Optimal. Leaf size=43 \[ -\frac{8 \cot (x)}{15 \sqrt{\csc ^2(x)}}-\frac{4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac{\cot (x)}{5 \csc ^2(x)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0147414, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4122, 192, 191} \[ -\frac{8 \cot (x)}{15 \sqrt{\csc ^2(x)}}-\frac{4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac{\cot (x)}{5 \csc ^2(x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4122
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\csc ^2(x)^{5/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^{7/2}} \, dx,x,\cot (x)\right )\\ &=-\frac{\cot (x)}{5 \csc ^2(x)^{5/2}}-\frac{4}{5} \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\\ &=-\frac{\cot (x)}{5 \csc ^2(x)^{5/2}}-\frac{4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac{8}{15} \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\\ &=-\frac{\cot (x)}{5 \csc ^2(x)^{5/2}}-\frac{4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac{8 \cot (x)}{15 \sqrt{\csc ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0269639, size = 31, normalized size = 0.72 \[ -\frac{(150 \cos (x)-25 \cos (3 x)+3 \cos (5 x)) \csc (x)}{240 \sqrt{\csc ^2(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.077, size = 38, normalized size = 0.9 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) \left ( 3\, \left ( \cos \left ( x \right ) \right ) ^{2}-9\,\cos \left ( x \right ) +8 \right ) }{30\, \left ( -1+\cos \left ( x \right ) \right ) ^{3}} \left ( - \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{-1} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.66542, size = 23, normalized size = 0.53 \begin{align*} -\frac{1}{80} \, \cos \left (5 \, x\right ) + \frac{5}{48} \, \cos \left (3 \, x\right ) - \frac{5}{8} \, \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.471179, size = 53, normalized size = 1.23 \begin{align*} -\frac{1}{5} \, \cos \left (x\right )^{5} + \frac{2}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 19.7235, size = 46, normalized size = 1.07 \begin{align*} - \frac{8 \cot ^{5}{\left (x \right )}}{15 \left (\csc ^{2}{\left (x \right )}\right )^{\frac{5}{2}}} - \frac{4 \cot ^{3}{\left (x \right )}}{3 \left (\csc ^{2}{\left (x \right )}\right )^{\frac{5}{2}}} - \frac{\cot{\left (x \right )}}{\left (\csc ^{2}{\left (x \right )}\right )^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.30193, size = 82, normalized size = 1.91 \begin{align*} -\frac{16 \,{\left (\frac{5 \,{\left (\cos \left (x\right ) - 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} - \frac{10 \,{\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm{sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \mathrm{sgn}\left (\sin \left (x\right )\right )\right )}}{15 \,{\left (\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{5}} + \frac{16}{15} \, \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]