Optimal. Leaf size=712 \[ -\frac{3 f \text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac{6 c f}{d}\right )}{4 a^3 d^2}-\frac{3 f \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{2 a^3 d^2}-\frac{3 f \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{4 a^3 d^2}-\frac{3 i f \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{4 a^3 d^2}-\frac{3 i f \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \cos \left (4 e-\frac{4 c f}{d}\right )}{2 a^3 d^2}-\frac{3 i f \text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \cos \left (6 e-\frac{6 c f}{d}\right )}{4 a^3 d^2}+\frac{3 i f \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{4 a^3 d^2}+\frac{3 i f \sin \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{2 a^3 d^2}+\frac{3 i f \sin \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{4 a^3 d^2}-\frac{3 f \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{4 a^3 d^2}-\frac{3 f \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{2 a^3 d^2}-\frac{3 f \cos \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{4 a^3 d^2}-\frac{i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{15 i \sin (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac{3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}+\frac{3 i \sin (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{9 \cos (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac{3 \cos (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{1}{8 a^3 d (c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.73245, antiderivative size = 712, normalized size of antiderivative = 1., number of steps used = 60, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {3728, 3297, 3303, 3299, 3302, 3313, 12, 4406, 4428} \[ -\frac{3 f \text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac{6 c f}{d}\right )}{4 a^3 d^2}-\frac{3 f \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{2 a^3 d^2}-\frac{3 f \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{4 a^3 d^2}-\frac{3 i f \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{4 a^3 d^2}-\frac{3 i f \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \cos \left (4 e-\frac{4 c f}{d}\right )}{2 a^3 d^2}-\frac{3 i f \text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \cos \left (6 e-\frac{6 c f}{d}\right )}{4 a^3 d^2}+\frac{3 i f \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{4 a^3 d^2}+\frac{3 i f \sin \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{2 a^3 d^2}+\frac{3 i f \sin \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{4 a^3 d^2}-\frac{3 f \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{4 a^3 d^2}-\frac{3 f \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{2 a^3 d^2}-\frac{3 f \cos \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{4 a^3 d^2}-\frac{i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{15 i \sin (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac{3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}+\frac{3 i \sin (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{9 \cos (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac{3 \cos (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{1}{8 a^3 d (c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3728
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rule 3313
Rule 12
Rule 4406
Rule 4428
Rubi steps
\begin{align*} \int \frac{1}{(c+d x)^2 (a+i a \tan (e+f x))^3} \, dx &=\int \left (\frac{1}{8 a^3 (c+d x)^2}+\frac{3 \cos (2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac{3 \cos ^2(2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac{\cos ^3(2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac{3 i \sin (2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac{3 i \cos ^2(2 e+2 f x) \sin (2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac{3 \sin ^2(2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac{i \sin ^3(2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac{3 i \sin (4 e+4 f x)}{8 a^3 (c+d x)^2}-\frac{3 \sin (2 e+2 f x) \sin (4 e+4 f x)}{16 a^3 (c+d x)^2}\right ) \, dx\\ &=-\frac{1}{8 a^3 d (c+d x)}+\frac{i \int \frac{\sin ^3(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}-\frac{(3 i) \int \frac{\sin (2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}-\frac{(3 i) \int \frac{\cos ^2(2 e+2 f x) \sin (2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}-\frac{(3 i) \int \frac{\sin (4 e+4 f x)}{(c+d x)^2} \, dx}{8 a^3}+\frac{\int \frac{\cos ^3(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}-\frac{3 \int \frac{\sin (2 e+2 f x) \sin (4 e+4 f