Optimal. Leaf size=202 \[ \frac {\sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)}}{a^2 (1-i n) \sqrt {a^2 c x^2+c}}-\frac {i 2^{\frac {3}{2}-\frac {i n}{2}} n \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} \, _2F_1\left (\frac {1}{2} (i n-1),\frac {1}{2} (i n+1);\frac {1}{2} (i n+3);\frac {1}{2} (1-i a x)\right )}{a^2 \left (n^2+1\right ) \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.19, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5085, 5082, 79, 69} \[ \frac {\sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)}}{a^2 (1-i n) \sqrt {a^2 c x^2+c}}-\frac {i 2^{\frac {3}{2}-\frac {i n}{2}} n \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} \, _2F_1\left (\frac {1}{2} (i n-1),\frac {1}{2} (i n+1);\frac {1}{2} (i n+3);\frac {1}{2} (1-i a x)\right )}{a^2 \left (n^2+1\right ) \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 79
Rule 5082
Rule 5085
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 x (1-i a x)^{-\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{2}-\frac {i n}{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}}{a^2 (1-i n) \sqrt {c+a^2 c x^2}}-\frac {\left (n \sqrt {1+a^2 x^2}\right ) \int (1-i a x)^{-\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{2} i (i+n)} \, dx}{a (1-i n) \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}}{a^2 (1-i n) \sqrt {c+a^2 c x^2}}-\frac {i 2^{\frac {3}{2}-\frac {i n}{2}} n (1-i a x)^{\frac {1}{2} (1+i n)} \sqrt {1+a^2 x^2} \, _2F_1\left (\frac {1}{2} (-1+i n),\frac {1}{2} (1+i n);\frac {1}{2} (3+i n);\frac {1}{2} (1-i a x)\right )}{a^2 \left (1+n^2\right ) \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 187, normalized size = 0.93 \[ \frac {i 2^{-\frac {1}{2}-\frac {i n}{2}} \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \left (2^{\frac {1}{2}+\frac {i n}{2}} (n-i) \sqrt {1+i a x}-4 n (1+i a x)^{\frac {i n}{2}} \, _2F_1\left (\frac {1}{2} (i n+1),\frac {1}{2} i (n+i);\frac {1}{2} (i n+3);\frac {1}{2} (1-i a x)\right )\right )}{a^2 \left (n^2+1\right ) \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}}, x\right ) \]
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 \[ \int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x}{\sqrt {a^{2} c \,x^{2}+c}}\, dx \]
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 \[ \int \frac {x e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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 \[ \int \frac {x e^{n \operatorname {atan}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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