Optimal. Leaf size=55 \[ -\frac {8 \cot (x)}{15 a^2 \sqrt {a \csc ^2(x)}}-\frac {4 \cot (x)}{15 a \left (a \csc ^2(x)\right )^{3/2}}-\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4122, 192, 191} \[ -\frac {8 \cot (x)}{15 a^2 \sqrt {a \csc ^2(x)}}-\frac {4 \cot (x)}{15 a \left (a \csc ^2(x)\right )^{3/2}}-\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 191
Rule 192
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx &=-\left (a \operatorname {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{7/2}} \, dx,x,\cot (x)\right )\right )\\ &=-\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}}-\frac {4}{5} \operatorname {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}}-\frac {4 \cot (x)}{15 a \left (a \csc ^2(x)\right )^{3/2}}-\frac {8 \operatorname {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\cot (x)\right )}{15 a}\\ &=-\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}}-\frac {4 \cot (x)}{15 a \left (a \csc ^2(x)\right )^{3/2}}-\frac {8 \cot (x)}{15 a^2 \sqrt {a \csc ^2(x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 36, normalized size = 0.65 \[ -\frac {\sin (x) (150 \cos (x)-25 \cos (3 x)+3 \cos (5 x)) \sqrt {a \csc ^2(x)}}{240 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.60, size = 37, normalized size = 0.67 \[ -\frac {{\left (3 \, \cos \relax (x)^{5} - 10 \, \cos \relax (x)^{3} + 15 \, \cos \relax (x)\right )} \sqrt {-\frac {a}{\cos \relax (x)^{2} - 1}} \sin \relax (x)}{15 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.50, size = 56, normalized size = 1.02 \[ -\frac {16 \, {\left (\frac {10 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )^{4} + 5 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )^{2} + \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right )}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{5}} - \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right )\right )}}{15 \, a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.42, size = 39, normalized size = 0.71 \[ \frac {\sin \relax (x ) \left (3 \left (\cos ^{2}\relax (x )\right )-9 \cos \relax (x )+8\right ) \sqrt {4}}{30 \left (-1+\cos \relax (x )\right )^{3} \left (-\frac {a}{-1+\cos ^{2}\relax (x )}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \csc \relax (x)^{2}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (\frac {a}{{\sin \relax (x)}^2}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 9.41, size = 61, normalized size = 1.11 \[ - \frac {8 \cot ^{5}{\relax (x )}}{15 a^{\frac {5}{2}} \left (\csc ^{2}{\relax (x )}\right )^{\frac {5}{2}}} - \frac {4 \cot ^{3}{\relax (x )}}{3 a^{\frac {5}{2}} \left (\csc ^{2}{\relax (x )}\right )^{\frac {5}{2}}} - \frac {\cot {\relax (x )}}{a^{\frac {5}{2}} \left (\csc ^{2}{\relax (x )}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________