Optimal. Leaf size=198 \[ -\frac{16 a^3 (18 A-19 i B) \tan ^{\frac{5}{2}}(c+d x)}{315 d}+\frac{8 a^3 (B+i A) \tan ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{8 \sqrt [4]{-1} a^3 (A-i B) \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )}{d}-\frac{2 (9 A-13 i B) \tan ^{\frac{5}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{63 d}+\frac{8 a^3 (A-i B) \sqrt{\tan (c+d x)}}{d}+\frac{2 i a B \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2}{9 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.467445, antiderivative size = 198, 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} \[ -\frac{16 a^3 (18 A-19 i B) \tan ^{\frac{5}{2}}(c+d x)}{315 d}+\frac{8 a^3 (B+i A) \tan ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{8 \sqrt [4]{-1} a^3 (A-i B) \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )}{d}-\frac{2 (9 A-13 i B) \tan ^{\frac{5}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{63 d}+\frac{8 a^3 (A-i B) \sqrt{\tan (c+d x)}}{d}+\frac{2 i a B \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2}{9 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3594
Rule 3592
Rule 3528
Rule 3533
Rule 205
Rubi steps
\begin{align*} \int \tan ^{\frac{3}{2}}(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{5}{2}}(c+d x) (a+i a \tan (c+d x))^2}{9 d}+\frac{2}{9} \int \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2 \left (\frac{1}{2} a (9 A-5 i B)+\frac{1}{2} a (9 i A+13 B) \tan (c+d x)\right ) \, dx\\ &=\frac{2 i a B \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2}{9 d}-\frac{2 (9 A-13 i B) \tan ^{\frac{5}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{63 d}+\frac{4}{63} \int \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x)) \left (a^2 (27 A-25 i B)+2 a^2 (18 i A+19 B) \tan (c+d x)\right ) \, dx\\ &=-\frac{16 a^3 (18 A-19 i B) \tan ^{\frac{5}{2}}(c+d x)}{315 d}+\frac{2 i a B \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2}{9 d}-\frac{2 (9 A-13 i B) \tan ^{\frac{5}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{63 d}+\frac{4}{63} \int \tan ^{\frac{3}{2}}(c+d x) \left (63 a^3 (A-i B)+63 a^3 (i A+B) \tan (c+d x)\right ) \, dx\\ &=\frac{8 a^3 (i A+B) \tan ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{16 a^3 (18 A-19 i B) \tan ^{\frac{5}{2}}(c+d x)}{315 d}+\frac{2 i a B \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2}{9 d}-\frac{2 (9 A-13 i B) \tan ^{\frac{5}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{63 d}+\frac{4}{63} \int \sqrt{\tan (c+d x)} \left (-63 a^3 (i A+B)+63 a^3 (A-i B) \tan (c+d x)\right ) \, dx\\ &=\frac{8 a^3 (A-i B) \sqrt{\tan (c+d x)}}{d}+\frac{8 a^3 (i A+B) \tan ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{16 a^3 (18 A-19 i B) \tan ^{\frac{5}{2}}(c+d x)}{315 d}+\frac{2 i a B \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2}{9 d}-\frac{2 (9 A-13 i B) \tan ^{\frac{5}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{63 d}+\frac{4}{63} \int \frac{-63 a^3 (A-i B)-63 a^3 (i A+B) \tan (c+d x)}{\sqrt{\tan (c+d x)}} \, dx\\ &=\frac{8 a^3 (A-i B) \sqrt{\tan (c+d x)}}{d}+\frac{8 a^3 (i A+B) \tan ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{16 a^3 (18 A-19 i B) \tan ^{\frac{5}{2}}(c+d x)}{315 d}+\frac{2 i a B \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2}{9 d}-\frac{2 (9 A-13 i B) \tan ^{\frac{5}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{63 