Optimal. Leaf size=159 \[ \frac{(-5 B+3 i A) (a+i a \tan (c+d x))^{3/2}}{3 a^2 d}+\frac{(-B+i A) \tan ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}-\frac{4 (-B+i A) \sqrt{a+i a \tan (c+d x)}}{a d}+\frac{(B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{2} \sqrt{a} d} \]
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Rubi [A] time = 0.334669, antiderivative size = 159, 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, 3592, 3527, 3480, 206} \[ \frac{(-5 B+3 i A) (a+i a \tan (c+d x))^{3/2}}{3 a^2 d}+\frac{(-B+i A) \tan ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}-\frac{4 (-B+i A) \sqrt{a+i a \tan (c+d x)}}{a d}+\frac{(B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{2} \sqrt{a} d} \]
Antiderivative was successfully verified.
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Rule 3595
Rule 3592
Rule 3527
Rule 3480
Rule 206
Rubi steps
\begin{align*} \int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{\sqrt{a+i a \tan (c+d x)}} \, dx &=\frac{(i A-B) \tan ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}-\frac{\int \tan (c+d x) \sqrt{a+i a \tan (c+d x)} \left (2 a (i A-B)+\frac{1}{2} a (3 A+5 i B) \tan (c+d x)\right ) \, dx}{a^2}\\ &=\frac{(i A-B) \tan ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}+\frac{(3 i A-5 B) (a+i a \tan (c+d x))^{3/2}}{3 a^2 d}-\frac{\int \sqrt{a+i a \tan (c+d x)} \left (-\frac{1}{2} a (3 A+5 i B)+2 a (i A-B) \tan (c+d x)\right ) \, dx}{a^2}\\ &=\frac{(i A-B) \tan ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}-\frac{4 (i A-B) \sqrt{a+i a \tan (c+d x)}}{a d}+\frac{(3 i A-5 B) (a+i a \tan (c+d x))^{3/2}}{3 a^2 d}-\frac{(A-i B) \int \sqrt{a+i a \tan (c+d x)} \, dx}{2 a}\\ &=\frac{(i A-B) \tan ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}-\frac{4 (i A-B) \sqrt{a+i a \tan (c+d x)}}{a d}+\frac{(3 i A-5 B) (a+i a \tan (c+d x))^{3/2}}{3 a^2 d}+\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 )}{d}\\ &=\frac{(i A+B) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{2} \sqrt{a} d}+\frac{(i A-B) \tan ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}-\frac{4 (i A-B) \sqrt{a+i a \tan (c+d x)}}{a d}+\frac{(3 i A-5 B) (a+i a \tan (c+d x))^{3/2}}{3 a^2 d}\\ \end{align*}
Mathematica [A] time = 2.33153, size = 147, normalized size = 0.92 \[ \frac{(A+B \tan (c+d x)) \left ((B+i A) \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left (e^{i (c+d x)}\right )+\frac{1}{3} \sec (c+d x) ((6 A+2 i B) \sin (2 (c+d x))+(5 B-9 i A) \cos (2 (c+d x))+9 (B-i A))\right )}{2 d \sqrt{a+i a \tan (c+d x)} (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 127, normalized size = 0.8 \begin{align*}{\frac{-2\,i}{{a}^{2}d} \left ( -{\frac{i}{3}}B \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}+iBa\sqrt{a+ia\tan \left ( dx+c \right ) }+A\sqrt{a+ia\tan \left ( dx+c \right ) }a+{\frac{{a}^{2} \left ( A+iB \right ) }{2}{\frac{1}{\sqrt{a+ia\tan \left ( dx+c \right ) }}}}-{\frac{ \left ( A-iB \right ) \sqrt{2}}{4}{a}^{{\frac{3}{2}}}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{a+ia\tan \left ( dx+c \right ) }{\frac{1}{\sqrt{a}}}} \right ) } \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 = 2.03114, size = 1089, normalized size = 6.85 \begin{align*} \frac{\sqrt{2}{\left ({\left (-30 i \, A + 14 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (-36 i \, A + 36 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - 6 i \, A + 6 \, B\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )} + 3 \,{\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} + a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt{-\frac{2 \, A^{2} - 4 i \, A B - 2 \, B^{2}}{a d^{2}}} \log \left (\frac{{\left (a d \sqrt{-\frac{2 \, A^{2} - 4 i \, A B - 2 \, B^{2}}{a 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 ) - 3 \,{\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} + a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt{-\frac{2 \, A^{2} - 4 i \, A B - 2 \, B^{2}}{a d^{2}}} \log \left (-\frac{{\left (a d \sqrt{-\frac{2 \, A^{2} - 4 i \, A B - 2 \, B^{2}}{a 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 )}{12 \,{\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} + a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )}} \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{\left (A + B \tan{\left (c + d x \right )}\right ) \tan ^{2}{\left (c + d x \right )}}{\sqrt{a \left (i \tan{\left (c + d x \right )} + 1\right )}}\, 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{{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{2}}{\sqrt{i \, a \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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