Optimal. Leaf size=65 \[ -\frac {3}{8} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {3}{8} a^2 \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4122, 195, 217, 206} \[ -\frac {3}{8} a^2 \cot (x) \sqrt {a \csc ^2(x)}-\frac {3}{8} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 206
Rule 217
Rule 4122
Rubi steps
\begin {align*} \int \left (a \csc ^2(x)\right )^{5/2} \, dx &=-\left (a \operatorname {Subst}\left (\int \left (a+a x^2\right )^{3/2} \, dx,x,\cot (x)\right )\right )\\ &=-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac {1}{4} \left (3 a^2\right ) \operatorname {Subst}\left (\int \sqrt {a+a x^2} \, dx,x,\cot (x)\right )\\ &=-\frac {3}{8} a^2 \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac {1}{8} \left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+a x^2}} \, dx,x,\cot (x)\right )\\ &=-\frac {3}{8} a^2 \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac {1}{8} \left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a \csc ^2(x)}}\right )\\ &=-\frac {3}{8} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {3}{8} a^2 \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.21, size = 51, normalized size = 0.78 \[ \frac {1}{64} \sin (x) \left (a \csc ^2(x)\right )^{5/2} \left (6 \left (\cos (3 x)+4 \sin ^4(x) \left (\log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )\right )\right )\right )-22 \cos (x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.66, size = 80, normalized size = 1.23 \[ -\frac {{\left (6 \, a^{2} \cos \relax (x)^{3} - 10 \, a^{2} \cos \relax (x) + 3 \, {\left (a^{2} \cos \relax (x)^{4} - 2 \, a^{2} \cos \relax (x)^{2} + a^{2}\right )} \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1}\right )\right )} \sqrt {-\frac {a}{\cos \relax (x)^{2} - 1}}}{16 \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.29, size = 124, normalized size = 1.91 \[ \frac {1}{64} \, {\left (12 \, a^{2} \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1}\right ) \mathrm {sgn}\left (\sin \relax (x)\right ) - \frac {8 \, a^{2} {\left (\cos \relax (x) - 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )}{\cos \relax (x) + 1} + \frac {a^{2} {\left (\cos \relax (x) - 1\right )}^{2} \mathrm {sgn}\left (\sin \relax (x)\right )}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {{\left (a^{2} \mathrm {sgn}\left (\sin \relax (x)\right ) - \frac {8 \, a^{2} {\left (\cos \relax (x) - 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )}{\cos \relax (x) + 1} + \frac {18 \, a^{2} {\left (\cos \relax (x) - 1\right )}^{2} \mathrm {sgn}\left (\sin \relax (x)\right )}{{\left (\cos \relax (x) + 1\right )}^{2}}\right )} {\left (\cos \relax (x) + 1\right )}^{2}}{{\left (\cos \relax (x) - 1\right )}^{2}}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.44, size = 79, normalized size = 1.22 \[ \frac {\left (3 \left (\cos ^{4}\relax (x )\right ) \ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )+3 \left (\cos ^{3}\relax (x )\right )-6 \left (\cos ^{2}\relax (x )\right ) \ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )-5 \cos \relax (x )+3 \ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )\right ) \sin \relax (x ) \left (-\frac {a}{-1+\cos ^{2}\relax (x )}\right )^{\frac {5}{2}} \sqrt {4}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.87, size = 1113, normalized size = 17.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\frac {a}{{\sin \relax (x)}^2}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \csc ^{2}{\relax (x )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________