Optimal. Leaf size=20 \[ \frac {\tan ^{-1}(a x)^{n+1}}{a c (n+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {4884} \[ \frac {\tan ^{-1}(a x)^{n+1}}{a c (n+1)} \]
Antiderivative was successfully verified.
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Rule 4884
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^n}{c+a^2 c x^2} \, dx &=\frac {\tan ^{-1}(a x)^{1+n}}{a c (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 20, normalized size = 1.00 \[ \frac {\tan ^{-1}(a x)^{n+1}}{a c (n+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 21, normalized size = 1.05 \[ \frac {\arctan \left (a x\right )^{n} \arctan \left (a x\right )}{a c n + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 20, normalized size = 1.00 \[ \frac {\arctan \left (a x\right )^{n + 1}}{a c {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 21, normalized size = 1.05 \[ \frac {\arctan \left (a x \right )^{1+n}}{a c \left (1+n \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 20, normalized size = 1.00 \[ \frac {{\mathrm {atan}\left (a\,x\right )}^{n+1}}{a\,c\,\left (n+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\operatorname {atan}^{n}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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