Optimal. Leaf size=133 \[ -\frac{i}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}+\frac{i}{2 a d (a+i a \tan (c+d x))^{3/2}}-\frac{i}{5 d (a+i a \tan (c+d x))^{5/2}} \]
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Rubi [A] time = 0.162084, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {3540, 3526, 3479, 3480, 206} \[ -\frac{i}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}+\frac{i}{2 a d (a+i a \tan (c+d x))^{3/2}}-\frac{i}{5 d (a+i a \tan (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3540
Rule 3526
Rule 3479
Rule 3480
Rule 206
Rubi steps
\begin{align*} \int \frac{\tan ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=-\frac{i}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{\int \frac{a-2 i a \tan (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx}{2 a^2}\\ &=-\frac{i}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{i}{2 a d (a+i a \tan (c+d x))^{3/2}}-\frac{\int \frac{1}{\sqrt{a+i a \tan (c+d x)}} \, dx}{4 a^2}\\ &=-\frac{i}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{i}{2 a d (a+i a \tan (c+d x))^{3/2}}-\frac{i}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{\int \sqrt{a+i a \tan (c+d x)} \, dx}{8 a^3}\\ &=-\frac{i}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{i}{2 a d (a+i a \tan (c+d x))^{3/2}}-\frac{i}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{i \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+i a \tan (c+d x)}\right )}{4 a^2 d}\\ &=\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}-\frac{i}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{i}{2 a d (a+i a \tan (c+d x))^{3/2}}-\frac{i}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.03433, size = 135, normalized size = 1.02 \[ -\frac{i e^{-6 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{3/2} \sec ^2(c+d x) \left (\sqrt{1+e^{2 i (c+d x)}} \left (-3 e^{2 i (c+d x)}+e^{4 i (c+d x)}+1\right )-5 e^{5 i (c+d x)} \sinh ^{-1}\left (e^{i (c+d x)}\right )\right )}{80 a^2 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 94, normalized size = 0.7 \begin{align*}{\frac{-2\,i}{ad} \left ( -{\frac{\sqrt{2}}{16}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{a+ia\tan \left ( dx+c \right ) }{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}}-{\frac{1}{4} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{1}{8\,a}{\frac{1}{\sqrt{a+ia\tan \left ( dx+c \right ) }}}}+{\frac{a}{10} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.15437, size = 849, normalized size = 6.38 \begin{align*} \frac{{\left (5 i \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left ({\left (2 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 5 i \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-{\left (2 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{\frac{1}{a^{5} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 2 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - i\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{40 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{2}{\left (c + d x \right )}}{\left (a \left (i \tan{\left (c + d x \right )} + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (d x + c\right )^{2}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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