x)}{(c+d x)^2} \, dx}{16 a^3}+\frac{3 \int \frac{\cos (2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}+\frac{3 \int \frac{\cos ^2(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}-\frac{3 \int \frac{\sin ^2(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}\\ &=-\frac{1}{8 a^3 d (c+d x)}-\frac{3 \cos (2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 i \sin (2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac{(3 i) \int \left (\frac{\sin (2 e+2 f x)}{4 (c+d x)^2}+\frac{\sin (6 e+6 f x)}{4 (c+d x)^2}\right ) \, dx}{8 a^3}-\frac{3 \int \left (\frac{\cos (2 e+2 f x)}{2 (c+d x)^2}-\frac{\cos (6 e+6 f x)}{2 (c+d x)^2}\right ) \, dx}{16 a^3}-\frac{(3 i f) \int \frac{\cos (2 e+2 f x)}{c+d x} \, dx}{4 a^3 d}+\frac{(3 i f) \int \left (\frac{\cos (2 e+2 f x)}{4 (c+d x)}-\frac{\cos (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{4 a^3 d}-\frac{(3 i f) \int \frac{\cos (4 e+4 f x)}{c+d x} \, dx}{2 a^3 d}-\frac{(3 f) \int \frac{\sin (2 e+2 f x)}{c+d x} \, dx}{4 a^3 d}+\frac{(3 f) \int \left (-\frac{\sin (2 e+2 f x)}{4 (c+d x)}-\frac{\sin (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{4 a^3 d}+\frac{(3 f) \int -\frac{\sin (4 e+4 f x)}{2 (c+d x)} \, dx}{2 a^3 d}-\frac{(3 f) \int \frac{\sin (4 e+4 f x)}{2 (c+d x)} \, dx}{2 a^3 d}\\ &=-\frac{1}{8 a^3 d (c+d x)}-\frac{3 \cos (2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 i \sin (2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac{(3 i) \int \frac{\sin (2 e+2 f x)}{(c+d x)^2} \, dx}{32 a^3}-\frac{(3 i) \int \frac{\sin (6 e+6 f x)}{(c+d x)^2} \, dx}{32 a^3}-\frac{3 \int \frac{\cos (2 e+2 f x)}{(c+d x)^2} \, dx}{32 a^3}+\frac{3 \int \frac{\cos (6 e+6 f x)}{(c+d x)^2} \, dx}{32 a^3}+\frac{(3 i f) \int \frac{\cos (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}-\frac{(3 i f) \int \frac{\cos (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}-\frac{(3 f) \int \frac{\sin (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}-\frac{(3 f) \int \frac{\sin (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}-2 \frac{(3 f) \int \frac{\sin (4 e+4 f x)}{c+d x} \, dx}{4 a^3 d}-\frac{\left (3 i f \cos \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{2 a^3 d}-\frac{\left (3 i f \cos \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{4 a^3 d}-\frac{\left (3 f \cos \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{4 a^3 d}+\frac{\left (3 i f \sin \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{2 a^3 d}+\frac{\left (3 i f \sin \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{4 a^3 d}-\frac{\left (3 f \sin \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{4 a^3 d}\\ &=-\frac{1}{8 a^3 d (c+d x)}-\frac{9 \cos (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac{3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \cos (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{3 i f \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Ci}\left (\frac{2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac{3 i f \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Ci}\left (\frac{4 c f}{d}+4 f x\right )}{2 a^3 d^2}-\frac{3 f \text{Ci}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{4 a^3 d^2}+\frac{15 i \sin (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac{3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}+\frac{3 i \sin (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{3 f \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac{3 i f \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac{3 i f \sin \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{2 a^3 d^2}-\frac{(3 i f) \int \frac{\cos (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}-\frac{(9 i f) \int \frac{\cos (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}+\frac{(3 f) \int \frac{\sin (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}-\frac{(9 f) \int \frac{\sin (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}-\frac{\left (3 i f \cos \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}-\frac{\left (3 f \cos \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}+\frac{\left (3 i f \cos \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}-\frac{\left (3 f \cos \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}+\frac{\left (3 i f \sin \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}-\frac{\left (3 f \sin \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}-2 \left (\frac{\left (3 f \cos \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{4 a^3 d}+\frac{\left (3 f \sin \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{4 