Optimal. Leaf size=54 \[ \frac {x}{3 c \sqrt {a^2 c x^2+c}}-\frac {2 i (1+i a x)}{3 a \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5075, 653, 191} \[ \frac {x}{3 c \sqrt {a^2 c x^2+c}}-\frac {2 i (1+i a x)}{3 a \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 653
Rule 5075
Rubi steps
\begin {align*} \int \frac {e^{2 i \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=c \int \frac {(1+i a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\\ &=-\frac {2 i (1+i a x)}{3 a \left (c+a^2 c x^2\right )^{3/2}}+\frac {1}{3} \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 i (1+i a x)}{3 a \left (c+a^2 c x^2\right )^{3/2}}+\frac {x}{3 c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 78, normalized size = 1.44 \[ \frac {(2-i a x) \sqrt {1+i a x} \sqrt {a^2 x^2+1}}{3 a c \sqrt {1-i a x} (a x+i) \sqrt {a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.55, size = 47, normalized size = 0.87 \[ \frac {\sqrt {a^{2} c x^{2} + c} {\left (a x + 2 i\right )}}{3 \, a^{3} c^{2} x^{2} + 6 i \, a^{2} c^{2} x - 3 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 76, normalized size = 1.41 \[ -\frac {2 \, \sqrt {a^{2} c} {\left (\sqrt {c} i + 3 \, \sqrt {a^{2} c} x - 3 \, \sqrt {a^{2} c x^{2} + c}\right )}}{3 \, {\left (\sqrt {c} i + \sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )}^{3} a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 398, normalized size = 7.37 \[ -\frac {x}{c \sqrt {a^{2} c \,x^{2}+c}}+\frac {\left (i \sqrt {-a^{2}}+a \right ) \left (-\frac {1}{3 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right ) \sqrt {\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}-\frac {2 \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right ) a^{2} c +2 c \sqrt {-a^{2}}}{3 c^{2} \sqrt {-a^{2}}\, \sqrt {\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}\right )}{a \sqrt {-a^{2}}}+\frac {\left (i \sqrt {-a^{2}}-a \right ) \left (\frac {1}{3 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right ) \sqrt {\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}+\frac {2 \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right ) a^{2} c -2 c \sqrt {-a^{2}}}{3 c^{2} \sqrt {-a^{2}}\, \sqrt {\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}\right )}{a \sqrt {-a^{2}}} \]
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 \[ \int \frac {{\left (i \, a x + 1\right )}^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a^{2} x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 32, normalized size = 0.59 \[ \frac {a^3\,x^3+3\,a\,x-2{}\mathrm {i}}{3\,a\,{\left (c\,\left (a^2\,x^2+1\right )\right )}^{3/2}} \]
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 \[ - \int \frac {a^{2} x^{2}}{a^{4} c x^{4} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} c x^{2} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} c x^{2} + c}}\, dx - \int \left (- \frac {2 i a x}{a^{4} c x^{4} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} c x^{2} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} c x^{2} + c}}\right )\, dx - \int \left (- \frac {1}{a^{4} c x^{4} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} c x^{2} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} c x^{2} + c}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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