Optimal. Leaf size=143 \[ \frac {\sqrt {1+a^2 x^2}}{2 a^3 c (i-a x) \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \log (i-a x)}{4 a^3 c \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \log (i+a x)}{4 a^3 c \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {5193, 5190, 90}
\begin {gather*} \frac {\sqrt {a^2 x^2+1}}{2 a^3 c (-a x+i) \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \log (-a x+i)}{4 a^3 c \sqrt {a^2 c x^2+c}}-\frac {i \sqrt {a^2 x^2+1} \log (a x+i)}{4 a^3 c \sqrt {a^2 c x^2+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 5190
Rule 5193
Rubi steps
\begin {align*} \int \frac {e^{-i \tan ^{-1}(a x)} x^2}{\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)} x^2}{\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 {x^2}{(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 a^2 (-i+a x)^2}-\frac {3 i}{4 a^2 (-i+a x)}-\frac {i}{4 a^2 (i+a x)}\right ) \, dx}{c \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2}}{2 a^3 c (i-a x) \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \log (i-a x)}{4 a^3 c \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \log (i+a x)}{4 a^3 c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 75, normalized size = 0.52 \begin {gather*} \frac {\sqrt {1+a^2 x^2} \left (\frac {2}{i-a x}-3 i \log (i-a x)-i \log (i+a x)\right )}{4 a^3 c \sqrt {c+a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 86, normalized size = 0.60
method | result | size |
default | \(\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (3 i \ln \left (-a x +i\right ) a x +i \ln \left (a x +i\right ) a x +3 \ln \left (-a x +i\right )+\ln \left (a x +i\right )+2\right )}{4 \sqrt {a^{2} x^{2}+1}\, c^{2} a^{3} \left (-a x +i\right )}\) | \(86\) |
risch | \(-\frac {\sqrt {a^{2} x^{2}+1}}{2 c \sqrt {c \left (a^{2} x^{2}+1\right )}\, a^{3} \left (a x -i\right )}-\frac {3 i \sqrt {a^{2} x^{2}+1}\, \ln \left (-i a x -1\right )}{4 c \sqrt {c \left (a^{2} x^{2}+1\right )}\, a^{3}}-\frac {i \sqrt {a^{2} x^{2}+1}\, \ln \left (i a x -1\right )}{4 c \sqrt {c \left (a^{2} x^{2}+1\right )}\, a^{3}}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 54, normalized size = 0.38 \begin {gather*} -\frac {\sqrt {c}}{2 \, {\left (a^{4} c^{2} x - i \, a^{3} c^{2}\right )}} - \frac {3 i \, \log \left (a x - i\right )}{4 \, a^{3} c^{\frac {3}{2}}} - \frac {i \, \log \left (i \, a x - 1\right )}{4 \, a^{3} c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - i \int \frac {x^{2} \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 \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^2\,\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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