a^3 d}\right )-\frac{\left (3 i f \sin \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}-\frac{\left (3 f \sin \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}\\ &=-\frac{1}{8 a^3 d (c+d x)}-\frac{9 \cos (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac{3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \cos (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{9 i f \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Ci}\left (\frac{2 c f}{d}+2 f x\right )}{16 a^3 d^2}-\frac{3 i f \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Ci}\left (\frac{4 c f}{d}+4 f x\right )}{2 a^3 d^2}-\frac{3 i f \cos \left (6 e-\frac{6 c f}{d}\right ) \text{Ci}\left (\frac{6 c f}{d}+6 f x\right )}{16 a^3 d^2}-\frac{3 f \text{Ci}\left (\frac{6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac{6 c f}{d}\right )}{16 a^3 d^2}-\frac{15 f \text{Ci}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{16 a^3 d^2}+\frac{15 i \sin (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac{3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}+\frac{3 i \sin (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{15 f \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{16 a^3 d^2}+\frac{9 i f \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{16 a^3 d^2}+\frac{3 i f \sin \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{2 a^3 d^2}-2 \left (\frac{3 f \text{Ci}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{4 a^3 d^2}+\frac{3 f \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{4 a^3 d^2}\right )-\frac{3 f \cos \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (\frac{6 c f}{d}+6 f x\right )}{16 a^3 d^2}+\frac{3 i f \sin \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (\frac{6 c f}{d}+6 f x\right )}{16 a^3 d^2}-\frac{\left (9 i f \cos \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}-\frac{\left (9 f \cos \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}-\frac{\left (3 i f \cos \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}+\frac{\left (3 f \cos \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}+\frac{\left (9 i f \sin \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}-\frac{\left (9 f \sin \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}+\frac{\left (3 i f \sin \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}+\frac{\left (3 f \sin \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}\\ &=-\frac{1}{8 a^3 d (c+d x)}-\frac{9 \cos (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac{3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \cos (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{3 i f \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Ci}\left (\frac{2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac{3 i f \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Ci}\left (\frac{4 c f}{d}+4 f x\right )}{2 a^3 d^2}-\frac{3 i f \cos \left (6 e-\frac{6 c f}{d}\right ) \text{Ci}\left (\frac{6 c f}{d}+6 f x\right )}{4 a^3 d^2}-\frac{3 f \text{Ci}\left (\frac{6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac{6 c f}{d}\right )}{4 a^3 d^2}-\frac{3 f \text{Ci}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{4 a^3 d^2}+\frac{15 i \sin (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac{3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}+\frac{3 i \sin (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{3 f \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac{3 i f \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac{3 i f \sin \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{2 a^3 d^2}-2 \left (\frac{3 f \text{Ci}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{4 a^3 d^2}+\frac{3 f \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{4 a^3 d^2}\right )-\frac{3 f \cos \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (\frac{6 c f}{d}+6 f x\right )}{4 a^3 d^2}+\frac{3 i f \sin \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (\frac{6 c f}{d}+6 f x\right )}{4 a^3 d^2}\\ \end{align*}
