Optimal. Leaf size=73 \[ \frac {2 i \sqrt {1+i a x}}{a \sqrt {1-i a x}}-\frac {2 i (1+i a x)^{3/2}}{3 a (1-i a x)^{3/2}}+\frac {\sinh ^{-1}(a x)}{a} \]
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Rubi [A]
time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5181, 49, 41,
221} \begin {gather*} -\frac {2 i (1+i a x)^{3/2}}{3 a (1-i a x)^{3/2}}+\frac {2 i \sqrt {1+i a x}}{a \sqrt {1-i a x}}+\frac {\sinh ^{-1}(a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 49
Rule 221
Rule 5181
Rubi steps
\begin {align*} \int \frac {e^{4 i \tan ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx &=\int \frac {(1+i a x)^{3/2}}{(1-i a x)^{5/2}} \, dx\\ &=-\frac {2 i (1+i a x)^{3/2}}{3 a (1-i a x)^{3/2}}-\int \frac {\sqrt {1+i a x}}{(1-i a x)^{3/2}} \, dx\\ &=\frac {2 i \sqrt {1+i a x}}{a \sqrt {1-i a x}}-\frac {2 i (1+i a x)^{3/2}}{3 a (1-i a x)^{3/2}}+\int \frac {1}{\sqrt {1-i a x} \sqrt {1+i a x}} \, dx\\ &=\frac {2 i \sqrt {1+i a x}}{a \sqrt {1-i a x}}-\frac {2 i (1+i a x)^{3/2}}{3 a (1-i a x)^{3/2}}+\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {2 i \sqrt {1+i a x}}{a \sqrt {1-i a x}}-\frac {2 i (1+i a x)^{3/2}}{3 a (1-i a x)^{3/2}}+\frac {\sinh ^{-1}(a x)}{a}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 48, normalized size = 0.66 \begin {gather*} -\frac {4 i \sqrt {2} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {1}{2} (1-i a x)\right )}{3 a (1-i a x)^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 223 vs. \(2 (57 ) = 114\).
time = 0.09, size = 224, normalized size = 3.07
method | result | size |
meijerg | \(\frac {x \left (2 a^{2} x^{2}+3\right )}{3 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 i \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{3 a \sqrt {\pi }}-\frac {2 a^{2} x^{3}}{\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {8 i \left (\sqrt {\pi }-\frac {\sqrt {\pi }\, \left (12 a^{2} x^{2}+8\right )}{8 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{3 a \sqrt {\pi }}+\frac {-\frac {\sqrt {\pi }\, x \left (a^{2}\right )^{\frac {5}{2}} \left (20 a^{2} x^{2}+15\right )}{15 a^{4} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {\sqrt {\pi }\, \left (a^{2}\right )^{\frac {5}{2}} \arcsinh \left (a x \right )}{a^{5}}}{\sqrt {\pi }\, \sqrt {a^{2}}}\) | \(178\) |
default | \(\frac {x}{3 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {2 x}{3 \sqrt {a^{2} x^{2}+1}}+a^{4} \left (-\frac {x^{3}}{3 a^{2} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {-\frac {x}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{a^{2} \sqrt {a^{2}}}}{a^{2}}\right )-4 i a^{3} \left (-\frac {x^{2}}{a^{2} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {2}{3 a^{4} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )-6 a^{2} \left (-\frac {x}{2 a^{2} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {\frac {x}{3 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {2 x}{3 \sqrt {a^{2} x^{2}+1}}}{2 a^{2}}\right )-\frac {4 i}{3 a \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}\) | \(224\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 112 vs. \(2 (51) = 102\).
time = 0.27, size = 112, normalized size = 1.53 \begin {gather*} -\frac {1}{3} \, a^{4} x {\left (\frac {3 \, x^{2}}{{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}} + \frac {2}{{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{4}}\right )} + \frac {4 i \, a x^{2}}{{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {5 \, x}{3 \, \sqrt {a^{2} x^{2} + 1}} + \frac {\operatorname {arsinh}\left (a x\right )}{a} + \frac {7 \, x}{3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {4 i}{3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.73, size = 86, normalized size = 1.18 \begin {gather*} -\frac {8 \, a^{2} x^{2} + 16 i \, a x + 3 \, {\left (a^{2} x^{2} + 2 i \, a x - 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + 4 \, \sqrt {a^{2} x^{2} + 1} {\left (2 \, a x + i\right )} - 8}{3 \, {\left (a^{3} x^{2} + 2 i \, a^{2} x - a\right )}} \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*} \int \frac {\left (a x - i\right )^{4}}{\left (a^{2} x^{2} + 1\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 24, normalized size = 0.33 \begin {gather*} -\frac {\log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{{\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.53, size = 92, normalized size = 1.26 \begin {gather*} \frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{\sqrt {a^2}}-\frac {8\,\sqrt {a^2\,x^2+1}}{3\,\left (x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}}+\frac {a\,\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{3\,\left (a^4\,x^2+a^3\,x\,2{}\mathrm {i}-a^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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