Integrand size = 13, antiderivative size = 42 \[ \int \frac {1}{\left (a-a \cos ^2(x)\right )^{3/2}} \, dx=-\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}}-\frac {\text {arctanh}(\cos (x)) \sin (x)}{2 a \sqrt {a \sin ^2(x)}} \] Output:
-1/2*cot(x)/a/(a*sin(x)^2)^(1/2)-1/2*arctanh(cos(x))*sin(x)/a/(a*sin(x)^2) ^(1/2)
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.31 \[ \int \frac {1}{\left (a-a \cos ^2(x)\right )^{3/2}} \, dx=-\frac {\left (\csc ^2\left (\frac {x}{2}\right )+4 \log \left (\cos \left (\frac {x}{2}\right )\right )-4 \log \left (\sin \left (\frac {x}{2}\right )\right )-\sec ^2\left (\frac {x}{2}\right )\right ) \sin ^3(x)}{8 \left (a \sin ^2(x)\right )^{3/2}} \] Input:
Integrate[(a - a*Cos[x]^2)^(-3/2),x]
Output:
-1/8*((Csc[x/2]^2 + 4*Log[Cos[x/2]] - 4*Log[Sin[x/2]] - Sec[x/2]^2)*Sin[x] ^3)/(a*Sin[x]^2)^(3/2)
Time = 0.34 (sec) , antiderivative size = 42, 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, 3683, 3042, 3686, 3042, 4257}
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.
\(\displaystyle \int \frac {1}{\left (a-a \cos ^2(x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a-a \sin \left (x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3655 |
\(\displaystyle \int \frac {1}{\left (a \sin ^2(x)\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a \sin (x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3683 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {a \sin ^2(x)}}dx}{2 a}-\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {a \sin (x)^2}}dx}{2 a}-\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}}\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle \frac {\sin (x) \int \csc (x)dx}{2 a \sqrt {a \sin ^2(x)}}-\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin (x) \int \csc (x)dx}{2 a \sqrt {a \sin ^2(x)}}-\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -\frac {\sin (x) \text {arctanh}(\cos (x))}{2 a \sqrt {a \sin ^2(x)}}-\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}}\) |
Input:
Int[(a - a*Cos[x]^2)^(-3/2),x]
Output:
-1/2*Cot[x]/(a*Sqrt[a*Sin[x]^2]) - (ArcTanh[Cos[x]]*Sin[x])/(2*a*Sqrt[a*Si n[x]^2])
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 + 1)/(b*f*(2*p + 1))), x] + Simp[2*((p + 1)/(b*(2*p + 1))) Int[(b*Sin[e + f*x]^2)^(p + 1), x], x] /; FreeQ[{b, e, f}, x] && !IntegerQ[p] && LtQ[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]])
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.98
method | result | size |
default | \(\frac {-2 \cos \left (x \right )+\left (-\ln \left (1+\cos \left (x \right )\right )+\ln \left (-1+\cos \left (x \right )\right )\right ) \sin \left (x \right )^{2}}{4 a \sin \left (x \right ) \sqrt {a \sin \left (x \right )^{2}}}\) | \(41\) |
risch | \(-\frac {i \left ({\mathrm e}^{2 i x}+1\right )}{a \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )}{a \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )}{a \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}\) | \(110\) |
Input:
int(1/(a-a*cos(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/4/a*(-2*cos(x)+(-ln(1+cos(x))+ln(-1+cos(x)))*sin(x)^2)/sin(x)/(a*sin(x)^ 2)^(1/2)
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\left (a-a \cos ^2(x)\right )^{3/2}} \, dx=\frac {\sqrt {-a \cos \left (x\right )^{2} + a} {\left ({\left (\cos \left (x\right )^{2} - 1\right )} \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) + 2 \, \cos \left (x\right )\right )}}{4 \, {\left (a^{2} \cos \left (x\right )^{2} - a^{2}\right )} \sin \left (x\right )} \] Input:
integrate(1/(a-a*cos(x)^2)^(3/2),x, algorithm="fricas")
Output:
1/4*sqrt(-a*cos(x)^2 + a)*((cos(x)^2 - 1)*log(-(cos(x) - 1)/(cos(x) + 1)) + 2*cos(x))/((a^2*cos(x)^2 - a^2)*sin(x))
\[ \int \frac {1}{\left (a-a \cos ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (- a \cos ^{2}{\left (x \right )} + a\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(a-a*cos(x)**2)**(3/2),x)
Output:
Integral((-a*cos(x)**2 + a)**(-3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (34) = 68\).
