Optimal. Leaf size=183 \[ -\frac{2 a^2 (9 A-11 i B) \tan ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{4 a^2 (B+i A) \tan ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{4 a^2 (A-i B) \tan ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{4 \sqrt [4]{-1} a^2 (B+i A) \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )}{d}-\frac{4 a^2 (B+i A) \sqrt{\tan (c+d x)}}{d}+\frac{2 i B \tan ^{\frac{7}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d} \]
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Rubi [A] time = 0.3568, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{2 a^2 (9 A-11 i B) \tan ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{4 a^2 (B+i A) \tan ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{4 a^2 (A-i B) \tan ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{4 \sqrt [4]{-1} a^2 (B+i A) \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )}{d}-\frac{4 a^2 (B+i A) \sqrt{\tan (c+d x)}}{d}+\frac{2 i B \tan ^{\frac{7}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d} \]
Antiderivative was successfully verified.
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Rule 3594
Rule 3592
Rule 3528
Rule 3533
Rule 205
Rubi steps
\begin{align*} \int \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx &=\frac{2 i B \tan ^{\frac{7}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d}+\frac{2}{9} \int \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x)) \left (\frac{1}{2} a (9 A-7 i B)+\frac{1}{2} a (9 i A+11 B) \tan (c+d x)\right ) \, dx\\ &=-\frac{2 a^2 (9 A-11 i B) \tan ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 i B \tan ^{\frac{7}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d}+\frac{2}{9} \int \tan ^{\frac{5}{2}}(c+d x) \left (9 a^2 (A-i B)+9 a^2 (i A+B) \tan (c+d x)\right ) \, dx\\ &=\frac{4 a^2 (i A+B) \tan ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{2 a^2 (9 A-11 i B) \tan ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 i B \tan ^{\frac{7}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d}+\frac{2}{9} \int \tan ^{\frac{3}{2}}(c+d x) \left (-9 a^2 (i A+B)+9 a^2 (A-i B) \tan (c+d x)\right ) \, dx\\ &=\frac{4 a^2 (A-i B) \tan ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 a^2 (i A+B) \tan ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{2 a^2 (9 A-11 i B) \tan ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 i B \tan ^{\frac{7}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d}+\frac{2}{9} \int \sqrt{\tan (c+d x)} \left (-9 a^2 (A-i B)-9 a^2 (i A+B) \tan (c+d x)\right ) \, dx\\ &=-\frac{4 a^2 (i A+B) \sqrt{\tan (c+d x)}}{d}+\frac{4 a^2 (A-i B) \tan ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 a^2 (i A+B) \tan ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{2 a^2 (9 A-11 i B) \tan ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 i B \tan ^{\frac{7}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d}+\frac{2}{9} \int \frac{9 a^2 (i A+B)-9 a^2 (A-i B) \tan (c+d x)}{\sqrt{\tan (c+d x)}} \, dx\\ &=-\frac{4 a^2 (i A+B) \sqrt{\tan (c+d x)}}{d}+\frac{4 a^2 (A-i B) \tan ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 a^2 (i A+B) \tan ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{2 a^2 (9 A-11 i B) \tan ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 i B \tan ^{\frac{7}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d}+\frac{\left (36 a^4 (i A+B)^2\right ) \operatorname{Subst}\left (\int \frac{1}{9 a^2 (i A+B)+9 a^2 (A-i B) x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=-\frac{4 \sqrt [4]{-1} a^2 (i A+B) \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )}{d}-\frac{4 a^2 (i A+B) \sqrt{\tan (c+d x)}}{d}+\frac{4 a^2 (A-i B) \tan ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 a^2 (i A+B) \tan ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{2 a^2 (9 A-11 i B) \tan ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 i B \tan ^{\frac{7}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d}\\ \end{align*}
Mathematica [A] time = 6.12168, size = 315, normalized size = 1.72 \[ \frac{\cos ^3(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \left (\frac{4 e^{-2 i c} (B+i A) \sqrt{-\frac{i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left (\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )}{\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}-\frac{i (\cos (2 c)-i \sin (2 c)) \sqrt{\tan (c+d x)} \sec ^4(c+d x) (30 (8 B+11 i A) \sin (2 (c+d x))+15 (20 B+17 i A) \sin (4 (c+d x))+140 (18 A-17 i B) \cos (2 (c+d x))+(756 A-791 i B) \cos (4 (c+d x))+21 (84 A-89 i B))}{1260}\right )}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 607, normalized size = 3.3 \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 = 1.84791, size = 316, normalized size = 1.73 \begin{align*} -\frac{140 \, B a^{2} \tan \left (d x + c\right )^{\frac{9}{2}} + 4 \,{\left (45 \, A - 90 i \, B\right )} a^{2} \tan \left (d x + c\right )^{\frac{7}{2}} + 504 \,{\left (-i \, A - B\right )} a^{2} \tan \left (d x + c\right )^{\frac{5}{2}} - 4 \,{\left (210 \, A - 210 i \, B\right )} a^{2} \tan \left (d x + c\right )^{\frac{3}{2}} + 2520 \,{\left (i \, A + B\right )} a^{2} \sqrt{\tan \left (d x + c\right )} - 315 \,{\left (2 \, \sqrt{2}{\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt{2}{\left (\left (i - 1\right ) \, A + \left (i + 1\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^{2}}{630 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.65759, size = 1555, normalized size = 8.5 \begin{align*} \text{result too large to display} \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 = 1.28495, size = 263, normalized size = 1.44 \begin{align*} \frac{\left (i - 1\right ) \, \sqrt{2}{\left (8 \, A a^{2} - 8 i \, B a^{2}\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{70 \, B a^{2} d^{8} \tan \left (d x + c\right )^{\frac{9}{2}} + 90 \, A a^{2} d^{8} \tan \left (d x + c\right )^{\frac{7}{2}} - 180 i \, B a^{2} d^{8} \tan \left (d x + c\right )^{\frac{7}{2}} - 252 i \, A a^{2} d^{8} \tan \left (d x + c\right )^{\frac{5}{2}} - 252 \, B a^{2} d^{8} \tan \left (d x + c\right )^{\frac{5}{2}} - 420 \, A a^{2} d^{8} \tan \left (d x + c\right )^{\frac{3}{2}} + 420 i \, B a^{2} d^{8} \tan \left (d x + c\right )^{\frac{3}{2}} + 1260 i \, A a^{2} d^{8} \sqrt{\tan \left (d x + c\right )} + 1260 \, B a^{2} d^{8} \sqrt{\tan \left (d x + c\right )}}{315 \, d^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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