\right )^{2} + a}}{3 \, \sin \left (x\right )} \] Input:
integrate((a-a*cos(x)^2)^(3/2),x, algorithm="fricas")
Output:
1/3*(a*cos(x)^3 - 3*a*cos(x))*sqrt(-a*cos(x)^2 + a)/sin(x)
Timed out. \[ \int \left (a-a \cos ^2(x)\right )^{3/2} \, dx=\text {Timed out} \] Input:
integrate((a-a*cos(x)**2)**(3/2),x)
Output:
Timed out
\[ \int \left (a-a \cos ^2(x)\right )^{3/2} \, dx=\int { {\left (-a \cos \left (x\right )^{2} + a\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a-a*cos(x)^2)^(3/2),x, algorithm="maxima")
Output:
integrate((-a*cos(x)^2 + a)^(3/2), x)
Time = 0.14 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.53 \[ \int \left (a-a \cos ^2(x)\right )^{3/2} \, dx=-\frac {4 \, {\left (3 \, a^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )^{2} + a^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )\right )\right )}}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3}} \] Input:
integrate((a-a*cos(x)^2)^(3/2),x, algorithm="giac")
Output:
-4/3*(3*a^(3/2)*sgn(tan(1/2*x)^3 + tan(1/2*x))*tan(1/2*x)^2 + a^(3/2)*sgn( tan(1/2*x)^3 + tan(1/2*x)))/(tan(1/2*x)^2 + 1)^3
Timed out. \[ \int \left (a-a \cos ^2(x)\right )^{3/2} \, dx=\int {\left (a-a\,{\cos \left (x\right )}^2\right )}^{3/2} \,d x \] Input:
int((a - a*cos(x)^2)^(3/2),x)
Output:
int((a - a*cos(x)^2)^(3/2), x)
\[ \int \left (a-a \cos ^2(x)\right )^{3/2} \, dx=\sqrt {a}\, a \left (\int \sqrt {-\cos \left (x \right )^{2}+1}d x -\left (\int \sqrt {-\cos \left (x \right )^{2}+1}\, \cos \left (x \right )^{2}d x \right )\right ) \] Input:
int((a-a*cos(x)^2)^(3/2),x)
Output:
sqrt(a)*a*(int(sqrt( - cos(x)**2 + 1),x) - int(sqrt( - cos(x)**2 + 1)*cos( x)**2,x))