Optimal. Leaf size=186 \[ \frac{b^2 \left (a^2 B+3 a A b+b^2 B\right ) \tan (c+d x)}{d}-\frac{a^2 \left (a^2 A-4 a b B-6 A b^2\right ) \log (\sin (c+d x))}{d}-x \left (4 a^3 A b-6 a^2 b^2 B+a^4 B-4 a A b^3+b^4 B\right )-\frac{b^3 (4 a B+A b) \log (\cos (c+d x))}{d}-\frac{a (2 a B+5 A b) \cot (c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{a A \cot ^2(c+d x) (a+b \tan (c+d x))^3}{2 d} \]
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Rubi [A] time = 0.505494, antiderivative size = 186, normalized size of antiderivative = 1., 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 = {3605, 3645, 3637, 3624, 3475} \[ \frac{b^2 \left (a^2 B+3 a A b+b^2 B\right ) \tan (c+d x)}{d}-\frac{a^2 \left (a^2 A-4 a b B-6 A b^2\right ) \log (\sin (c+d x))}{d}-x \left (4 a^3 A b-6 a^2 b^2 B+a^4 B-4 a A b^3+b^4 B\right )-\frac{b^3 (4 a B+A b) \log (\cos (c+d x))}{d}-\frac{a (2 a B+5 A b) \cot (c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{a A \cot ^2(c+d x) (a+b \tan (c+d x))^3}{2 d} \]
Antiderivative was successfully verified.
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Rule 3605
Rule 3645
Rule 3637
Rule 3624
Rule 3475
Rubi steps
\begin{align*} \int \cot ^3(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=-\frac{a A \cot ^2(c+d x) (a+b \tan (c+d x))^3}{2 d}+\frac{1}{2} \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \left (a (5 A b+2 a B)-2 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+b (a A+2 b B) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{a (5 A b+2 a B) \cot (c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{a A \cot ^2(c+d x) (a+b \tan (c+d x))^3}{2 d}+\frac{1}{2} \int \cot (c+d x) (a+b \tan (c+d x)) \left (-2 a \left (a^2 A-6 A b^2-4 a b B\right )-2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)+2 b \left (3 a A b+a^2 B+b^2 B\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\frac{b^2 \left (3 a A b+a^2 B+b^2 B\right ) \tan (c+d x)}{d}-\frac{a (5 A b+2 a B) \cot (c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{a A \cot ^2(c+d x) (a+b \tan (c+d x))^3}{2 d}-\frac{1}{2} \int \cot (c+d x) \left (2 a^2 \left (a^2 A-6 A b^2-4 a b B\right )+2 \left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \tan (c+d x)-2 b^3 (A b+4 a B) \tan ^2(c+d x)\right ) \, dx\\ &=-\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) x+\frac{b^2 \left (3 a A b+a^2 B+b^2 B\right ) \tan (c+d x)}{d}-\frac{a (5 A b+2 a B) \cot (c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{a A \cot ^2(c+d x) (a+b \tan (c+d x))^3}{2 d}+\left (b^3 (A b+4 a B)\right ) \int \tan (c+d x) \, dx-\left (a^2 \left (a^2 A-6 A b^2-4 a b B\right )\right ) \int \cot (c+d x) \, dx\\ &=-\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) x-\frac{b^3 (A b+4 a B) \log (\cos (c+d x))}{d}-\frac{a^2 \left (a^2 A-6 A b^2-4 a b B\right ) \log (\sin (c+d x))}{d}+\frac{b^2 \left (3 a A b+a^2 B+b^2 B\right ) \tan (c+d x)}{d}-\frac{a (5 A b+2 a B) \cot (c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{a A \cot ^2(c+d x) (a+b \tan (c+d x))^3}{2 d}\\ \end{align*}
Mathematica [C] time = 0.673258, size = 140, normalized size = 0.75 \[ \frac{-2 a^2 \left (a^2 A-4 a b B-6 A b^2\right ) \log (\tan (c+d x))-2 a^3 (a B+4 A b) \cot (c+d x)+a^4 (-A) \cot ^2(c+d x)+(a+i b)^4 (A+i B) \log (-\tan (c+d x)+i)+(a-i b)^4 (A-i B) \log (\tan (c+d x)+i)+2 b^4 B \tan (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.094, size = 244, normalized size = 1.3 \begin{align*} -{\frac{A{b}^{4}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-B{b}^{4}x+{\frac{B{b}^{4}\tan \left ( dx+c \right ) }{d}}-{\frac{B{b}^{4}c}{d}}+4\,Aa{b}^{3}x+4\,{\frac{Aa{b}^{3}c}{d}}-4\,{\frac{Ba{b}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+6\,{\frac{A{a}^{2}{b}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+6\,B{a}^{2}{b}^{2}x+6\,{\frac{B{a}^{2}{b}^{2}c}{d}}-4\,Ax{a}^{3}b-4\,{\frac{A\cot \left ( dx+c \right ){a}^{3}b}{d}}-4\,{\frac{A{a}^{3}bc}{d}}+4\,{\frac{B{a}^{3}b\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{A{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{A{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-B{a}^{4}x-{\frac{B\cot \left ( dx+c \right ){a}^{4}}{d}}-{\frac{B{a}^{4}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46268, size = 234, normalized size = 1.26 \begin{align*} \frac{2 \, B b^{4} \tan \left (d x + c\right ) - 2 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )}{\left (d x + c\right )} +{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \,{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac{A a^{4} + 2 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14768, size = 456, normalized size = 2.45 \begin{align*} \frac{2 \, B b^{4} \tan \left (d x + c\right )^{3} - A a^{4} -{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} 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 )^{2} -{\left (4 \, B a b^{3} + A b^{4}\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{2} -{\left (A a^{4} + 2 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{2} - 2 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{2 \, d \tan \left (d x + c\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.3322, size = 309, normalized size = 1.66 \begin{align*} \begin{cases} \tilde{\infty } A a^{4} 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{\left (c \right )}\right ) \left (a + b \tan{\left (c \right )}\right )^{4} \cot ^{3}{\left (c \right )} & \text{for}\: d = 0 \\\frac{A a^{4} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac{A a^{4} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} - \frac{A a^{4}}{2 d \tan ^{2}{\left (c + d x \right )}} - 4 A a^{3} b x - \frac{4 A a^{3} b}{d \tan{\left (c + d x \right )}} - \frac{3 A a^{2} b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac{6 A a^{2} b^{2} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + 4 A a b^{3} x + \frac{A b^{4} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - B a^{4} x - \frac{B a^{4}}{d \tan{\left (c + d x \right )}} - \frac{2 B a^{3} b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac{4 B a^{3} b \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + 6 B a^{2} b^{2} x + \frac{2 B a b^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - B b^{4} x + \frac{B b^{4} \tan{\left (c + d x \right )}}{d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.79531, size = 302, normalized size = 1.62 \begin{align*} \frac{2 \, B b^{4} \tan \left (d x + c\right ) - 2 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )}{\left (d x + c\right )} +{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \,{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + \frac{3 \, A a^{4} \tan \left (d x + c\right )^{2} - 12 \, B a^{3} b \tan \left (d x + c\right )^{2} - 18 \, A a^{2} b^{2} \tan \left (d x + c\right )^{2} - 2 \, B a^{4} \tan \left (d x + c\right ) - 8 \, A a^{3} b \tan \left (d x + c\right ) - A a^{4}}{\tan \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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