Optimal. Leaf size=88 \[ \frac {\sqrt {a^2 x^2+1} \tan ^{-1}(a x)}{2 a c \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1}}{2 a c (a x+i) \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.08, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5076, 5073, 44, 203} \[ \frac {\sqrt {a^2 x^2+1} \tan ^{-1}(a x)}{2 a c \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1}}{2 a c (a x+i) \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 203
Rule 5073
Rule 5076
Rubi steps
\begin {align*} \int \frac {e^{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^{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)^2 (1+i a x)} \, dx}{c \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \int \left (-\frac {1}{2 (i+a x)^2}+\frac {1}{2 \left (1+a^2 x^2\right )}\right ) \, dx}{c \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2}}{2 a c (i+a x) \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \int \frac {1}{1+a^2 x^2} \, dx}{2 c \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2}}{2 a c (i+a x) \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \tan ^{-1}(a x)}{2 a c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 51, normalized size = 0.58 \[ \frac {\sqrt {a^2 x^2+1} \left (\tan ^{-1}(a x)+\frac {1}{a x+i}\right )}{2 a c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 317, normalized size = 3.60 \[ \frac {{\left (-i \, a^{3} c^{2} x^{3} + a^{2} c^{2} x^{2} - i \, a c^{2} x + c^{2}\right )} \sqrt {\frac {1}{a^{2} c^{3}}} \log \left (\frac {8 \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} a^{6} x + {\left (4 i \, a^{10} c^{2} x^{4} - 4 i \, a^{6} c^{2}\right )} \sqrt {\frac {1}{a^{2} c^{3}}}}{2 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )}}\right ) + {\left (i \, a^{3} c^{2} x^{3} - a^{2} c^{2} x^{2} + i \, a c^{2} x - c^{2}\right )} \sqrt {\frac {1}{a^{2} c^{3}}} \log \left (\frac {8 \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} a^{6} x + {\left (-4 i \, a^{10} c^{2} x^{4} + 4 i \, a^{6} c^{2}\right )} \sqrt {\frac {1}{a^{2} c^{3}}}}{2 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )}}\right ) + 4 i \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} x}{2 \, {\left (4 \, a^{3} c^{2} x^{3} + 4 i \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x + 4 i \, c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 54, normalized size = 0.61 \[ \frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (\arctan \left (a x \right ) x^{2} a^{2}+a x +\arctan \left (a x \right )-i\right )}{2 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} a \,c^{2}} \]
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 [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1+a\,x\,1{}\mathrm {i}}{{\left (c\,a^2\,x^2+c\right )}^{3/2}\,\sqrt {a^2\,x^2+1}} \,d x \]
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 \[ i \left (\int \left (- \frac {i}{a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {a x}{a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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