Time = 0.16 (sec) , antiderivative size = 314, normalized size of antiderivative = 7.48 \[ \int \frac {1}{\left (a-a \cos ^2(x)\right )^{3/2}} \, dx=-\frac {{\left ({\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + 2 \, {\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \cos \left (4 \, x\right ) - 2 \, {\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\right )} \sin \left (4 \, x\right ) - 2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + 4 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \cos \left (x\right ) \sin \left (2 \, x\right ) - 4 \, \cos \left (2 \, x\right ) \sin \left (x\right ) + 2 \, \sin \left (x\right )\right )} \sqrt {-a}}{2 \, {\left (a^{2} \cos \left (4 \, x\right )^{2} + 4 \, a^{2} \cos \left (2 \, x\right )^{2} + a^{2} \sin \left (4 \, x\right )^{2} - 4 \, a^{2} \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a^{2} \sin \left (2 \, x\right )^{2} - 4 \, a^{2} \cos \left (2 \, x\right ) + a^{2} - 2 \, {\left (2 \, a^{2} \cos \left (2 \, x\right ) - a^{2}\right )} \cos \left (4 \, x\right )\right )}} \] Input:
integrate(1/(a-a*cos(x)^2)^(3/2),x, algorithm="maxima")
Output:
-1/2*((2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)^ 2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)*arctan2(sin(x), c os(x) + 1) - (2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - si n(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)*arctan2(si n(x), cos(x) - 1) + 2*(sin(3*x) + sin(x))*cos(4*x) - 2*(cos(3*x) + cos(x)) *sin(4*x) - 2*(2*cos(2*x) - 1)*sin(3*x) + 4*cos(3*x)*sin(2*x) + 4*cos(x)*s in(2*x) - 4*cos(2*x)*sin(x) + 2*sin(x))*sqrt(-a)/(a^2*cos(4*x)^2 + 4*a^2*c os(2*x)^2 + a^2*sin(4*x)^2 - 4*a^2*sin(4*x)*sin(2*x) + 4*a^2*sin(2*x)^2 - 4*a^2*cos(2*x) + a^2 - 2*(2*a^2*cos(2*x) - a^2)*cos(4*x))
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (34) = 68\).
Time = 0.15 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.07 \[ \int \frac {1}{\left (a-a \cos ^2(x)\right )^{3/2}} \, dx=\frac {\tan \left (\frac {1}{2} \, x\right )^{2}}{8 \, a^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )\right )} + \frac {\log \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right )}{4 \, a^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )\right )} - \frac {2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}{8 \, a^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )^{2}} \] Input:
integrate(1/(a-a*cos(x)^2)^(3/2),x, algorithm="giac")
Output:
1/8*tan(1/2*x)^2/(a^(3/2)*sgn(tan(1/2*x)^3 + tan(1/2*x))) + 1/4*log(tan(1/ 2*x)^2)/(a^(3/2)*sgn(tan(1/2*x)^3 + tan(1/2*x))) - 1/8*(2*tan(1/2*x)^2 + 1 )/(a^(3/2)*sgn(tan(1/2*x)^3 + tan(1/2*x))*tan(1/2*x)^2)
Timed out. \[ \int \frac {1}{\left (a-a \cos ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (a-a\,{\cos \left (x\right )}^2\right )}^{3/2}} \,d x \] Input:
int(1/(a - a*cos(x)^2)^(3/2),x)
Output:
int(1/(a - a*cos(x)^2)^(3/2), x)
\[ \int \frac {1}{\left (a-a \cos ^2(x)\right )^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {-\cos \left (x \right )^{2}+1}}{\cos \left (x \right )^{4}-2 \cos \left (x \right )^{2}+1}d x \right )}{a^{2}} \] Input:
int(1/(a-a*cos(x)^2)^(3/2),x)
Output:
(sqrt(a)*int(sqrt( - cos(x)**2 + 1)/(cos(x)**4 - 2*cos(x)**2 + 1),x))/a**2