Optimal. Leaf size=449 \[ -\frac{3 i \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{i \text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}+\frac{3 i \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{3 \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}+\frac{3 \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \cos \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{\text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \cos \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}+\frac{3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 \sin \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{\sin \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}+\frac{3 i \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{i \cos \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.73156, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 53, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3728, 3303, 3299, 3302, 3312, 4406, 4428} \[ -\frac{3 i \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{i \text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}+\frac{3 i \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{3 \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}+\frac{3 \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \cos \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{\text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \cos \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}+\frac{3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 \sin \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{\sin \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}+\frac{3 i \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{i \cos \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3728
Rule 3303
Rule 3299
Rule 3302
Rule 3312
Rule 4406
Rule 4428
Rubi steps
\begin{align*} \int \frac{1}{(c+d x) (a+i a \cot (e+f x))^3} \, dx &=\int \left (\frac{1}{8 a^3 (c+d x)}-\frac{3 \cos (2 e+2 f x)}{8 a^3 (c+d x)}+\frac{3 \cos ^2(2 e+2 f x)}{8 a^3 (c+d x)}-\frac{\cos ^3(2 e+2 f x)}{8 a^3 (c+d x)}-\frac{3 i \sin (2 e+2 f x)}{8 a^3 (c+d x)}-\frac{3 i \cos ^2(2 e+2 f x) \sin (2 e+2 f x)}{8 a^3 (c+d x)}-\frac{3 \sin ^2(2 e+2 f x)}{8 a^3 (c+d x)}+\frac{i \sin ^3(2 e+2 f x)}{8 a^3 (c+d x)}+\frac{3 i \sin (4 e+4 f x)}{8 a^3 (c+d x)}+\frac{3 \sin (2 e+2 f x) \sin (4 e+4 f x)}{16 a^3 (c+d x)}\right ) \, dx\\ &=\frac{\log (c+d x)}{8 a^3 d}+\frac{i \int \frac{\sin ^3(2 e+2 f x)}{c+d x} \, dx}{8 a^3}-\frac{(3 i) \int \frac{\sin (2 e+2 f x)}{c+d x} \, 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} \, dx}{8 a^3}+\frac{(3 i) \int \frac{\sin (4 e+4 f x)}{c+d x} \, dx}{8 a^3}-\frac{\int \frac{\cos ^3(2 e+2 f x)}{c+d x} \, dx}{8 a^3}+\frac{3 \int \frac{\sin (2 e+2 f x) \sin (4 e+4 f x)}{c+d x} \, dx}{16 a^3}-\frac{3 \int \frac{\cos (2 e+2 f x)}{c+d x} \, dx}{8 a^3}+\frac{3 \int \frac{\cos ^2(2 e+2 f x)}{c+d x} \, dx}{8 a^3}-\frac{3 \int \frac{\sin ^2(2 e+2 f x)}{c+d x} \, dx}{8 a^3}\\ &=\frac{\log (c+d x)}{8 a^3 d}+\frac{i \int \left (\frac{3 \sin (2 e+2 f x)}{4 (c+d x)}-\frac{\sin (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{8 a^3}-\frac{(3 i) \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}{8 a^3}-\frac{\int \left (\frac{3 \cos (2 e+2 f x)}{4 (c+d x)}+\frac{\cos (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{8 a^3}+\frac{3 \int \left (\frac{\cos (2 e+2 f x)}{2 (c+d x)}-\frac{\cos (6 e+6 f x)}{2 (c+d x)}\right ) \, dx}{16 a^3}-\frac{3 \int \left (\frac{1}{2 (c+d x)}-\frac{\cos (4 e+4 f x)}{2 (c+d x)}\right ) \, dx}{8 a^3}+\frac{3 \int \left (\frac{1}{2 (c+d x)}+\frac{\cos (4 e+4 f x)}{2 (c+d x)}\right ) \, dx}{8 a^3}+\frac{\left (3 i \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}{8 a^3}-\frac{\left (3 i \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}{8 a^3}-\frac{\left (3 \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}{8 a^3}+\frac{\left (3 i \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}{8 a^3}-\frac{\left (3 i \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}{8 a^3}+\frac{\left (3 \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}{8 a^3}\\ &=-\frac{3 \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Ci}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d}+\frac{3 i \text{Ci}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{3 i \text{Ci}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac{3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac{3 i \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{8 a^3 d}-\frac{i \int \frac{\sin (6 e+6 f x)}{c+d x} \, dx}{32 a^3}-\frac{(3 i) \int \frac{\sin (6 e+6 f x)}{c+d x} \, dx}{32 a^3}-\frac{\int \frac{\cos (6 e+6 f x)}{c+d x} \, dx}{32 a^3}-\frac{3 \int \frac{\cos (6 e+6 f x)}{c+d x} \, dx}{32 a^3}+2 \frac{3 \int \frac{\cos (4 e+4 f x)}{c+d x} \, dx}{16 a^3}\\ &=-\frac{3 \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Ci}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d}+\frac{3 i \text{Ci}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{3 i \text{Ci}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac{3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac{3 i \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{8 a^3 d}-\frac{\left (i \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}{32 a^3}-\frac{\left (3 i \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}{32 a^3}-\frac{\cos \left (6 e-\frac{6 c f}{d}\right ) \int \frac{\cos \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}-\frac{\left (3 \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}{32 a^3}-\frac{\left (i \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}{32 a^3}-\frac{\left (3 i \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}{32 a^3}+\frac{\sin \left (6 e-\frac{6 c f}{d}\right ) \int \frac{\sin \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}+\frac{\left (3 \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}{32 a^3}+2 \left (\frac{\left (3 \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}{16 a^3}-\frac{\left (3 \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}{16 a^3}\right )\\ &=-\frac{3 \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Ci}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac{\cos \left (6 e-\frac{6 c f}{d}\right ) \text{Ci}\left (\frac{6 c f}{d}+6 f x\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d}-\frac{i \text{Ci}\left (\frac{6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}+\frac{3 i \text{Ci}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{3 i \text{Ci}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac{3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac{3 i \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{8 a^3 d}+2 \left (\frac{3 \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Ci}\left (\frac{4 c f}{d}+4 f x\right )}{16 a^3 d}-\frac{3 \sin \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{16 a^3 d}\right )-\frac{i \cos \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (\frac{6 c f}{d}+6 f x\right )}{8 a^3 d}+\frac{\sin \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (\frac{6 c f}{d}+6 f x\right )}{8 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.527879, size = 197, normalized size = 0.44 \[ \frac{-3 \left (\text{CosIntegral}\left (\frac{2 f (c+d x)}{d}\right )+i \text{Si}\left (\frac{2 f (c+d