Optimal. Leaf size=49 \[ -\frac {i \sqrt {a^2 x^2+1}}{2 a c (1-i a x)^2 \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.07, antiderivative size = 49, 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 = {5076, 5073, 32} \[ -\frac {i \sqrt {a^2 x^2+1}}{2 a c (1-i a x)^2 \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 32
Rule 5073
Rule 5076
Rubi steps
\begin {align*} \int \frac {e^{3 i \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {e^{3 i \tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{c \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \int \frac {1}{(1-i a x)^3} \, dx}{c \sqrt {c+a^2 c x^2}}\\ &=-\frac {i \sqrt {1+a^2 x^2}}{2 a c (1-i a x)^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 1.16 \[ \frac {i \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}{2 a c^2 (a x-i) (a x+i)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 71, normalized size = 1.45 \[ \frac {\sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} {\left (i \, a x^{2} - 2 \, x\right )}}{2 \, a^{4} c^{2} x^{4} + 4 i \, a^{3} c^{2} x^{3} + 4 i \, a c^{2} x - 2 \, c^{2}} \]
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 \[ \int \frac {{\left (i \, a x + 1\right )}^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 44, normalized size = 0.90 \[ -\frac {\left (a x +i\right ) \left (i a x +1\right )^{3}}{2 a \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 41, normalized size = 0.84 \[ \frac {\sqrt {c\,\left (a^2\,x^2+1\right )}\,1{}\mathrm {i}}{2\,a\,c^2\,\sqrt {a^2\,x^2+1}\,{\left (a\,x+1{}\mathrm {i}\right )}^2} \]
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 \[ - i \left (\int \frac {i}{a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {3 a x}{a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {a^{3} x^{3}}{a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {3 i a^{2} x^{2}}{a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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