Optimal. Leaf size=170 \[ -\frac {3 a^3 b \cot ^4(c+d x)}{5 d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{15 d}+\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}+\frac {4 a b \left (a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)}{d}-x \left (a^4-6 a^2 b^2+b^4\right ) \]
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Rubi [A] time = 0.38, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3565, 3635, 3628, 3529, 3531, 3475} \[ \frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{15 d}+\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}-\frac {\left (-6 a^2 b^2+a^4+b^4\right ) \cot (c+d x)}{d}+\frac {4 a b \left (a^2-b^2\right ) \log (\sin (c+d x))}{d}-x \left (-6 a^2 b^2+a^4+b^4\right )-\frac {3 a^3 b \cot ^4(c+d x)}{5 d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3529
Rule 3531
Rule 3565
Rule 3628
Rule 3635
Rubi steps
\begin {align*} \int \cot ^6(c+d x) (a+b \tan (c+d x))^4 \, dx &=-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^5(c+d x) (a+b \tan (c+d x)) \left (12 a^2 b-5 a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (3 a^2-5 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac {3 a^3 b \cot ^4(c+d x)}{5 d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^4(c+d x) \left (-a^2 \left (5 a^2-27 b^2\right )-20 a b \left (a^2-b^2\right ) \tan (c+d x)-b^2 \left (3 a^2-5 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{15 d}-\frac {3 a^3 b \cot ^4(c+d x)}{5 d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^3(c+d x) \left (-20 a b \left (a^2-b^2\right )+5 \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{15 d}-\frac {3 a^3 b \cot ^4(c+d x)}{5 d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^2(c+d x) \left (5 \left (a^4-6 a^2 b^2+b^4\right )+20 a b \left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)}{d}+\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{15 d}-\frac {3 a^3 b \cot ^4(c+d x)}{5 d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot (c+d x) \left (20 a b \left (a^2-b^2\right )-5 \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right ) \, dx\\ &=-\left (a^4-6 a^2 b^2+b^4\right ) x-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)}{d}+\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{15 d}-\frac {3 a^3 b \cot ^4(c+d x)}{5 d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\left (4 a b \left (a^2-b^2\right )\right ) \int \cot (c+d x) \, dx\\ &=-\left (a^4-6 a^2 b^2+b^4\right ) x-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)}{d}+\frac {2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}+\frac {a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{15 d}-\frac {3 a^3 b \cot ^4(c+d x)}{5 d}+\frac {4 a b \left (a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\\ \end {align*}
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Mathematica [C] time = 0.42, size = 154, normalized size = 0.91 \[ -\frac {\frac {1}{5} a^4 \cot ^5(c+d x)+a^3 b \cot ^4(c+d x)-\frac {1}{3} a^2 \left (a^2-6 b^2\right ) \cot ^3(c+d x)+\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)-2 a b (a-b) (a+b) \cot ^2(c+d x)+\frac {1}{2} i (a-i b)^4 \log (-\cot (c+d x)+i)-\frac {1}{2} i (a+i b)^4 \log (\cot (c+d x)+i)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 186, normalized size = 1.09 \[ \frac {30 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{5} + 15 \, {\left (3 \, a^{3} b - 2 \, a b^{3} - {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{5} - 15 \, a^{3} b \tan \left (d x + c\right ) - 15 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} - 3 \, a^{4} + 30 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} + 5 \, {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{15 \, d \tan \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 15.11, size = 416, normalized size = 2.45 \[ \frac {3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 360 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 240 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 330 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1800 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 240 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 480 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )} - 1920 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 1920 \, {\left (a^{3} b - a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {4384 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4384 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 330 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1800 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 240 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 360 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 30 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 232, normalized size = 1.36 \[ -\frac {a^{4} \left (\cot ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a^{4} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}-\frac {a^{4} \cot \left (d x +c \right )}{d}-a^{4} x -\frac {a^{4} c}{d}-\frac {a^{3} b \left (\cot ^{4}\left (d x +c \right )\right )}{d}+\frac {2 a^{3} b \left (\cot ^{2}\left (d x +c \right )\right )}{d}+\frac {4 a^{3} b \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {2 a^{2} b^{2} \left (\cot ^{3}\left (d x +c \right )\right )}{d}+6 a^{2} b^{2} x +\frac {6 \cot \left (d x +c \right ) a^{2} b^{2}}{d}+\frac {6 a^{2} b^{2} c}{d}-\frac {2 a \,b^{3} \left (\cot ^{2}\left (d x +c \right )\right )}{d}-\frac {4 a \,b^{3} \ln \left (\sin \left (d x +c \right )\right )}{d}-b^{4} x -\frac {\cot \left (d x +c \right ) b^{4}}{d}-\frac {b^{4} c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 170, normalized size = 1.00 \[ -\frac {15 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )} + 30 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {15 \, a^{3} b \tan \left (d x + c\right ) + 15 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} + 3 \, a^{4} - 30 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} - 5 \, {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{5}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.06, size = 174, normalized size = 1.02 \[ \frac {4\,a\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2-b^2\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^5\,\left ({\mathrm {tan}\left (c+d\,x\right )}^3\,\left (2\,a\,b^3-2\,a^3\,b\right )-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {a^4}{3}-2\,a^2\,b^2\right )+\frac {a^4}{5}+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^4-6\,a^2\,b^2+b^4\right )+a^3\,b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (a-b\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.33, size = 265, normalized size = 1.56 \[ \begin {cases} \tilde {\infty } a^{4} x & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (a + b \tan {\relax (c )}\right )^{4} \cot ^{6}{\relax (c )} & \text {for}\: d = 0 \\- a^{4} x - \frac {a^{4}}{d \tan {\left (c + d x \right )}} + \frac {a^{4}}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {a^{4}}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {2 a^{3} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {4 a^{3} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {2 a^{3} b}{d \tan ^{2}{\left (c + d x \right )}} - \frac {a^{3} b}{d \tan ^{4}{\left (c + d x \right )}} + 6 a^{2} b^{2} x + \frac {6 a^{2} b^{2}}{d \tan {\left (c + d x \right )}} - \frac {2 a^{2} b^{2}}{d \tan ^{3}{\left (c + d x \right )}} + \frac {2 a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac {4 a b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {2 a b^{3}}{d \tan ^{2}{\left (c + d x \right )}} - b^{4} x - \frac {b^{4}}{d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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