Optimal. Leaf size=89 \[ \frac {\sqrt {a^2 x^2+1} \tan ^{-1}(a x)}{2 a c \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 x^2+1}}{2 a c (-a x+i) \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.08, antiderivative size = 89, 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) (1+i a x)^2} \, 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 = 60, normalized size = 0.67 \[ \frac {\sqrt {a^2 x^2+1} \left (\frac {\tan ^{-1}(a x)}{2 a}-\frac {1}{2 a (-a x+i)}\right )}{c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 317, normalized size = 3.56 \[ \frac {{\left (-i \, a^{3} c^{2} x^{3} - a^{2} c^{2} x^{2} - i \, a c^{2} x - c^{2}\right )} \sqrt {\frac {1}{a^{2} c^{3}}} \log \left (\frac {8 \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} a^{6} x + {\left (4 i \, a^{10} c^{2} x^{4} - 4 i \, a^{6} c^{2}\right )} \sqrt {\frac {1}{a^{2} c^{3}}}}{2 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )}}\right ) + {\left (i \, a^{3} c^{2} x^{3} + a^{2} c^{2} x^{2} + i \, a c^{2} x + c^{2}\right )} \sqrt {\frac {1}{a^{2} c^{3}}} \log \left (\frac {8 \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} a^{6} x + {\left (-4 i \, a^{10} c^{2} x^{4} + 4 i \, a^{6} c^{2}\right )} \sqrt {\frac {1}{a^{2} c^{3}}}}{2 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )}}\right ) - 4 i \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} x}{2 \, {\left (4 \, a^{3} c^{2} x^{3} - 4 i \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x - 4 i \, c^{2}\right )}} \]
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 = 0.20, size = 86, normalized size = 0.97 \[ -\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (i \ln \left (a x +i\right ) x a -i \ln \left (-a x +i\right ) x a +\ln \left (a x +i\right )-\ln \left (-a x +i\right )+2\right )}{4 \sqrt {a^{2} x^{2}+1}\, c^{2} a \left (-a x +i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 52, normalized size = 0.58 \[ \frac {\sqrt {c}}{2 \, a^{2} c^{2} x - 2 i \, a c^{2}} - \frac {i \, \log \left (a x - i\right )}{4 \, a c^{\frac {3}{2}}} + \frac {i \, \log \left (i \, a x - 1\right )}{4 \, a c^{\frac {3}{2}}} \]
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 \[ \int \frac {\sqrt {a^2\,x^2+1}}{{\left (c\,a^2\,x^2+c\right )}^{3/2}\,\left (1+a\,x\,1{}\mathrm {i}\right )} \,d x \]
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 \[ - i \int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} c x^{3} \sqrt {a^{2} c x^{2} + c} - i a^{2} c x^{2} \sqrt {a^{2} c x^{2} + c} + a c x \sqrt {a^{2} c x^{2} + c} - i c \sqrt {a^{2} c x^{2} + c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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