Optimal. Leaf size=306 \[ -\frac{\log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{4 \sqrt{2} a^2 d e^{3/2}}+\frac{\log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{4 \sqrt{2} a^2 d e^{3/2}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 a^2 d e^{3/2}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 \sqrt{2} a^2 d e^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{2 \sqrt{2} a^2 d e^{3/2}}+\frac{5}{2 a^2 d e \sqrt{e \cot (c+d x)}}-\frac{1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt{e \cot (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.795635, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 15, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {3569, 3649, 3653, 12, 3476, 329, 211, 1165, 628, 1162, 617, 204, 3634, 63, 205} \[ -\frac{\log \left (\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{4 \sqrt{2} a^2 d e^{3/2}}+\frac{\log \left (\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}+\sqrt{e}\right )}{4 \sqrt{2} a^2 d e^{3/2}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 a^2 d e^{3/2}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 \sqrt{2} a^2 d e^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}+1\right )}{2 \sqrt{2} a^2 d e^{3/2}}+\frac{5}{2 a^2 d e \sqrt{e \cot (c+d x)}}-\frac{1}{2 d e \left (a^2 \cot (c+d x)+a^2\right ) \sqrt{e \cot (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3569
Rule 3649
Rule 3653
Rule 12
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rule 3634
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2} \, dx &=-\frac{1}{2 d e \sqrt{e \cot (c+d x)} \left (a^2+a^2 \cot (c+d x)\right )}-\frac{\int \frac{-\frac{5 a^2 e}{2}+a^2 e \cot (c+d x)-\frac{3}{2} a^2 e \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx}{2 a^3 e}\\ &=\frac{5}{2 a^2 d e \sqrt{e \cot (c+d x)}}-\frac{1}{2 d e \sqrt{e \cot (c+d x)} \left (a^2+a^2 \cot (c+d x)\right )}-\frac{\int \frac{\frac{7 a^3 e^3}{4}+\frac{1}{2} a^3 e^3 \cot (c+d x)+\frac{5}{4} a^3 e^3 \cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{a^4 e^4}\\ &=\frac{5}{2 a^2 d e \sqrt{e \cot (c+d x)}}-\frac{1}{2 d e \sqrt{e \cot (c+d x)} \left (a^2+a^2 \cot (c+d x)\right )}-\frac{\int \frac{a^4 e^3}{\sqrt{e \cot (c+d x)}} \, dx}{2 a^6 e^4}-\frac{5 \int \frac{1+\cot ^2(c+d x)}{\sqrt{e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{4 a e}\\ &=\frac{5}{2 a^2 d e \sqrt{e \cot (c+d x)}}-\frac{1}{2 d e \sqrt{e \cot (c+d x)} \left (a^2+a^2 \cot (c+d x)\right )}-\frac{\int \frac{1}{\sqrt{e \cot (c+d x)}} \, dx}{2 a^2 e}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{4 a d e}\\ &=\frac{5}{2 a^2 d e \sqrt{e \cot (c+d x)}}-\frac{1}{2 d e \sqrt{e \cot (c+d x)} \left (a^2+a^2 \cot (c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (e^2+x^2\right )} \, dx,x,e \cot (c+d x)\right )}{2 a^2 d}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{a+\frac{a x^2}{e}} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 a d e^2}\\ &=\frac{5 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 a^2 d e^{3/2}}+\frac{5}{2 a^2 d e \sqrt{e \cot (c+d x)}}-\frac{1}{2 d e \sqrt{e \cot (c+d x)} \left (a^2+a^2 \cot (c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{a^2 d}\\ &=\frac{5 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 a^2 d e^{3/2}}+\frac{5}{2 a^2 d e \sqrt{e \cot (c+d x)}}-\frac{1}{2 d e \sqrt{e \cot (c+d x)} \left (a^2+a^2 \cot (c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{e-x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 a^2 d e}+\frac{\operatorname{Subst}\left (\int \frac{e+x^2}{e^2+x^4} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{2 a^2 d e}\\ &=\frac{5 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 a^2 d e^{3/2}}+\frac{5}{2 a^2 d e \sqrt{e \cot (c+d x)}}-\frac{1}{2 d e \sqrt{e \cot (c+d x)} \left (a^2+a^2 \cot (c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}+2 x}{-e-\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{4 \sqrt{2} a^2 d e^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}-2 x}{-e+\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{4 \sqrt{2} a^2 d e^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{e-\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{4 a^2 d e}+\frac{\operatorname{Subst}\left (\int \frac{1}{e+\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \cot (c+d x)}\right )}{4 a^2 d e}\\ &=\frac{5 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 a^2 d e^{3/2}}+\frac{5}{2 a^2 d e \sqrt{e \cot (c+d x)}}-\frac{1}{2 d e \sqrt{e \cot (c+d x)} \left (a^2+a^2 \cot (c+d x)\right )}-\frac{\log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{4 \sqrt{2} a^2 d e^{3/2}}+\frac{\log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{4 \sqrt{2} a^2 d e^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 \sqrt{2} a^2 d e^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 \sqrt{2} a^2 d e^{3/2}}\\ &=\frac{5 \tan ^{-1}\left (\frac{\sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 a^2 d e^{3/2}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 \sqrt{2} a^2 d e^{3/2}}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \cot (c+d x)}}{\sqrt{e}}\right )}{2 \sqrt{2} a^2 d e^{3/2}}+\frac{5}{2 a^2 d e \sqrt{e \cot (c+d x)}}-\frac{1}{2 d e \sqrt{e \cot (c+d x)} \left (a^2+a^2 \cot (c+d x)\right )}-\frac{\log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)-\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{4 \sqrt{2} a^2 d e^{3/2}}+\frac{\log \left (\sqrt{e}+\sqrt{e} \cot (c+d x)+\sqrt{2} \sqrt{e \cot (c+d x)}\right )}{4 \sqrt{2} a^2 d e^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.28337, size = 203, normalized size = 0.66 \[ \frac{\cot ^{\frac{3}{2}}(c+d x) \left (\frac{\log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-\log \left (-\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}-1\right )}{\sqrt{2}}-\sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )+\sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+10 \tan ^{-1}\left (\sqrt{\cot (c+d x)}\right )+\frac{2 (4 \sin (c+d x)+5 \cos (c+d x))}{\sqrt{\cot (c+d x)} (\sin (c+d x)+\cos (c+d x))}\right )}{4 a^2 d (e \cot (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 255, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}}{8\,d{a}^{2}{e}^{2}}\sqrt [4]{{e}^{2}}\ln \left ({ \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{4\,d{a}^{2}{e}^{2}}\sqrt [4]{{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }-{\frac{\sqrt{2}}{4\,d{a}^{2}{e}^{2}}\sqrt [4]{{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }+2\,{\frac{1}{d{a}^{2}e\sqrt{e\cot \left ( dx+c \right ) }}}+{\frac{1}{2\,d{a}^{2}e \left ( e\cot \left ( dx+c \right ) +e \right ) }\sqrt{e\cot \left ( dx+c \right ) }}+{\frac{5}{2\,d{a}^{2}}\arctan \left ({\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt{e}}}} \right ){e}^{-{\frac{3}{2}}}} \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*} \frac{\int \frac{1}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}} \cot ^{2}{\left (c + d x \right )} + 2 \left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}} \cot{\left (c + d x \right )} + \left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx}{a^{2}} \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{1}{{\left (a \cot \left (d x + c\right ) + a\right )}^{2} \left (e \cot \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]