Optimal. Leaf size=121 \[ -\frac {2 (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (-1-i n)} \sqrt {1+a^2 x^2} \, _2F_1\left (1,\frac {1}{2} (1+i n);\frac {1}{2} (3+i n);\frac {1-i a x}{1+i a x}\right )}{(1+i n) \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5193, 5190,
133} \begin {gather*} -\frac {2 \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (-1-i n)} \, _2F_1\left (1,\frac {1}{2} (i n+1);\frac {1}{2} (i n+3);\frac {1-i a x}{i a x+1}\right )}{(1+i n) \sqrt {a^2 c x^2+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 133
Rule 5190
Rule 5193
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)}}{x \sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {e^{n \tan ^{-1}(a x)}}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \int \frac {(1-i a x)^{-\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{2}-\frac {i n}{2}}}{x} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {2 (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (-1-i n)} \sqrt {1+a^2 x^2} \, _2F_1\left (1,\frac {1}{2} (1+i n);\frac {1}{2} (3+i n);\frac {1-i a x}{1+i a x}\right )}{(1+i n) \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 120, normalized size = 0.99 \begin {gather*} \frac {2 (1-i a x)^{\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{2}-\frac {i n}{2}} \sqrt {1+a^2 x^2} \, _2F_1\left (1,\frac {1}{2}+\frac {i n}{2};\frac {3}{2}+\frac {i n}{2};\frac {i+a x}{i-a x}\right )}{(-1-i n) \sqrt {c+a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{x \sqrt {a^{2} c \,x^{2}+c}}\, dx\]
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*} \int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{x \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \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 \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{x\,\sqrt {c\,a^2\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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