Optimal. Leaf size=84 \[ -\frac {2^{2-\frac {i n}{2}} c (1-i a x)^{2+\frac {i n}{2}} \, _2F_1\left (-1+\frac {i n}{2},2+\frac {i n}{2};3+\frac {i n}{2};\frac {1}{2} (1-i a x)\right )}{a (4 i-n)} \]
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Rubi [A]
time = 0.03, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5181, 71}
\begin {gather*} -\frac {c 2^{2-\frac {i n}{2}} (1-i a x)^{2+\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2}-1,\frac {i n}{2}+2;\frac {i n}{2}+3;\frac {1}{2} (1-i a x)\right )}{a (-n+4 i)} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 5181
Rubi steps
\begin {align*} \int e^{n \tan ^{-1}(a x)} \left (c+a^2 c x^2\right ) \, dx &=c \int (1-i a x)^{1+\frac {i n}{2}} (1+i a x)^{1-\frac {i n}{2}} \, dx\\ &=-\frac {2^{2-\frac {i n}{2}} c (1-i a x)^{2+\frac {i n}{2}} \, _2F_1\left (-1+\frac {i n}{2},2+\frac {i n}{2};3+\frac {i n}{2};\frac {1}{2} (1-i a x)\right )}{a (4 i-n)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 88, normalized size = 1.05 \begin {gather*} \frac {i 2^{1-\frac {i n}{2}} c (1-i a x)^{2+\frac {i n}{2}} \, _2F_1\left (-1+\frac {i n}{2},2+\frac {i n}{2};3+\frac {i n}{2};\frac {1}{2} (1-i a x)\right )}{a \left (2+\frac {i n}{2}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} c \left (\int a^{2} x^{2} e^{n \operatorname {atan}{\left (a x \right )}}\, dx + \int e^{n \operatorname {atan}{\left (a x \right )}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,\left (c\,a^2\,x^2+c\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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