Optimal. Leaf size=182 \[ \frac{n \left (3 a^2 d-e\right ) \text{PolyLog}\left (2,-a^2 x^2\right )}{12 a^3}+\frac{\left (3 a^2 d-e\right ) \log \left (a^2 x^2+1\right ) \log \left (c x^n\right )}{6 a^3}-\frac{d n \log \left (a^2 x^2+1\right )}{2 a}+\frac{e n \log \left (a^2 x^2+1\right )}{18 a^3}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac{e x^2 \log \left (c x^n\right )}{6 a}+\frac{1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )-d n x \cot ^{-1}(a x)-\frac{5 e n x^2}{36 a}-\frac{1}{9} e n x^3 \cot ^{-1}(a x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.147888, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {4913, 1593, 444, 43, 2388, 4847, 260, 4853, 266, 2391} \[ \frac{n \left (3 a^2 d-e\right ) \text{PolyLog}\left (2,-a^2 x^2\right )}{12 a^3}+\frac{\left (3 a^2 d-e\right ) \log \left (a^2 x^2+1\right ) \log \left (c x^n\right )}{6 a^3}-\frac{d n \log \left (a^2 x^2+1\right )}{2 a}+\frac{e n \log \left (a^2 x^2+1\right )}{18 a^3}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac{e x^2 \log \left (c x^n\right )}{6 a}+\frac{1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )-d n x \cot ^{-1}(a x)-\frac{5 e n x^2}{36 a}-\frac{1}{9} e n x^3 \cot ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4913
Rule 1593
Rule 444
Rule 43
Rule 2388
Rule 4847
Rule 260
Rule 4853
Rule 266
Rule 2391
Rubi steps
\begin{align*} \int \left (d+e x^2\right ) \cot ^{-1}(a x) \log \left (c x^n\right ) \, dx &=\frac{e x^2 \log \left (c x^n\right )}{6 a}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )+\frac{\left (3 a^2 d-e\right ) \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )}{6 a^3}-n \int \left (\frac{e x}{6 a}+d \cot ^{-1}(a x)+\frac{1}{3} e x^2 \cot ^{-1}(a x)+\frac{\left (3 a^2 d-e\right ) \log \left (1+a^2 x^2\right )}{6 a^3 x}\right ) \, dx\\ &=-\frac{e n x^2}{12 a}+\frac{e x^2 \log \left (c x^n\right )}{6 a}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )+\frac{\left (3 a^2 d-e\right ) \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )}{6 a^3}-(d n) \int \cot ^{-1}(a x) \, dx-\frac{\left (\left (3 a^2 d-e\right ) n\right ) \int \frac{\log \left (1+a^2 x^2\right )}{x} \, dx}{6 a^3}-\frac{1}{3} (e n) \int x^2 \cot ^{-1}(a x) \, dx\\ &=-\frac{e n x^2}{12 a}-d n x \cot ^{-1}(a x)-\frac{1}{9} e n x^3 \cot ^{-1}(a x)+\frac{e x^2 \log \left (c x^n\right )}{6 a}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )+\frac{\left (3 a^2 d-e\right ) \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )}{6 a^3}+\frac{\left (3 a^2 d-e\right ) n \text{Li}_2\left (-a^2 x^2\right )}{12 a^3}-(a d n) \int \frac{x}{1+a^2 x^2} \, dx-\frac{1}{9} (a e n) \int \frac{x^3}{1+a^2 x^2} \, dx\\ &=-\frac{e n x^2}{12 a}-d n x \cot ^{-1}(a x)-\frac{1}{9} e n x^3 \cot ^{-1}(a x)+\frac{e x^2 \log \left (c x^n\right )}{6 a}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )-\frac{d n \log \left (1+a^2 x^2\right )}{2 a}+\frac{\left (3 a^2 d-e\right ) \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )}{6 a^3}+\frac{\left (3 a^2 d-e\right ) n \text{Li}_2\left (-a^2 x^2\right )}{12 a^3}-\frac{1}{18} (a e n) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{e n x^2}{12 a}-d n x \cot ^{-1}(a x)-\frac{1}{9} e n x^3 \cot ^{-1}(a x)+\frac{e x^2 \log \left (c x^n\right )}{6 a}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )-\frac{d n \log \left (1+a^2 x^2\right )}{2 a}+\frac{\left (3 a^2 d-e\right ) \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )}{6 a^3}+\frac{\left (3 a^2 d-e\right ) n \text{Li}_2\left (-a^2 x^2\right )}{12 a^3}-\frac{1}{18} (a e n) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{5 e n x^2}{36 a}-d n x \cot ^{-1}(a x)-\frac{1}{9} e n x^3 \cot ^{-1}(a x)+\frac{e x^2 \log \left (c x^n\right )}{6 a}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )-\frac{d n \log \left (1+a^2 x^2\right )}{2 a}+\frac{e n \log \left (1+a^2 x^2\right )}{18 a^3}+\frac{\left (3 a^2 d-e\right ) \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )}{6 a^3}+\frac{\left (3 a^2 d-e\right ) n \text{Li}_2\left (-a^2 x^2\right )}{12 a^3}\\ \end{align*}
Mathematica [A] time = 0.12069, size = 178, normalized size = 0.98 \[ \frac{\text{PolyLog}\left (2,-a^2 x^2\right ) \left (9 a^2 d n-3 e n\right )-4 a^3 x \cot ^{-1}(a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )+18 a^2 d \log \left (a^2 x^2+1\right ) \log \left (c x^n\right )+6 a^2 e x^2 \log \left (c x^n\right )-6 e \log \left (a^2 x^2+1\right ) \log \left (c x^n\right )+36 a^2 d n \log \left (\frac{1}{a x \sqrt{\frac{1}{a^2 x^2}+1}}\right )-5 a^2 e n x^2+2 e n \log \left (a^2 x^2+1\right )}{36 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 11.066, size = 0, normalized size = 0. \begin{align*} \int \left ({x}^{2}e+d \right ){\rm arccot} \left (ax\right )\ln \left ( c{x}^{n} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (e x^{2} + d\right )} \operatorname{arccot}\left (a x\right ) \log \left (c x^{n}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [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]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )} \operatorname{arccot}\left (a x\right ) \log \left (c x^{n}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]