Optimal. Leaf size=196 \[ -\frac {2 a n \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (-1-i n)} \, _2F_1\left (1,\frac {1}{2} (i n+1);\frac {1}{2} (i n+3);\frac {1-i a x}{i a x+1}\right )}{(1+i n) \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)}}{x \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.23, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5085, 5082, 96, 131} \[ -\frac {2 a n \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (-1-i n)} \, _2F_1\left (1,\frac {1}{2} (i n+1);\frac {1}{2} (i n+3);\frac {1-i a x}{i a x+1}\right )}{(1+i n) \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)}}{x \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 96
Rule 131
Rule 5082
Rule 5085
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)}}{x^2 \sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {e^{n \tan ^{-1}(a x)}}{x^2 \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^2} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {(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}}{x \sqrt {c+a^2 c x^2}}+\frac {\left (a n \sqrt {1+a^2 x^2}\right ) \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 {(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}}{x \sqrt {c+a^2 c x^2}}-\frac {2 a n (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 = 0.07, size = 142, normalized size = 0.72 \[ \frac {\sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{2}-\frac {i n}{2}} \left (2 a n x \, _2F_1\left (1,\frac {i n}{2}+\frac {1}{2};\frac {i n}{2}+\frac {3}{2};\frac {a x+i}{i-a x}\right )-(n-i) (a x-i)\right )}{(-1-i n) x \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{4} + c x^{2}}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{x^{2} \sqrt {a^{2} c \,x^{2}+c}}\, dx \]
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 \[ \int \frac {e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c} x^{2}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{x^2\,\sqrt {c\,a^2\,x^2+c}} \,d x \]
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 \[ \int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{x^{2} \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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