d}+\frac{\left (504 a^6 (A-i B)^2\right ) \operatorname{Subst}\left (\int \frac{1}{-63 a^3 (A-i B)+63 a^3 (i A+B) x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=\frac{8 \sqrt [4]{-1} a^3 (A-i B) \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )}{d}+\frac{8 a^3 (A-i B) \sqrt{\tan (c+d x)}}{d}+\frac{8 a^3 (i A+B) \tan ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{16 a^3 (18 A-19 i B) \tan ^{\frac{5}{2}}(c+d x)}{315 d}+\frac{2 i a B \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2}{9 d}-\frac{2 (9 A-13 i B) \tan ^{\frac{5}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{63 d}\\ \end{align*}
Mathematica [B] time = 10.0018, size = 496, normalized size = 2.51 \[ \frac{\cos ^4(c+d x) \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \left (\sec (c) \left (\frac{2}{7} \cos (3 c)-\frac{2}{7} i \sin (3 c)\right ) \sec ^3(c+d x) (-3 B \sin (d x)-i A \sin (d x))+\sec (c) \left (-\frac{2}{315} \cos (3 c)+\frac{2}{315} i \sin (3 c)\right ) \sec ^2(c+d x) (45 i A \sin (c)+189 A \cos (c)+135 B \sin (c)-322 i B \cos (c))+\sec (c) \left (\frac{2}{21} \cos (3 c)-\frac{2}{21} i \sin (3 c)\right ) \sec (c+d x) (37 B \sin (d x)+31 i A \sin (d x))+\sec (c) \left (\frac{2}{315} \cos (3 c)-\frac{2}{315} i \sin (3 c)\right ) (465 i A \sin (c)+1449 A \cos (c)+555 B \sin (c)-1547 i B \cos (c))+\left (-\frac{2}{9} B \sin (3 c)-\frac{2}{9} i B \cos (3 c)\right ) \sec ^4(c+d x)\right )}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}-\frac{8 e^{-3 i c} (A-i B) \sqrt{-\frac{i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \cos ^4(c+d x) \tanh ^{-1}\left (\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right ) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{d \sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.014, size = 610, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.89566, size = 316, normalized size = 1.6 \begin{align*} -\frac{70 i \, B a^{3} \tan \left (d x + c\right )^{\frac{9}{2}} + 90 \,{\left (i \, A + 3 \, B\right )} a^{3} \tan \left (d x + c\right )^{\frac{7}{2}} + 2 \,{\left (189 \, A - 252 i \, B\right )} a^{3} \tan \left (d x + c\right )^{\frac{5}{2}} + 840 \,{\left (-i \, A - B\right )} a^{3} \tan \left (d x + c\right )^{\frac{3}{2}} - 2 \,{\left (1260 \, A - 1260 i \, B\right )} a^{3} \sqrt{\tan \left (d x + c\right )} - 315 \,{\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}}{315 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.41392, size = 1569, normalized size = 7.92 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.34052, size = 262, normalized size = 1.32 \begin{align*} \frac{\left (4 i - 4\right ) \, \sqrt{2}{\left (i \, A a^{3} + 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 )}{d} - \frac{70 i \, B a^{3} d^{8} \tan \left (d x + c\right )^{\frac{9}{2}} + 90 i \, A a^{3} d^{8} \tan \left (d x + c\right )^{\frac{7}{2}} + 270 \, B a^{3} d^{8} \tan \left (d x + c\right )^{\frac{7}{2}} + 378 \, A a^{3} d^{8} \tan \left (d x + c\right )^{\frac{5}{2}} - 504 i \, B a^{3} d^{8} \tan \left (d x + c\right )^{\frac{5}{2}} - 840 i \, A a^{3} d^{8} \tan \left (d x + c\right )^{\frac{3}{2}} - 840 \, B a^{3} d^{8} \tan \left (d x + c\right )^{\frac{3}{2}} - 2520 \, A a^{3} d^{8} \sqrt{\tan \left (d x + c\right )} + 2520 i \, B a^{3} d^{8} \sqrt{\tan \left (d x + c\right )}}{315 \, d^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]