Optimal. Leaf size=168 \[ \frac{d x^2 \left (35 a^4 c^2-21 a^2 c d+5 d^2\right )}{70 a^5}+\frac{\left (-35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2-5 d^3\right ) \log \left (a^2 x^2+1\right )}{70 a^7}+\frac{d^2 x^4 \left (21 a^2 c-5 d\right )}{140 a^3}+c^2 d x^3 \cot ^{-1}(a x)+c^3 x \cot ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac{d^3 x^6}{42 a}+\frac{1}{7} d^3 x^7 \cot ^{-1}(a x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.120067, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {194, 4913, 1810, 260} \[ \frac{d x^2 \left (35 a^4 c^2-21 a^2 c d+5 d^2\right )}{70 a^5}+\frac{\left (-35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2-5 d^3\right ) \log \left (a^2 x^2+1\right )}{70 a^7}+\frac{d^2 x^4 \left (21 a^2 c-5 d\right )}{140 a^3}+c^2 d x^3 \cot ^{-1}(a x)+c^3 x \cot ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac{d^3 x^6}{42 a}+\frac{1}{7} d^3 x^7 \cot ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 194
Rule 4913
Rule 1810
Rule 260
Rubi steps
\begin{align*} \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx &=c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cot ^{-1}(a x)+a \int \frac{c^3 x+c^2 d x^3+\frac{3}{5} c d^2 x^5+\frac{d^3 x^7}{7}}{1+a^2 x^2} \, dx\\ &=c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cot ^{-1}(a x)+a \int \left (\frac{d \left (35 a^4 c^2-21 a^2 c d+5 d^2\right ) x}{35 a^6}+\frac{\left (21 a^2 c-5 d\right ) d^2 x^3}{35 a^4}+\frac{d^3 x^5}{7 a^2}+\frac{\left (35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right ) x}{35 a^6 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac{d \left (35 a^4 c^2-21 a^2 c d+5 d^2\right ) x^2}{70 a^5}+\frac{\left (21 a^2 c-5 d\right ) d^2 x^4}{140 a^3}+\frac{d^3 x^6}{42 a}+c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cot ^{-1}(a x)+\frac{\left (35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right ) \int \frac{x}{1+a^2 x^2} \, dx}{35 a^5}\\ &=\frac{d \left (35 a^4 c^2-21 a^2 c d+5 d^2\right ) x^2}{70 a^5}+\frac{\left (21 a^2 c-5 d\right ) d^2 x^4}{140 a^3}+\frac{d^3 x^6}{42 a}+c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cot ^{-1}(a x)+\frac{\left (35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right ) \log \left (1+a^2 x^2\right )}{70 a^7}\\ \end{align*}
Mathematica [A] time = 0.103544, size = 149, normalized size = 0.89 \[ \frac{a^2 d x^2 \left (a^4 \left (210 c^2+63 c d x^2+10 d^2 x^4\right )-3 a^2 d \left (42 c+5 d x^2\right )+30 d^2\right )+6 \left (-35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2-5 d^3\right ) \log \left (a^2 x^2+1\right )+12 a^7 x \cot ^{-1}(a x) \left (35 c^2 d x^2+35 c^3+21 c d^2 x^4+5 d^3 x^6\right )}{420 a^7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 191, normalized size = 1.1 \begin{align*}{\frac{{d}^{3}{x}^{7}{\rm arccot} \left (ax\right )}{7}}+{\frac{3\,c{d}^{2}{x}^{5}{\rm arccot} \left (ax\right )}{5}}+{c}^{2}d{x}^{3}{\rm arccot} \left (ax\right )+{c}^{3}x{\rm arccot} \left (ax\right )+{\frac{{c}^{2}d{x}^{2}}{2\,a}}+{\frac{3\,c{x}^{4}{d}^{2}}{20\,a}}+{\frac{{d}^{3}{x}^{6}}{42\,a}}-{\frac{3\,c{d}^{2}{x}^{2}}{10\,{a}^{3}}}-{\frac{{d}^{3}{x}^{4}}{28\,{a}^{3}}}+{\frac{{d}^{3}{x}^{2}}{14\,{a}^{5}}}+{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ){c}^{3}}{2\,a}}-{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ){c}^{2}d}{2\,{a}^{3}}}+{\frac{3\,\ln \left ({a}^{2}{x}^{2}+1 \right ) c{d}^{2}}{10\,{a}^{5}}}-{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ){d}^{3}}{14\,{a}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.971695, size = 215, normalized size = 1.28 \begin{align*} \frac{1}{420} \, a{\left (\frac{10 \, a^{4} d^{3} x^{6} + 3 \,{\left (21 \, a^{4} c d^{2} - 5 \, a^{2} d^{3}\right )} x^{4} + 6 \,{\left (35 \, a^{4} c^{2} d - 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} x^{2}}{a^{6}} + \frac{6 \,{\left (35 \, a^{6} c^{3} - 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} - 5 \, d^{3}\right )} \log \left (a^{2} x^{2} + 1\right )}{a^{8}}\right )} + \frac{1}{35} \,{\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \operatorname{arccot}\left (a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.83765, size = 366, normalized size = 2.18 \begin{align*} \frac{10 \, a^{6} d^{3} x^{6} + 3 \,{\left (21 \, a^{6} c d^{2} - 5 \, a^{4} d^{3}\right )} x^{4} + 6 \,{\left (35 \, a^{6} c^{2} d - 21 \, a^{4} c d^{2} + 5 \, a^{2} d^{3}\right )} x^{2} + 12 \,{\left (5 \, a^{7} d^{3} x^{7} + 21 \, a^{7} c d^{2} x^{5} + 35 \, a^{7} c^{2} d x^{3} + 35 \, a^{7} c^{3} x\right )} \operatorname{arccot}\left (a x\right ) + 6 \,{\left (35 \, a^{6} c^{3} - 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} - 5 \, d^{3}\right )} \log \left (a^{2} x^{2} + 1\right )}{420 \, a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.53637, size = 243, normalized size = 1.45 \begin{align*} \begin{cases} c^{3} x \operatorname{acot}{\left (a x \right )} + c^{2} d x^{3} \operatorname{acot}{\left (a x \right )} + \frac{3 c d^{2} x^{5} \operatorname{acot}{\left (a x \right )}}{5} + \frac{d^{3} x^{7} \operatorname{acot}{\left (a x \right )}}{7} + \frac{c^{3} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{2 a} + \frac{c^{2} d x^{2}}{2 a} + \frac{3 c d^{2} x^{4}}{20 a} + \frac{d^{3} x^{6}}{42 a} - \frac{c^{2} d \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{2 a^{3}} - \frac{3 c d^{2} x^{2}}{10 a^{3}} - \frac{d^{3} x^{4}}{28 a^{3}} + \frac{3 c d^{2} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{10 a^{5}} + \frac{d^{3} x^{2}}{14 a^{5}} - \frac{d^{3} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{14 a^{7}} & \text{for}\: a \neq 0 \\\frac{\pi \left (c^{3} x + c^{2} d x^{3} + \frac{3 c d^{2} x^{5}}{5} + \frac{d^{3} x^{7}}{7}\right )}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13143, size = 221, normalized size = 1.32 \begin{align*} \frac{1}{35} \,{\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \arctan \left (\frac{1}{a x}\right ) + \frac{10 \, a^{5} d^{3} x^{6} + 63 \, a^{5} c d^{2} x^{4} + 210 \, a^{5} c^{2} d x^{2} - 15 \, a^{3} d^{3} x^{4} - 126 \, a^{3} c d^{2} x^{2} + 30 \, a d^{3} x^{2}}{420 \, a^{6}} + \frac{{\left (35 \, a^{6} c^{3} - 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} - 5 \, d^{3}\right )} \log \left (a^{2} x^{2} + 1\right )}{70 \, a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]