Optimal. Leaf size=151 \[ \frac {\left (a^2 A-2 a b B-A b^2\right ) \cot ^2(c+d x)}{2 d}+\frac {\left (a^2 A-2 a b B-A b^2\right ) \log (\sin (c+d x))}{d}+x \left (a^2 B+2 a A b-b^2 B\right )-\frac {a^2 A \cot ^4(c+d x)}{4 d}-\frac {\left (b^2 B-a (a B+2 A b)\right ) \cot (c+d x)}{d}-\frac {a (a B+2 A b) \cot ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.30, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3604, 3628, 3529, 3531, 3475} \[ \frac {\left (a^2 A-2 a b B-A b^2\right ) \cot ^2(c+d x)}{2 d}+\frac {\left (a^2 A-2 a b B-A b^2\right ) \log (\sin (c+d x))}{d}+x \left (a^2 B+2 a A b-b^2 B\right )-\frac {a^2 A \cot ^4(c+d x)}{4 d}-\frac {\left (b^2 B-a (a B+2 A b)\right ) \cot (c+d x)}{d}-\frac {a (a B+2 A b) \cot ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3529
Rule 3531
Rule 3604
Rule 3628
Rubi steps
\begin {align*} \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx &=-\frac {a^2 A \cot ^4(c+d x)}{4 d}+\int \cot ^4(c+d x) \left (a (2 A b+a B)-\left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+b^2 B \tan ^2(c+d x)\right ) \, dx\\ &=-\frac {a (2 A b+a B) \cot ^3(c+d x)}{3 d}-\frac {a^2 A \cot ^4(c+d x)}{4 d}+\int \cot ^3(c+d x) \left (-a^2 A+A b^2+2 a b B+\left (b^2 B-a (2 A b+a B)\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {\left (a^2 A-A b^2-2 a b B\right ) \cot ^2(c+d x)}{2 d}-\frac {a (2 A b+a B) \cot ^3(c+d x)}{3 d}-\frac {a^2 A \cot ^4(c+d x)}{4 d}+\int \cot ^2(c+d x) \left (b^2 B-a (2 A b+a B)+\left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)\right ) \, dx\\ &=-\frac {\left (b^2 B-a (2 A b+a B)\right ) \cot (c+d x)}{d}+\frac {\left (a^2 A-A b^2-2 a b B\right ) \cot ^2(c+d x)}{2 d}-\frac {a (2 A b+a B) \cot ^3(c+d x)}{3 d}-\frac {a^2 A \cot ^4(c+d x)}{4 d}+\int \cot (c+d x) \left (a^2 A-A b^2-2 a b B+\left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)\right ) \, dx\\ &=\left (2 a A b+a^2 B-b^2 B\right ) x-\frac {\left (b^2 B-a (2 A b+a B)\right ) \cot (c+d x)}{d}+\frac {\left (a^2 A-A b^2-2 a b B\right ) \cot ^2(c+d x)}{2 d}-\frac {a (2 A b+a B) \cot ^3(c+d x)}{3 d}-\frac {a^2 A \cot ^4(c+d x)}{4 d}+\left (a^2 A-A b^2-2 a b B\right ) \int \cot (c+d x) \, dx\\ &=\left (2 a A b+a^2 B-b^2 B\right ) x-\frac {\left (b^2 B-a (2 A b+a B)\right ) \cot (c+d x)}{d}+\frac {\left (a^2 A-A b^2-2 a b B\right ) \cot ^2(c+d x)}{2 d}-\frac {a (2 A b+a B) \cot ^3(c+d x)}{3 d}-\frac {a^2 A \cot ^4(c+d x)}{4 d}+\frac {\left (a^2 A-A b^2-2 a b B\right ) \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [C] time = 2.92, size = 180, normalized size = 1.19 \[ \frac {6 \left (a^2 A-2 a b B-A b^2\right ) \cot ^2(c+d x)+12 \left (a^2 B+2 a A b-b^2 B\right ) \cot (c+d x)-6 \left (\left (-2 a^2 A+4 a b B+2 A b^2\right ) \log (\tan (c+d x))+(a-i b)^2 (A-i B) \log (\tan (c+d x)+i)+(a+i b)^2 (A+i B) \log (-\tan (c+d x)+i)\right )-3 a^2 A \cot ^4(c+d x)-4 a (a B+2 A b) \cot ^3(c+d x)}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 191, normalized size = 1.26 \[ \frac {6 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{4} + 3 \, {\left (3 \, A a^{2} - 4 \, B a b - 2 \, A b^{2} + 4 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} d x\right )} \tan \left (d x + c\right )^{4} + 12 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \tan \left (d x + c\right )^{3} - 3 \, A a^{2} + 6 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \tan \left (d x + c\right )^{2} - 4 \, {\left (B a^{2} + 2 \, A a b\right )} \tan \left (d x + c\right )}{12 \, d \tan \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.88, size = 435, normalized size = 2.88 \[ -\frac {3 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 48 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 240 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 192 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} {\left (d x + c\right )} + 192 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 192 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {400 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 800 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 400 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 240 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 96 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 48 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 16 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A a^{2}}{\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 = 0.42, size = 238, normalized size = 1.58 \[ -\frac {a^{2} A \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a^{2} A \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a^{2} A \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {a^{2} B \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {B \cot \left (d x +c \right ) a^{2}}{d}+a^{2} B x +\frac {B \,a^{2} c}{d}-\frac {2 A a b \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {2 A \cot \left (d x +c \right ) a b}{d}+2 A x a b +\frac {2 A a b c}{d}-\frac {B a b \left (\cot ^{2}\left (d x +c \right )\right )}{d}-\frac {2 B a b \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {A \,b^{2} \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {A \,b^{2} \ln \left (\sin \left (d x +c \right )\right )}{d}-B x \,b^{2}-\frac {B \cot \left (d x +c \right ) b^{2}}{d}-\frac {B \,b^{2} c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 175, normalized size = 1.16 \[ \frac {12 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} {\left (d x + c\right )} - 6 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {12 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \tan \left (d x + c\right )^{3} - 3 \, A a^{2} + 6 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \tan \left (d x + c\right )^{2} - 4 \, {\left (B a^{2} + 2 \, A a b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.34, size = 182, normalized size = 1.21 \[ -\frac {{\mathrm {cot}\left (c+d\,x\right )}^4\,\left (\frac {A\,a^2}{4}+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-\frac {A\,a^2}{2}+B\,a\,b+\frac {A\,b^2}{2}\right )-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (B\,a^2+2\,A\,a\,b-B\,b^2\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^2}{3}+\frac {2\,A\,b\,a}{3}\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-A\,a^2+2\,B\,a\,b+A\,b^2\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^2}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^2}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.66, size = 313, normalized size = 2.07 \[ \begin {cases} \tilde {\infty } A a^{2} x & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (A + B \tan {\relax (c )}\right ) \left (a + b \tan {\relax (c )}\right )^{2} \cot ^{5}{\relax (c )} & \text {for}\: d = 0 \\- \frac {A a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {A a^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {A a^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {A a^{2}}{4 d \tan ^{4}{\left (c + d x \right )}} + 2 A a b x + \frac {2 A a b}{d \tan {\left (c + d x \right )}} - \frac {2 A a b}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {A b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {A b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {A b^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} + B a^{2} x + \frac {B a^{2}}{d \tan {\left (c + d x \right )}} - \frac {B a^{2}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {B a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac {2 B a b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {B a b}{d \tan ^{2}{\left (c + d x \right )}} - B b^{2} x - \frac {B b^{2}}{d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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