x)}{d}\right )\right ) \left (\cos \left (2 e-\frac{2 c f}{d}\right )+i \sin \left (2 e-\frac{2 c f}{d}\right )\right )+3 \left (\text{CosIntegral}\left (\frac{4 f (c+d x)}{d}\right )+i \text{Si}\left (\frac{4 f (c+d x)}{d}\right )\right ) \left (\cos \left (4 e-\frac{4 c f}{d}\right )+i \sin \left (4 e-\frac{4 c f}{d}\right )\right )-\left (\text{CosIntegral}\left (\frac{6 f (c+d x)}{d}\right )+i \text{Si}\left (\frac{6 f (c+d x)}{d}\right )\right ) \left (\cos \left (6 e-\frac{6 c f}{d}\right )+i \sin \left (6 e-\frac{6 c f}{d}\right )\right )+\log (c+d x)}{8 a^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.08, size = 560, normalized size = 1.3 \begin{align*}{\frac{-{\frac{i}{8}}}{{a}^{3}d}{\it Si} \left ( 6\,fx+6\,e+6\,{\frac{cf-de}{d}} \right ) \cos \left ( 6\,{\frac{cf-de}{d}} \right ) }+{\frac{{\frac{i}{8}}}{{a}^{3}d}{\it Ci} \left ( 6\,fx+6\,e+6\,{\frac{cf-de}{d}} \right ) \sin \left ( 6\,{\frac{cf-de}{d}} \right ) }-{\frac{{\frac{3\,i}{8}}}{{a}^{3}d}{\it Si} \left ( 2\,fx+2\,e+2\,{\frac{cf-de}{d}} \right ) \cos \left ( 2\,{\frac{cf-de}{d}} \right ) }+{\frac{{\frac{3\,i}{8}}}{{a}^{3}d}{\it Ci} \left ( 2\,fx+2\,e+2\,{\frac{cf-de}{d}} \right ) \sin \left ( 2\,{\frac{cf-de}{d}} \right ) }+{\frac{3}{8\,{a}^{3}d}{\it Si} \left ( 4\,fx+4\,e+4\,{\frac{cf-de}{d}} \right ) \sin \left ( 4\,{\frac{cf-de}{d}} \right ) }+{\frac{3}{8\,{a}^{3}d}{\it Ci} \left ( 4\,fx+4\,e+4\,{\frac{cf-de}{d}} \right ) \cos \left ( 4\,{\frac{cf-de}{d}} \right ) }+{\frac{\ln \left ( \left ( fx+e \right ) d+cf-de \right ) }{8\,{a}^{3}d}}-{\frac{3}{8\,{a}^{3}d}{\it Si} \left ( 2\,fx+2\,e+2\,{\frac{cf-de}{d}} \right ) \sin \left ( 2\,{\frac{cf-de}{d}} \right ) }-{\frac{3}{8\,{a}^{3}d}{\it Ci} \left ( 2\,fx+2\,e+2\,{\frac{cf-de}{d}} \right ) \cos \left ( 2\,{\frac{cf-de}{d}} \right ) }-{\frac{1}{8\,{a}^{3}d}{\it Si} \left ( 6\,fx+6\,e+6\,{\frac{cf-de}{d}} \right ) \sin \left ( 6\,{\frac{cf-de}{d}} \right ) }-{\frac{1}{8\,{a}^{3}d}{\it Ci} \left ( 6\,fx+6\,e+6\,{\frac{cf-de}{d}} \right ) \cos \left ( 6\,{\frac{cf-de}{d}} \right ) }+{\frac{{\frac{3\,i}{8}}}{{a}^{3}d}{\it Si} \left ( 4\,fx+4\,e+4\,{\frac{cf-de}{d}} \right ) \cos \left ( 4\,{\frac{cf-de}{d}} \right ) }-{\frac{{\frac{3\,i}{8}}}{{a}^{3}d}{\it Ci} \left ( 4\,fx+4\,e+4\,{\frac{cf-de}{d}} \right ) \sin \left ( 4\,{\frac{cf-de}{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.59168, size = 371, normalized size = 0.83 \begin{align*} \frac{f \cos \left (-\frac{6 \,{\left (d e - c f\right )}}{d}\right ) E_{1}\left (-\frac{6 i \,{\left (f x + e\right )} d - 6 i \, d e + 6 i \, c f}{d}\right ) - 3 \, f \cos \left (-\frac{4 \,{\left (d e - c f\right )}}{d}\right ) E_{1}\left (-\frac{4 i \,{\left (f x + e\right )} d - 4 i \, d e + 4 i \, c f}{d}\right ) + 3 \, f \cos \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) E_{1}\left (-\frac{2 i \,{\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) - 3 i \, f E_{1}\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 ) + 3 i \, f E_{1}\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 ) - i \, f E_{1}\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 ) + f \log \left ({\left (f x + e\right )} d - d e + c f\right )}{8 \, a^{3} d f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.69982, size = 277, normalized size = 0.62 \begin{align*} -\frac{{\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 )} - 3 \,{\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 )} + 3 \,{\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 )} - \log \left (\frac{d x + c}{d}\right )}{8 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.39601, size = 2547, normalized size = 5.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]