Optimal. Leaf size=111 \[ -\frac {a (B+i A) \cot ^3(c+d x)}{3 d}+\frac {a (A-i B) \cot ^2(c+d x)}{2 d}+\frac {a (B+i A) \cot (c+d x)}{d}+\frac {a (A-i B) \log (\sin (c+d x))}{d}+a x (B+i A)-\frac {a A \cot ^4(c+d x)}{4 d} \]
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Rubi [A] time = 0.19, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3591, 3529, 3531, 3475} \[ -\frac {a (B+i A) \cot ^3(c+d x)}{3 d}+\frac {a (A-i B) \cot ^2(c+d x)}{2 d}+\frac {a (B+i A) \cot (c+d x)}{d}+\frac {a (A-i B) \log (\sin (c+d x))}{d}+a x (B+i A)-\frac {a A \cot ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3529
Rule 3531
Rule 3591
Rubi steps
\begin {align*} \int \cot ^5(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx &=-\frac {a A \cot ^4(c+d x)}{4 d}+\int \cot ^4(c+d x) (a (i A+B)-a (A-i B) \tan (c+d x)) \, dx\\ &=-\frac {a (i A+B) \cot ^3(c+d x)}{3 d}-\frac {a A \cot ^4(c+d x)}{4 d}+\int \cot ^3(c+d x) (-a (A-i B)-a (i A+B) \tan (c+d x)) \, dx\\ &=\frac {a (A-i B) \cot ^2(c+d x)}{2 d}-\frac {a (i A+B) \cot ^3(c+d x)}{3 d}-\frac {a A \cot ^4(c+d x)}{4 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-i B) \cot ^2(c+d x)}{2 d}-\frac {a (i A+B) \cot ^3(c+d x)}{3 d}-\frac {a A \cot ^4(c+d x)}{4 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-i B) \cot ^2(c+d x)}{2 d}-\frac {a (i A+B) \cot ^3(c+d x)}{3 d}-\frac {a A \cot ^4(c+d x)}{4 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-i B) \cot ^2(c+d x)}{2 d}-\frac {a (i A+B) \cot ^3(c+d x)}{3 d}-\frac {a A \cot ^4(c+d x)}{4 d}+\frac {a (A-i B) \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [C] time = 0.88, size = 96, normalized size = 0.86 \[ -\frac {a \left (4 (B+i A) \cot ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(c+d x)\right )-6 (A-i B) \cot ^2(c+d x)-12 (A-i B) (\log (\tan (c+d x))+\log (\cos (c+d x)))+3 A \cot ^4(c+d x)\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 206, normalized size = 1.86 \[ -\frac {6 \, {\left (4 \, A - 3 i \, B\right )} a e^{\left (6 i \, d x + 6 i \, c\right )} - 36 \, {\left (A - i \, B\right )} a e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\left (16 \, A - 13 i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} - 8 \, {\left (A - i \, B\right )} a - 3 \, {\left ({\left (A - i \, B\right )} a e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, {\left (A - i \, B\right )} a e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, {\left (A - i \, B\right )} a e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, {\left (A - i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (A - i \, B\right )} a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.21, size = 282, normalized size = 2.54 \[ -\frac {3 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 i \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 i \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 i \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 384 \, {\left (A a - i \, B a\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 192 \, {\left (A a - i \, B a\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {400 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 400 i \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 i \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 i \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 i \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 159, normalized size = 1.43 \[ -\frac {i A a \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {i A \cot \left (d x +c \right ) a}{d}+i A a x +\frac {i A a c}{d}-\frac {i B a \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {i B a \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {a A \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a A \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a A \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {a B \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {B \cot \left (d x +c \right ) a}{d}+a B x +\frac {B a c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 115, normalized size = 1.04 \[ \frac {12 \, {\left (d x + c\right )} {\left (i \, A + B\right )} a - 6 \, {\left (A - i \, B\right )} a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \, {\left (A - i \, B\right )} a \log \left (\tan \left (d x + c\right )\right ) + \frac {{\left (12 i \, A + 12 \, B\right )} a \tan \left (d x + c\right )^{3} + 6 \, {\left (A - i \, B\right )} a \tan \left (d x + c\right )^{2} + {\left (-4 i \, A - 4 \, B\right )} a \tan \left (d x + c\right ) - 3 \, A a}{\tan \left (d x + c\right )^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.48, size = 100, normalized size = 0.90 \[ -\frac {\left (-B\,a-A\,a\,1{}\mathrm {i}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^3+\left (-\frac {A\,a}{2}+\frac {B\,a\,1{}\mathrm {i}}{2}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^2+\left (\frac {B\,a}{3}+\frac {A\,a\,1{}\mathrm {i}}{3}\right )\,\mathrm {tan}\left (c+d\,x\right )+\frac {A\,a}{4}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^4}+\frac {a\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.77, size = 218, normalized size = 1.96 \[ \frac {a \left (A - i B\right ) \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 8 A a + 8 i B a + \left (32 A a e^{2 i c} - 26 i B a e^{2 i c}\right ) e^{2 i d x} + \left (- 36 A a e^{4 i c} + 36 i B a e^{4 i c}\right ) e^{4 i d x} + \left (24 A a e^{6 i c} - 18 i B a e^{6 i c}\right ) e^{6 i d x}}{- 3 d e^{8 i c} e^{8 i d x} + 12 d e^{6 i c} e^{6 i d x} - 18 d e^{4 i c} e^{4 i d x} + 12 d e^{2 i c} e^{2 i d x} - 3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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