Mathematica [A] time = 2.80523, size = 833, normalized size = 1.17 \[ \frac{\sec ^3(e+f x) \left (\sin \left (\frac{3 c f}{d}\right )-i \cos \left (\frac{3 c f}{d}\right )\right ) \left (3 d \cos \left (e+f \left (x-\frac{3 c}{d}\right )\right )+d \cos \left (3 \left (e+f \left (x-\frac{c}{d}\right )\right )\right )+d \cos \left (3 \left (e+f \left (\frac{c}{d}+x\right )\right )\right )+3 d \cos \left (e+f \left (\frac{3 c}{d}+x\right )\right )+6 i c f \cos \left (3 e-\frac{3 f (c+d x)}{d}\right ) \text{CosIntegral}\left (\frac{6 f (c+d x)}{d}\right )+6 i d f x \cos \left (3 e-\frac{3 f (c+d x)}{d}\right ) \text{CosIntegral}\left (\frac{6 f (c+d x)}{d}\right )+6 i f (c+d x) \text{CosIntegral}\left (\frac{2 f (c+d x)}{d}\right ) \left (\cos \left (e-\frac{c f}{d}+3 f x\right )+i \sin \left (e-\frac{c f}{d}+3 f x\right )\right )+3 i d \sin \left (e+f \left (x-\frac{3 c}{d}\right )\right )+i d \sin \left (3 \left (e+f \left (x-\frac{c}{d}\right )\right )\right )-i d \sin \left (3 \left (e+f \left (\frac{c}{d}+x\right )\right )\right )-3 i d \sin \left (e+f \left (\frac{3 c}{d}+x\right )\right )+6 c f \text{CosIntegral}\left (\frac{6 f (c+d x)}{d}\right ) \sin \left (3 e-\frac{3 f (c+d x)}{d}\right )+6 d f x \text{CosIntegral}\left (\frac{6 f (c+d x)}{d}\right ) \sin \left (3 e-\frac{3 f (c+d x)}{d}\right )+12 f (c+d x) \text{CosIntegral}\left (\frac{4 f (c+d x)}{d}\right ) \left (i \cos \left (e-\frac{f (c+3 d x)}{d}\right )+\sin \left (e-\frac{f (c+3 d x)}{d}\right )\right )+6 c f \cos \left (e-\frac{c f}{d}+3 f x\right ) \text{Si}\left (\frac{2 f (c+d x)}{d}\right )+6 d f x \cos \left (e-\frac{c f}{d}+3 f x\right ) \text{Si}\left (\frac{2 f (c+d x)}{d}\right )+6 i c f \sin \left (e-\frac{c f}{d}+3 f x\right ) \text{Si}\left (\frac{2 f (c+d x)}{d}\right )+6 i d f x \sin \left (e-\frac{c f}{d}+3 f x\right ) \text{Si}\left (\frac{2 f (c+d x)}{d}\right )+12 c f \cos \left (e-\frac{f (c+3 d x)}{d}\right ) \text{Si}\left (\frac{4 f (c+d x)}{d}\right )+12 d f x \cos \left (e-\frac{f (c+3 d x)}{d}\right ) \text{Si}\left (\frac{4 f (c+d x)}{d}\right )-12 i c f \sin \left (e-\frac{f (c+3 d x)}{d}\right ) \text{Si}\left (\frac{4 f (c+d x)}{d}\right )-12 i d f x \sin \left (e-\frac{f (c+3 d x)}{d}\right ) \text{Si}\left (\frac{4 f (c+d x)}{d}\right )+6 c f \cos \left (3 e-\frac{3 f (c+d x)}{d}\right ) \text{Si}\left (\frac{6 f (c+d x)}{d}\right )+6 d f x \cos \left (3 e-\frac{3 f (c+d x)}{d}\right ) \text{Si}\left (\frac{6 f (c+d x)}{d}\right )-6 i c f \sin \left (3 e-\frac{3 f (c+d x)}{d}\right ) \text{Si}\left (\frac{6 f (c+d x)}{d}\right )-6 i d f x \sin \left (3 e-\frac{3 f (c+d x)}{d}\right ) \text{Si}\left (\frac{6 f (c+d x)}{d}\right )\right )}{8 a^3 d^2 (c+d x) (\tan (e+f x)-i)^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.142, size = 787, normalized size = 1.1 \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 = 1.78887, size = 397, normalized size = 0.56 \begin{align*} -\frac{8192 \, f^{2} \cos \left (-\frac{6 \,{\left (d e - c f\right )}}{d}\right ) E_{2}\left (\frac{6 i \,{\left (f x + e\right )} d - 6 i \, d e + 6 i \, c f}{d}\right ) + 24576 \, f^{2} \cos \left (-\frac{4 \,{\left (d e - c f\right )}}{d}\right ) E_{2}\left (\frac{4 i \,{\left (f x + e\right )} d - 4 i \, d e + 4 i \, c f}{d}\right ) + 24576 \, f^{2} \cos \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) E_{2}\left (\frac{2 i \,{\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) + 24576 i \, f^{2} E_{2}\left (\frac{2 i \,{\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) \sin \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) + 24576 i \, f^{2} E_{2}\left (\frac{4 i \,{\left (f x + e\right )} d - 4 i \, d e + 4 i \, c f}{d}\right ) \sin \left (-\frac{4 \,{\left (d e - c f\right )}}{d}\right ) + 8192 i \, f^{2} E_{2}\left (\frac{6 i \,{\left (f x + e\right )} d - 6 i \, d e + 6 i \, c f}{d}\right ) \sin \left (-\frac{6 \,{\left (d e - c f\right )}}{d}\right ) + 8192 \, f^{2}}{65536 \,{\left ({\left (f x + e\right )} a^{3} d^{2} - a^{3} d^{2} e + a^{3} c d f\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62105, size = 509, normalized size = 0.71 \begin{align*} \frac{{\left ({\left ({\left (-6 i \, d f x - 6 i \, c f\right )}{\rm Ei}\left (\frac{-2 i \, d f x - 2 i \, c f}{d}\right ) e^{\left (\frac{-2 i \, d e + 2 i \, c f}{d}\right )} +{\left (-12 i \, d f x - 12 i \, c f\right )}{\rm Ei}\left (\frac{-4 i \, d f x - 4 i \, c f}{d}\right ) e^{\left (\frac{-4 i \, d e + 4 i \, c f}{d}\right )} +{\left (-6 i \, d f x - 6 i \, c f\right )}{\rm Ei}\left (\frac{-6 i \, d f x - 6 i \, c f}{d}\right ) e^{\left (\frac{-6 i \, d e + 6 i \, c f}{d}\right )} - d\right )} e^{\left (6 i \, f x + 6 i \, e\right )} - 3 \, d e^{\left (4 i \, f x + 4 i \, e\right )} - 3 \, d e^{\left (2 i \, f x + 2 i \, e\right )} - d\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{8 \,{\left (a^{3} d^{3} x + a^{3} c d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.35632, size = 2302, normalized size = 3.23 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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