Optimal. Leaf size=234 \[ -\frac{2 (d \cot (e+f x))^{5/2}}{5 d^4 f}+\frac{2 \sqrt{d \cot (e+f x)}}{d^2 f}+\frac{\log \left (\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} d^{3/2} f}-\frac{\log \left (\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} d^{3/2} f}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} d^{3/2} f}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right )}{\sqrt{2} d^{3/2} f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.188422, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {16, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{2 (d \cot (e+f x))^{5/2}}{5 d^4 f}+\frac{2 \sqrt{d \cot (e+f x)}}{d^2 f}+\frac{\log \left (\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} d^{3/2} f}-\frac{\log \left (\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} d^{3/2} f}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} d^{3/2} f}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right )}{\sqrt{2} d^{3/2} f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 3473
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{\cot ^5(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx &=\frac{\int (d \cot (e+f x))^{7/2} \, dx}{d^5}\\ &=-\frac{2 (d \cot (e+f x))^{5/2}}{5 d^4 f}-\frac{\int (d \cot (e+f x))^{3/2} \, dx}{d^3}\\ &=\frac{2 \sqrt{d \cot (e+f x)}}{d^2 f}-\frac{2 (d \cot (e+f x))^{5/2}}{5 d^4 f}+\frac{\int \frac{1}{\sqrt{d \cot (e+f x)}} \, dx}{d}\\ &=\frac{2 \sqrt{d \cot (e+f x)}}{d^2 f}-\frac{2 (d \cot (e+f x))^{5/2}}{5 d^4 f}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (d^2+x^2\right )} \, dx,x,d \cot (e+f x)\right )}{f}\\ &=\frac{2 \sqrt{d \cot (e+f x)}}{d^2 f}-\frac{2 (d \cot (e+f x))^{5/2}}{5 d^4 f}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{d^2+x^4} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{f}\\ &=\frac{2 \sqrt{d \cot (e+f x)}}{d^2 f}-\frac{2 (d \cot (e+f x))^{5/2}}{5 d^4 f}-\frac{\operatorname{Subst}\left (\int \frac{d-x^2}{d^2+x^4} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{d f}-\frac{\operatorname{Subst}\left (\int \frac{d+x^2}{d^2+x^4} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{d f}\\ &=\frac{2 \sqrt{d \cot (e+f x)}}{d^2 f}-\frac{2 (d \cot (e+f x))^{5/2}}{5 d^4 f}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}+2 x}{-d-\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} d^{3/2} f}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}-2 x}{-d+\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} d^{3/2} f}-\frac{\operatorname{Subst}\left (\int \frac{1}{d-\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 d f}-\frac{\operatorname{Subst}\left (\int \frac{1}{d+\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 d f}\\ &=\frac{2 \sqrt{d \cot (e+f x)}}{d^2 f}-\frac{2 (d \cot (e+f x))^{5/2}}{5 d^4 f}+\frac{\log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} d^{3/2} f}-\frac{\log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} d^{3/2} f}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} d^{3/2} f}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} d^{3/2} f}\\ &=\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} d^{3/2} f}-\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} d^{3/2} f}+\frac{2 \sqrt{d \cot (e+f x)}}{d^2 f}-\frac{2 (d \cot (e+f x))^{5/2}}{5 d^4 f}+\frac{\log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} d^{3/2} f}-\frac{\log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} d^{3/2} f}\\ \end{align*}
Mathematica [A] time = 0.292425, size = 172, normalized size = 0.74 \[ \frac{\cot ^{\frac{3}{2}}(e+f x) \left (-8 \cot ^{\frac{5}{2}}(e+f x)+40 \sqrt{\cot (e+f x)}+5 \sqrt{2} \log \left (\cot (e+f x)-\sqrt{2} \sqrt{\cot (e+f x)}+1\right )-5 \sqrt{2} \log \left (\cot (e+f x)+\sqrt{2} \sqrt{\cot (e+f x)}+1\right )+10 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (e+f x)}\right )-10 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (e+f x)}+1\right )\right )}{20 f (d \cot (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.016, size = 202, normalized size = 0.9 \begin{align*} -{\frac{2}{5\,{d}^{4}f} \left ( d\cot \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}}+2\,{\frac{\sqrt{d\cot \left ( fx+e \right ) }}{{d}^{2}f}}+{\frac{\sqrt{2}}{2\,{d}^{2}f}\sqrt [4]{{d}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{d\cot \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }-{\frac{\sqrt{2}}{4\,{d}^{2}f}\sqrt [4]{{d}^{2}}\ln \left ({ \left ( d\cot \left ( fx+e \right ) +\sqrt [4]{{d}^{2}}\sqrt{d\cot \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) \left ( d\cot \left ( fx+e \right ) -\sqrt [4]{{d}^{2}}\sqrt{d\cot \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{2\,{d}^{2}f}\sqrt [4]{{d}^{2}}\arctan \left ({\sqrt{2}\sqrt{d\cot \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{5}{\left (e + f x \right )}}{\left (d \cot{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (f x + e\right )^{5}}{\left (d \cot \left (f x + e\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]