Optimal. Leaf size=118 \[ \frac {\left (a^2 A-2 a b B-A b^2\right ) \cot (c+d x)}{d}+x \left (a^2 A-2 a b B-A b^2\right )-\frac {a^2 A \cot ^3(c+d x)}{3 d}+\frac {\left (b^2 B-a (a B+2 A b)\right ) \log (\sin (c+d x))}{d}-\frac {a (a B+2 A b) \cot ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.24, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ \frac {\left (a^2 A-2 a b B-A b^2\right ) \cot (c+d x)}{d}+x \left (a^2 A-2 a b B-A b^2\right )-\frac {a^2 A \cot ^3(c+d x)}{3 d}+\frac {\left (b^2 B-a (a B+2 A b)\right ) \log (\sin (c+d x))}{d}-\frac {a (a B+2 A b) \cot ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3529
Rule 3531
Rule 3604
Rule 3628
Rubi steps
\begin {align*} \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx &=-\frac {a^2 A \cot ^3(c+d x)}{3 d}+\int \cot ^3(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 ^2(c+d x)}{2 d}-\frac {a^2 A \cot ^3(c+d x)}{3 d}+\int \cot ^2(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 (c+d x)}{d}-\frac {a (2 A b+a B) \cot ^2(c+d x)}{2 d}-\frac {a^2 A \cot ^3(c+d x)}{3 d}+\int \cot (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\\ &=\left (a^2 A-A b^2-2 a b B\right ) x+\frac {\left (a^2 A-A b^2-2 a b B\right ) \cot (c+d x)}{d}-\frac {a (2 A b+a B) \cot ^2(c+d x)}{2 d}-\frac {a^2 A \cot ^3(c+d x)}{3 d}+\left (b^2 B-a (2 A b+a B)\right ) \int \cot (c+d x) \, dx\\ &=\left (a^2 A-A b^2-2 a b B\right ) x+\frac {\left (a^2 A-A b^2-2 a b B\right ) \cot (c+d x)}{d}-\frac {a (2 A b+a B) \cot ^2(c+d x)}{2 d}-\frac {a^2 A \cot ^3(c+d x)}{3 d}+\frac {\left (b^2 B-a (2 A b+a B)\right ) \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [C] time = 1.40, size = 152, normalized size = 1.29 \[ \frac {6 \left (a^2 A-2 a b B-A b^2\right ) \cot (c+d x)-6 \left (a^2 B+2 a A b-b^2 B\right ) \log (\tan (c+d x))-2 a^2 A \cot ^3(c+d x)-3 a (a B+2 A b) \cot ^2(c+d x)+3 (a+i b)^2 (B-i A) \log (-\tan (c+d x)+i)+3 (a-i b)^2 (B+i A) \log (\tan (c+d x)+i)}{6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 157, normalized size = 1.33 \[ -\frac {3 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{3} + 3 \, {\left (B a^{2} + 2 \, A a b - 2 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 2 \, A a^{2} - 6 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} \tan \left (d x + c\right )}{6 \, d \tan \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.49, size = 334, normalized size = 2.83 \[ \frac {A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} {\left (d x + c\right )} + 24 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 24 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {44 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 88 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 44 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 188, normalized size = 1.59 \[ -\frac {a^{2} A \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a^{2} A \cot \left (d x +c \right )}{d}+a^{2} A x +\frac {A \,a^{2} c}{d}-\frac {a^{2} B \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a^{2} B \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {A a b \left (\cot ^{2}\left (d x +c \right )\right )}{d}-\frac {2 A a b \ln \left (\sin \left (d x +c \right )\right )}{d}-2 B x a b -\frac {2 B \cot \left (d x +c \right ) a b}{d}-\frac {2 B a b c}{d}-A x \,b^{2}-\frac {A \cot \left (d x +c \right ) b^{2}}{d}-\frac {A \,b^{2} c}{d}+\frac {b^{2} B \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 149, normalized size = 1.26 \[ \frac {6 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} {\left (d x + c\right )} + 3 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac {2 \, A a^{2} - 6 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.37, size = 156, normalized size = 1.32 \[ -\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left (\frac {A\,a^2}{3}+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-A\,a^2+2\,B\,a\,b+A\,b^2\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^2}{2}+A\,b\,a\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,a^2+2\,A\,a\,b-B\,b^2\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^2}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^2}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.50, size = 260, normalized size = 2.20 \[ \begin {cases} \tilde {\infty } A a^{2} x & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (A + B \tan {\relax (c )}\right ) \left (a + b \tan {\relax (c )}\right )^{2} \cot ^{4}{\relax (c )} & \text {for}\: d = 0 \\A a^{2} x + \frac {A a^{2}}{d \tan {\left (c + d x \right )}} - \frac {A a^{2}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {A a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac {2 A a b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {A a b}{d \tan ^{2}{\left (c + d x \right )}} - A b^{2} x - \frac {A b^{2}}{d \tan {\left (c + d x \right )}} + \frac {B a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {B a^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {B a^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - 2 B a b x - \frac {2 B a b}{d \tan {\left (c + d x \right )}} - \frac {B b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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