Optimal. Leaf size=96 \[ \frac {2 i c (1-i a x)^3}{3 a \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 i (1-i a x)}{a \sqrt {a^2 c x^2+c}}+\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a \sqrt {c}} \]
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Rubi [A] time = 0.08, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5074, 669, 653, 217, 206} \[ \frac {2 i c (1-i a x)^3}{3 a \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 i (1-i a x)}{a \sqrt {a^2 c x^2+c}}+\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 653
Rule 669
Rule 5074
Rubi steps
\begin {align*} \int \frac {e^{-4 i \tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx &=c^2 \int \frac {(1-i a x)^4}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac {2 i c (1-i a x)^3}{3 a \left (c+a^2 c x^2\right )^{3/2}}-c \int \frac {(1-i a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac {2 i c (1-i a x)^3}{3 a \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 i (1-i a x)}{a \sqrt {c+a^2 c x^2}}+\int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {2 i c (1-i a x)^3}{3 a \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 i (1-i a x)}{a \sqrt {c+a^2 c x^2}}+\operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )\\ &=\frac {2 i c (1-i a x)^3}{3 a \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 i (1-i a x)}{a \sqrt {c+a^2 c x^2}}+\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 132, normalized size = 1.38 \[ \frac {2 \sqrt {a^2 x^2+1} \left (2 i \sqrt {1+i a x} \left (2 a^2 x^2+i a x+1\right )+3 i \sqrt {1-i a x} (a x-i)^2 \sin ^{-1}\left (\frac {\sqrt {1-i a x}}{\sqrt {2}}\right )\right )}{3 a \sqrt {1-i a x} (a x-i)^2 \sqrt {a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.51, size = 187, normalized size = 1.95 \[ \frac {{\left (3 \, a^{3} c x^{2} - 6 i \, a^{2} c x - 3 \, a c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {2 \, {\left (a^{2} c x + \sqrt {a^{2} c x^{2} + c} a^{2} c \sqrt {\frac {1}{a^{2} c}}\right )}}{x}\right ) - {\left (3 \, a^{3} c x^{2} - 6 i \, a^{2} c x - 3 \, a c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {2 \, {\left (a^{2} c x - \sqrt {a^{2} c x^{2} + c} a^{2} c \sqrt {\frac {1}{a^{2} c}}\right )}}{x}\right ) - \sqrt {a^{2} c x^{2} + c} {\left (16 \, a x - 8 i\right )}}{6 \, a^{3} c x^{2} - 12 i \, a^{2} c x - 6 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 137, normalized size = 1.43 \[ -\frac {\log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} + c} \right |}\right )}{a \sqrt {c}} + \frac {8 \, {\left (3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )}^{2} i - 2 \, c i + 3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )} \sqrt {c}\right )}}{3 \, {\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )} i + \sqrt {c}\right )}^{3} a i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 136, normalized size = 1.42 \[ \frac {\ln \left (\frac {x \,a^{2} c}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+c}\right )}{\sqrt {a^{2} c}}-\frac {8 \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2} c +2 i a c \left (x -\frac {i}{a}\right )}}{3 a^{2} c \left (x -\frac {i}{a}\right )}-\frac {4 i \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2} c +2 i a c \left (x -\frac {i}{a}\right )}}{3 a^{3} c \left (x -\frac {i}{a}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 76, normalized size = 0.79 \[ -\frac {4 i \, \sqrt {a^{2} c x^{2} + c}}{3 \, {\left (a^{3} c x^{2} - 2 i \, a^{2} c x - a c\right )}} - \frac {8 i \, \sqrt {a^{2} c x^{2} + c}}{3 i \, a^{2} c x + 3 \, a c} + \frac {\operatorname {arsinh}\left (a x\right )}{a \sqrt {c}} \]
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 \[ \int \frac {{\left (a^2\,x^2+1\right )}^2}{\sqrt {c\,a^2\,x^2+c}\,{\left (1+a\,x\,1{}\mathrm {i}\right )}^4} \,d x \]
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 \[ \int \frac {\left (a^{2} x^{2} + 1\right )^{2}}{\sqrt {c \left (a^{2} x^{2} + 1\right )} \left (a x - i\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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