Optimal. Leaf size=171 \[ -\frac{8 a^3 (21 A-23 i B) \tan ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{8 \sqrt [4]{-1} a^3 (B+i A) \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )}{d}-\frac{2 (7 A-11 i B) \tan ^{\frac{3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}+\frac{8 a^3 (B+i A) \sqrt{\tan (c+d x)}}{d}+\frac{2 i a B \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2}{7 d} \]
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Rubi [A] time = 0.421445, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.139, Rules used = {3594, 3592, 3528, 3533, 205} \[ -\frac{8 a^3 (21 A-23 i B) \tan ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{8 \sqrt [4]{-1} a^3 (B+i A) \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )}{d}-\frac{2 (7 A-11 i B) \tan ^{\frac{3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}+\frac{8 a^3 (B+i A) \sqrt{\tan (c+d x)}}{d}+\frac{2 i a B \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2}{7 d} \]
Antiderivative was successfully verified.
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Rule 3594
Rule 3592
Rule 3528
Rule 3533
Rule 205
Rubi steps
\begin{align*} \int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx &=\frac{2 i a B \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2}{7 d}+\frac{2}{7} \int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^2 \left (\frac{1}{2} a (7 A-3 i B)+\frac{1}{2} a (7 i A+11 B) \tan (c+d x)\right ) \, dx\\ &=\frac{2 i a B \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2}{7 d}-\frac{2 (7 A-11 i B) \tan ^{\frac{3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}+\frac{4}{35} \int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x)) \left (2 a^2 (7 A-6 i B)+a^2 (21 i A+23 B) \tan (c+d x)\right ) \, dx\\ &=-\frac{8 a^3 (21 A-23 i B) \tan ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 i a B \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2}{7 d}-\frac{2 (7 A-11 i B) \tan ^{\frac{3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}+\frac{4}{35} \int \sqrt{\tan (c+d x)} \left (35 a^3 (A-i B)+35 a^3 (i A+B) \tan (c+d x)\right ) \, dx\\ &=\frac{8 a^3 (i A+B) \sqrt{\tan (c+d x)}}{d}-\frac{8 a^3 (21 A-23 i B) \tan ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 i a B \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2}{7 d}-\frac{2 (7 A-11 i B) \tan ^{\frac{3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}+\frac{4}{35} \int \frac{-35 a^3 (i A+B)+35 a^3 (A-i B) \tan (c+d x)}{\sqrt{\tan (c+d x)}} \, dx\\ &=\frac{8 a^3 (i A+B) \sqrt{\tan (c+d x)}}{d}-\frac{8 a^3 (21 A-23 i B) \tan ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 i a B \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2}{7 d}-\frac{2 (7 A-11 i B) \tan ^{\frac{3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}+\frac{\left (280 a^6 (i A+B)^2\right ) \operatorname{Subst}\left (\int \frac{1}{-35 a^3 (i A+B)-35 a^3 (A-i B) x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=\frac{8 \sqrt [4]{-1} a^3 (i A+B) \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )}{d}+\frac{8 a^3 (i A+B) \sqrt{\tan (c+d x)}}{d}-\frac{8 a^3 (21 A-23 i B) \tan ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 i a B \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2}{7 d}-\frac{2 (7 A-11 i B) \tan ^{\frac{3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}\\ \end{align*}
