Optimal. Leaf size=68 \[ -\frac {a (B+i A) \cot (c+d x)}{d}-\frac {a (A-i B) \log (\sin (c+d x))}{d}-a x (B+i A)-\frac {a A \cot ^2(c+d x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3591, 3529, 3531, 3475} \[ -\frac {a (B+i A) \cot (c+d x)}{d}-\frac {a (A-i B) \log (\sin (c+d x))}{d}-a x (B+i A)-\frac {a A \cot ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3475
Rule 3529
Rule 3531
Rule 3591
Rubi steps
\begin {align*} \int \cot ^3(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx &=-\frac {a A \cot ^2(c+d x)}{2 d}+\int \cot ^2(c+d x) (a (i A+B)-a (A-i B) \tan (c+d x)) \, dx\\ &=-\frac {a (i A+B) \cot (c+d x)}{d}-\frac {a A \cot ^2(c+d x)}{2 d}+\int \cot (c+d x) (-a (A-i B)-a (i A+B) \tan (c+d x)) \, dx\\ &=-a (i A+B) x-\frac {a (i A+B) \cot (c+d x)}{d}-\frac {a A \cot ^2(c+d x)}{2 d}-(a (A-i B)) \int \cot (c+d x) \, dx\\ &=-a (i A+B) x-\frac {a (i A+B) \cot (c+d x)}{d}-\frac {a A \cot ^2(c+d x)}{2 d}-\frac {a (A-i B) \log (\sin (c+d x))}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.37, size = 76, normalized size = 1.12 \[ -\frac {a \left (2 (B+i A) \cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )+2 (A-i B) (\log (\tan (c+d x))+\log (\cos (c+d x)))+A \cot ^2(c+d x)\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 111, normalized size = 1.63 \[ \frac {2 \, {\left (2 \, A - i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \, {\left (A - i \, B\right )} a - {\left ({\left (A - i \, B\right )} a e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, {\left (A - i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (A - i \, B\right )} a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.79, size = 162, normalized size = 2.38 \[ -\frac {A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 i \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 16 \, {\left (A a - i \, B a\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 8 \, {\left (A a - i \, B a\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - \frac {12 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 i \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 i \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.45, size = 101, normalized size = 1.49 \[ -i A a x -\frac {i A \cot \left (d x +c \right ) a}{d}-\frac {i A a c}{d}+\frac {i B a \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {a A \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a A \ln \left (\sin \left (d x +c \right )\right )}{d}-a B x -\frac {B \cot \left (d x +c \right ) a}{d}-\frac {B a c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.66, size = 83, normalized size = 1.22 \[ \frac {2 \, {\left (d x + c\right )} {\left (-i \, A - B\right )} a + {\left (A - i \, B\right )} a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, {\left (A - i \, B\right )} a \log \left (\tan \left (d x + c\right )\right ) - \frac {{\left (2 i \, A + 2 \, B\right )} a \tan \left (d x + c\right ) + A a}{\tan \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.21, size = 60, normalized size = 0.88 \[ -\frac {\frac {A\,a}{2}+\mathrm {tan}\left (c+d\,x\right )\,\left (B\,a+A\,a\,1{}\mathrm {i}\right )}{d\,{\mathrm {tan}\left (c+d\,x\right )}^2}-\frac {a\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.80, size = 114, normalized size = 1.68 \[ - \frac {a \left (A - i B\right ) \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 2 i A a - 2 B a + \left (4 i A a e^{2 i c} + 2 B a e^{2 i c}\right ) e^{2 i d x}}{i d e^{4 i c} e^{4 i d x} - 2 i d e^{2 i c} e^{2 i d x} + i d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________