Optimal. Leaf size=92 \[ -\frac {2 \sqrt {2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {2 a \sqrt {a+i a \tan (c+d x)}}{d}+\frac {2 (a+i a \tan (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3527, 3478, 3480, 206} \[ -\frac {2 \sqrt {2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {2 a \sqrt {a+i a \tan (c+d x)}}{d}+\frac {2 (a+i a \tan (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3478
Rule 3480
Rule 3527
Rubi steps
\begin {align*} \int \tan (c+d x) (a+i a \tan (c+d x))^{3/2} \, dx &=\frac {2 (a+i a \tan (c+d x))^{3/2}}{3 d}-i \int (a+i a \tan (c+d x))^{3/2} \, dx\\ &=\frac {2 a \sqrt {a+i a \tan (c+d x)}}{d}+\frac {2 (a+i a \tan (c+d x))^{3/2}}{3 d}-(2 i a) \int \sqrt {a+i a \tan (c+d x)} \, dx\\ &=\frac {2 a \sqrt {a+i a \tan (c+d x)}}{d}+\frac {2 (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {\left (4 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{d}\\ &=-\frac {2 \sqrt {2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {2 a \sqrt {a+i a \tan (c+d x)}}{d}+\frac {2 (a+i a \tan (c+d x))^{3/2}}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.93, size = 148, normalized size = 1.61 \[ \frac {\sqrt {2} a e^{-\frac {1}{2} i (2 c+3 d x)} \sqrt {1+e^{2 i (c+d x)}} \sqrt {\frac {a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left (\cos \left (\frac {d x}{2}\right )+i \sin \left (\frac {d x}{2}\right )\right ) \left (\sqrt {1+e^{2 i (c+d x)}} (4+i \tan (c+d x)) \sec (c+d x)-6 \sinh ^{-1}\left (e^{i (c+d x)}\right )\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 258, normalized size = 2.80 \[ -\frac {3 \, \sqrt {2} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a^{3}}{d^{2}}} \log \left (\frac {4 \, {\left (a^{2} e^{\left (i \, d x + i \, c\right )} + {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{a}\right ) - 3 \, \sqrt {2} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a^{3}}{d^{2}}} \log \left (\frac {4 \, {\left (a^{2} e^{\left (i \, d x + i \, c\right )} - {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{a}\right ) - 2 \, \sqrt {2} {\left (5 \, a e^{\left (3 i \, d x + 3 i \, c\right )} + 3 \, a e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 70, normalized size = 0.76 \[ \frac {\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 a \sqrt {a +i a \tan \left (d x +c \right )}-2 a^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 102, normalized size = 1.11 \[ \frac {3 \, \sqrt {2} a^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{2} + 6 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{3}}{3 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.06, size = 74, normalized size = 0.80 \[ \frac {2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{3\,d}+\frac {2\,a\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{d}-\frac {2\,\sqrt {2}\,a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {a}}\right )}{d} \]
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 \[ \int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}} \tan {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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