Optimal. Leaf size=69 \[ \frac {i c (1-i a x)^4}{3 a \left (a^2 c x^2+c\right )^{5/2}}-\frac {i c (1-i a x)^5}{15 a \left (a^2 c x^2+c\right )^{5/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 69, 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 = {5074, 659, 651} \[ \frac {i c (1-i a x)^4}{3 a \left (a^2 c x^2+c\right )^{5/2}}-\frac {i c (1-i a x)^5}{15 a \left (a^2 c x^2+c\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 651
Rule 659
Rule 5074
Rubi steps
\begin {align*} \int \frac {e^{-4 i \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=c^2 \int \frac {(1-i a x)^4}{\left (c+a^2 c x^2\right )^{7/2}} \, dx\\ &=\frac {i c (1-i a x)^4}{3 a \left (c+a^2 c x^2\right )^{5/2}}-\frac {1}{3} c^2 \int \frac {(1-i a x)^5}{\left (c+a^2 c x^2\right )^{7/2}} \, dx\\ &=\frac {i c (1-i a x)^4}{3 a \left (c+a^2 c x^2\right )^{5/2}}-\frac {i c (1-i a x)^5}{15 a \left (c+a^2 c x^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 77, normalized size = 1.12 \[ \frac {(1-i a x)^{3/2} (a x-4 i) \sqrt {a^2 x^2+1}}{15 a c \sqrt {1+i a x} (a x-i)^2 \sqrt {a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.72, size = 67, normalized size = 0.97 \[ -\frac {\sqrt {a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 3 i \, a x + 4\right )}}{15 \, a^{4} c^{2} x^{3} - 45 i \, a^{3} c^{2} x^{2} - 45 \, a^{2} c^{2} x + 15 i \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 134, normalized size = 1.94 \[ -\frac {2 \, {\left (5 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )}^{2} c i + 15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )}^{3} \sqrt {c} + c^{2} i - 5 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )} c^{\frac {3}{2}}\right )}}{15 \, {\left (\sqrt {c} i - \sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} + c}\right )}^{5} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 307, normalized size = 4.45 \[ \frac {x}{c \sqrt {a^{2} c \,x^{2}+c}}-\frac {4 \left (\frac {i}{5 a c \left (x -\frac {i}{a}\right )^{2} \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2} c +2 i a c \left (x -\frac {i}{a}\right )}}+\frac {3 i a \left (\frac {i}{3 a c \left (x -\frac {i}{a}\right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2} c +2 i a c \left (x -\frac {i}{a}\right )}}+\frac {i \left (2 \left (x -\frac {i}{a}\right ) a^{2} c +2 i a c \right )}{3 a \,c^{2} \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2} c +2 i a c \left (x -\frac {i}{a}\right )}}\right )}{5}\right )}{a^{2}}+\frac {4 i \left (\frac {i}{3 a c \left (x -\frac {i}{a}\right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2} c +2 i a c \left (x -\frac {i}{a}\right )}}+\frac {i \left (2 \left (x -\frac {i}{a}\right ) a^{2} c +2 i a c \right )}{3 a \,c^{2} \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2} c +2 i a c \left (x -\frac {i}{a}\right )}}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 120, normalized size = 1.74 \[ -\frac {x}{15 \, \sqrt {a^{2} c x^{2} + c} c} - \frac {4 i}{5 \, \sqrt {a^{2} c x^{2} + c} a^{3} c x^{2} - 10 i \, \sqrt {a^{2} c x^{2} + c} a^{2} c x - 5 \, \sqrt {a^{2} c x^{2} + c} a c} - \frac {8 i}{15 i \, \sqrt {a^{2} c x^{2} + c} a^{2} c x + 15 \, \sqrt {a^{2} c x^{2} + c} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 45, normalized size = 0.65 \[ \frac {\sqrt {c\,\left (a^2\,x^2+1\right )}\,\left (a^2\,x^2-a\,x\,3{}\mathrm {i}+4\right )\,1{}\mathrm {i}}{15\,a\,c^2\,{\left (1+a\,x\,1{}\mathrm {i}\right )}^3} \]
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 \[ \int \frac {\left (a^{2} x^{2} + 1\right )^{2}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a x - i\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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