Optimal. Leaf size=168 \[ \frac{(A-i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\frac{4 a^4 (B+i A) \tan (c+d x)}{d}-\frac{8 a^4 (A-i B) \log (\cos (c+d x))}{d}-8 a^4 x (B+i A)+\frac{a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac{A (a+i a \tan (c+d x))^4}{4 d}-\frac{i B (a+i a \tan (c+d x))^5}{5 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.157747, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {3592, 3527, 3478, 3477, 3475} \[ \frac{(A-i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\frac{4 a^4 (B+i A) \tan (c+d x)}{d}-\frac{8 a^4 (A-i B) \log (\cos (c+d x))}{d}-8 a^4 x (B+i A)+\frac{a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac{A (a+i a \tan (c+d x))^4}{4 d}-\frac{i B (a+i a \tan (c+d x))^5}{5 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3592
Rule 3527
Rule 3478
Rule 3477
Rule 3475
Rubi steps
\begin{align*} \int \tan (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=-\frac{i B (a+i a \tan (c+d x))^5}{5 a d}+\int (a+i a \tan (c+d x))^4 (-B+A \tan (c+d x)) \, dx\\ &=\frac{A (a+i a \tan (c+d x))^4}{4 d}-\frac{i B (a+i a \tan (c+d x))^5}{5 a d}-(i A+B) \int (a+i a \tan (c+d x))^4 \, dx\\ &=\frac{a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac{A (a+i a \tan (c+d x))^4}{4 d}-\frac{i B (a+i a \tan (c+d x))^5}{5 a d}-(2 a (i A+B)) \int (a+i a \tan (c+d x))^3 \, dx\\ &=\frac{a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac{A (a+i a \tan (c+d x))^4}{4 d}-\frac{i B (a+i a \tan (c+d x))^5}{5 a d}+\frac{(A-i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\left (4 a^2 (i A+B)\right ) \int (a+i a \tan (c+d x))^2 \, dx\\ &=-8 a^4 (i A+B) x+\frac{4 a^4 (i A+B) \tan (c+d x)}{d}+\frac{a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac{A (a+i a \tan (c+d x))^4}{4 d}-\frac{i B (a+i a \tan (c+d x))^5}{5 a d}+\frac{(A-i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\left (8 a^4 (A-i B)\right ) \int \tan (c+d x) \, dx\\ &=-8 a^4 (i A+B) x-\frac{8 a^4 (A-i B) \log (\cos (c+d x))}{d}+\frac{4 a^4 (i A+B) \tan (c+d x)}{d}+\frac{a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac{A (a+i a \tan (c+d x))^4}{4 d}-\frac{i B (a+i a \tan (c+d x))^5}{5 a d}+\frac{(A-i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}\\ \end{align*}
Mathematica [B] time = 4.52209, size = 589, normalized size = 3.51 \[ \frac{a^4 \sec (c) \sec ^5(c+d x) \left (-15 i \cos (d x) \left (-10 i (A-i B) \log \left (\cos ^2(c+d x)\right )+20 A d x-11 i A-20 i B d x-14 B\right )-15 i \cos (2 c+d x) \left (-10 i (A-i B) \log \left (\cos ^2(c+d x)\right )+20 A d x-11 i A-20 i B d x-14 B\right )-300 i A \sin (2 c+d x)+260 i A \sin (2 c+3 d x)-90 i A \sin (4 c+3 d x)+70 i A \sin (4 c+5 d x)-60 A \cos (2 c+3 d x)-150 i A d x \cos (2 c+3 d x)-60 A \cos (4 c+3 d x)-150 i A d x \cos (4 c+3 d x)-30 i A d x \cos (4 c+5 d x)-30 i A d x \cos (6 c+5 d x)-75 A \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )-75 A \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )-15 A \cos (4 c+5 d x) \log \left (\cos ^2(c+d x)\right )-15 A \cos (6 c+5 d x) \log \left (\cos ^2(c+d x)\right )+400 i A \sin (d x)-345 B \sin (2 c+d x)+275 B \sin (2 c+3 d x)-120 B \sin (4 c+3 d x)+79 B \sin (4 c+5 d x)+90 i B \cos (2 c+3 d x)-150 B d x \cos (2 c+3 d x)+90 i B \cos (4 c+3 d x)-150 B d x \cos (4 c+3 d x)-30 B d x \cos (4 c+5 d x)-30 B d x \cos (6 c+5 d x)+75 i B \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )+75 i B \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )+15 i B \cos (4 c+5 d x) \log \left (\cos ^2(c+d x)\right )+15 i B \cos (6 c+5 d x) \log \left (\cos ^2(c+d x)\right )+445 B \sin (d x)\right )}{120 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 229, normalized size = 1.4 \begin{align*}{\frac{-i{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{d}}+{\frac{{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{{\frac{4\,i}{3}}{a}^{4}A \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{{a}^{4}A \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{4\,i{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{7\,{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{8\,i{a}^{4}A\tan \left ( dx+c \right ) }{d}}-{\frac{7\,{a}^{4}A \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+8\,{\frac{{a}^{4}B\tan \left ( dx+c \right ) }{d}}-{\frac{4\,i{a}^{4}B\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}+4\,{\frac{{a}^{4}A\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}-{\frac{8\,i{a}^{4}A\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}}-8\,{\frac{{a}^{4}B\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.20945, size = 182, normalized size = 1.08 \begin{align*} \frac{12 \, B a^{4} \tan \left (d x + c\right )^{5} +{\left (15 \, A - 60 i \, B\right )} a^{4} \tan \left (d x + c\right )^{4} - 20 \,{\left (4 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )^{3} -{\left (210 \, A - 240 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} - 480 \,{\left (d x + c\right )}{\left (i \, A + B\right )} a^{4} + 60 \,{\left (4 \, A - 4 i \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 480 \,{\left (-i \, A - B\right )} a^{4} \tan \left (d x + c\right )}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.76039, size = 790, normalized size = 4.7 \begin{align*} -\frac{4 \,{\left (30 \,{\left (5 \, A - 7 i \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 15 \,{\left (31 \, A - 37 i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 5 \,{\left (113 \, A - 131 i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \,{\left (64 \, A - 73 i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (70 \, A - 79 i \, B\right )} a^{4} + 30 \,{\left ({\left (A - i \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \,{\left (A - i \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \,{\left (A - i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \,{\left (A - i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \,{\left (A - i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (A - i \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{15 \,{\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, 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.
[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 [B] time = 1.62291, size = 680, normalized size = 4.05 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]