Optimal. Leaf size=167 \[ -\frac{-31 B+i A}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{(B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}+\frac{(-B+i A) \tan ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{-13 B+3 i A}{30 a d (a+i a \tan (c+d x))^{3/2}} \]
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Rubi [A] time = 0.39508, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.139, Rules used = {3595, 3590, 3526, 3480, 206} \[ -\frac{-31 B+i A}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{(B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}+\frac{(-B+i A) \tan ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{-13 B+3 i A}{30 a d (a+i a \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3595
Rule 3590
Rule 3526
Rule 3480
Rule 206
Rubi steps
\begin{align*} \int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx &=\frac{(i A-B) \tan ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}-\frac{\int \frac{\tan (c+d x) \left (2 a (i A-B)-\frac{1}{2} a (A-9 i B) \tan (c+d x)\right )}{(a+i a \tan (c+d x))^{3/2}} \, dx}{5 a^2}\\ &=\frac{(i A-B) \tan ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{3 i A-13 B}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{i \int \frac{\frac{1}{2} a^2 (3 i A-13 B)-a^2 (A-9 i B) \tan (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx}{10 a^4}\\ &=\frac{(i A-B) \tan ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{3 i A-13 B}{30 a d (a+i a \tan (c+d x))^{3/2}}-\frac{i A-31 B}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{(A-i B) \int \sqrt{a+i a \tan (c+d x)} \, dx}{8 a^3}\\ &=\frac{(i A-B) \tan ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{3 i A-13 B}{30 a d (a+i a \tan (c+d x))^{3/2}}-\frac{i A-31 B}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{(i A+B) \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 A+B) \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 A-B) \tan ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{3 i A-13 B}{30 a d (a+i a \tan (c+d x))^{3/2}}-\frac{i A-31 B}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 3.15901, size = 176, normalized size = 1.05 \[ \frac{e^{-6 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{3/2} \sec ^2(c+d x) \left (\sqrt{1+e^{2 i (c+d x)}} \left (B \left (-19 e^{2 i (c+d x)}+83 e^{4 i (c+d x)}+3\right )-3 i A \left (-3 e^{2 i (c+d x)}+e^{4 i (c+d x)}+1\right )\right )+15 (B+i A) e^{5 i (c+d x)} \sinh ^{-1}\left (e^{i (c+d x)}\right )\right )}{240 a^2 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 124, normalized size = 0.7 \begin{align*}{\frac{-2\,i}{{a}^{2}d} \left ( -{ \left ( -{\frac{7\,i}{8}}B-{\frac{A}{8}} \right ){\frac{1}{\sqrt{a+ia\tan \left ( dx+c \right ) }}}}-{\frac{a \left ( 3\,A+5\,iB \right ) }{12} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{{a}^{2} \left ( A+iB \right ) }{10} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{\sqrt{2}}{2} \left ({\frac{A}{8}}-{\frac{i}{8}}B \right ){\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{a+ia\tan \left ( dx+c \right ) }{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \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 = 2.02363, size = 1092, normalized size = 6.54 \begin{align*} \frac{{\left (15 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{-\frac{A^{2} - 2 i \, A B - B^{2}}{a^{5} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (\frac{{\left (2 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{-\frac{A^{2} - 2 i \, A B - B^{2}}{a^{5} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{2}{\left ({\left (i \, A + B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, A + B\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - 15 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{-\frac{A^{2} - 2 i \, A B - B^{2}}{a^{5} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac{{\left (2 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{-\frac{A^{2} - 2 i \, A B - B^{2}}{a^{5} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt{2}{\left ({\left (i \, A + B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, A + B\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) + \sqrt{2}{\left ({\left (-3 i \, A + 83 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} +{\left (6 i \, A + 64 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (6 i \, A - 16 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i \, A + 3 \, B\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{120 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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{{\left (B \tan \left (d x + c\right ) + A\right )} \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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