Mathematica [B] time = 9.84344, size = 452, normalized size = 2.64 \[ \frac{\cos ^4(c+d x) \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \left (\sec (c) \left (-\frac{2}{35} \sin (3 c)-\frac{2}{35} i \cos (3 c)\right ) \sec ^2(c+d x) (7 A \cos (c)+5 B \sin (c)-21 i B \cos (c))+\sec (c) \left (-\frac{2}{21} \cos (3 c)+\frac{2}{21} i \sin (3 c)\right ) \sec (c+d x) (21 A \sin (d x)-31 i B \sin (d x))+\sec (c) \left (\frac{2}{105} \cos (3 c)-\frac{2}{105} i \sin (3 c)\right ) (-105 A \sin (c)+441 i A \cos (c)+155 i B \sin (c)+483 B \cos (c))-i B \sec (c) \left (\frac{2}{7} \cos (3 c)-\frac{2}{7} i \sin (3 c)\right ) \sin (d x) \sec ^3(c+d x)\right )}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}-\frac{8 i e^{-3 i c} (A-i B) \sqrt{-\frac{i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \cos ^4(c+d x) \tanh ^{-1}\left (\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right ) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{d \sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.014, size = 574, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.10206, size = 289, normalized size = 1.69 \begin{align*} -\frac{30 i \, B a^{3} \tan \left (d x + c\right )^{\frac{7}{2}} + 42 \,{\left (i \, A + 3 \, B\right )} a^{3} \tan \left (d x + c\right )^{\frac{5}{2}} + 2 \,{\left (105 \, A - 140 i \, B\right )} a^{3} \tan \left (d x + c\right )^{\frac{3}{2}} + 840 \,{\left (-i \, A - B\right )} a^{3} \sqrt{\tan \left (d x + c\right )} - 105 \,{\left (\sqrt{2}{\left (-\left (2 i - 2\right ) \, A - \left (2 i + 2\right ) \, B\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right ) + \sqrt{2}{\left (-\left (2 i - 2\right ) \, A - \left (2 i + 2\right ) \, B\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right ) + \sqrt{2}{\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt{2}{\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (-\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )} a^{3}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.08939, size = 1381, normalized size = 8.08 \begin{align*} -\frac{105 \, \sqrt{\frac{{\left (64 i \, A^{2} + 128 \, A B - 64 i \, B^{2}\right )} a^{6}}{d^{2}}}{\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac{{\left (8 \,{\left (A - i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{\frac{{\left (64 i \, A^{2} + 128 \, A B - 64 i \, B^{2}\right )} a^{6}}{d^{2}}}{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt{\frac{-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (4 i \, A + 4 \, B\right )} a^{3}}\right ) - 105 \, \sqrt{\frac{{\left (64 i \, A^{2} + 128 \, A B - 64 i \, B^{2}\right )} a^{6}}{d^{2}}}{\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac{{\left (8 \,{\left (A - i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt{\frac{{\left (64 i \, A^{2} + 128 \, A B - 64 i \, B^{2}\right )} a^{6}}{d^{2}}}{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt{\frac{-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (4 i \, A + 4 \, B\right )} a^{3}}\right ) -{\left ({\left (4368 i \, A + 5104 \, B\right )} a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} +{\left (10752 i \, A + 10336 \, B\right )} a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (9072 i \, A + 8816 \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (2688 i \, A + 2624 \, B\right )} a^{3}\right )} \sqrt{\frac{-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{420 \,{\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27522, size = 217, normalized size = 1.27 \begin{align*} -\frac{\left (i - 1\right ) \, \sqrt{2}{\left (16 \, A a^{3} - 16 i \, B a^{3}\right )} \arctan \left (-\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\tan \left (d x + c\right )}\right )}{4 \, d} - \frac{30 i \, B a^{3} d^{6} \tan \left (d x + c\right )^{\frac{7}{2}} + 42 i \, A a^{3} d^{6} \tan \left (d x + c\right )^{\frac{5}{2}} + 126 \, B a^{3} d^{6} \tan \left (d x + c\right )^{\frac{5}{2}} + 210 \, A a^{3} d^{6} \tan \left (d x + c\right )^{\frac{3}{2}} - 280 i \, B a^{3} d^{6} \tan \left (d x + c\right )^{\frac{3}{2}} - 840 i \, A a^{3} d^{6} \sqrt{\tan \left (d x + c\right )} - 840 \, B a^{3} d^{6} \sqrt{\tan \left (d x + c\right )}}{105 \, d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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