Optimal. Leaf size=141 \[ -\frac {5 a^3 b \cot ^3(c+d x)}{6 d}+\frac {a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}+\frac {4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}+4 a b x \left (a^2-b^2\right )-\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.34, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}+\frac {4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac {\left (-6 a^2 b^2+a^4+b^4\right ) \log (\sin (c+d x))}{d}+4 a b x \left (a^2-b^2\right )-\frac {5 a^3 b \cot ^3(c+d x)}{6 d}-\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d} \]
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 ^5(c+d x) (a+b \tan (c+d x))^4 \, dx &=-\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac {1}{4} \int \cot ^4(c+d x) (a+b \tan (c+d x)) \left (10 a^2 b-4 a \left (a^2-3 b^2\right ) \tan (c+d x)-2 b \left (a^2-2 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac {5 a^3 b \cot ^3(c+d x)}{6 d}-\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac {1}{4} \int \cot ^3(c+d x) \left (-2 a^2 \left (2 a^2-11 b^2\right )-16 a b \left (a^2-b^2\right ) \tan (c+d x)-2 b^2 \left (a^2-2 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}-\frac {5 a^3 b \cot ^3(c+d x)}{6 d}-\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac {1}{4} \int \cot ^2(c+d x) \left (-16 a b \left (a^2-b^2\right )+4 \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac {a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}-\frac {5 a^3 b \cot ^3(c+d x)}{6 d}-\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac {1}{4} \int \cot (c+d x) \left (4 \left (a^4-6 a^2 b^2+b^4\right )+16 a b \left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=4 a b \left (a^2-b^2\right ) x+\frac {4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac {a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}-\frac {5 a^3 b \cot ^3(c+d x)}{6 d}-\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\left (a^4-6 a^2 b^2+b^4\right ) \int \cot (c+d x) \, dx\\ &=4 a b \left (a^2-b^2\right ) x+\frac {4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac {a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}-\frac {5 a^3 b \cot ^3(c+d x)}{6 d}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\\ \end {align*}
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Mathematica [C] time = 2.37, size = 147, normalized size = 1.04 \[ \frac {-3 a^4 \cot ^4(c+d x)-16 a^3 b \cot ^3(c+d x)+6 a^2 \left (a^2-6 b^2\right ) \cot ^2(c+d x)+48 a b \left (a^2-b^2\right ) \cot (c+d x)-6 \left (-2 \left (a^4-6 a^2 b^2+b^4\right ) \log (\tan (c+d x))+(a-i b)^4 \log (\tan (c+d x)+i)+(a+i b)^4 \log (-\tan (c+d x)+i)\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 162, normalized size = 1.15 \[ \frac {6 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{4} - 16 \, a^{3} b \tan \left (d x + c\right ) + 3 \, {\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 16 \, {\left (a^{3} b - a b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{4} - 3 \, a^{4} + 48 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} + 6 \, {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{12 \, d \tan \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 7.96, size = 335, normalized size = 2.38 \[ -\frac {3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 32 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 144 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 480 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 384 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 768 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )} + 192 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 192 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {400 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2400 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 400 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 480 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 384 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 144 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 32 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 180, normalized size = 1.28 \[ -\frac {a^{4} \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a^{4} \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a^{4} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {4 a^{3} b \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {4 a^{3} b \cot \left (d x +c \right )}{d}+4 a^{3} b x +\frac {4 a^{3} b c}{d}-\frac {3 a^{2} b^{2} \left (\cot ^{2}\left (d x +c \right )\right )}{d}-\frac {6 a^{2} b^{2} \ln \left (\sin \left (d x +c \right )\right )}{d}-4 a \,b^{3} x -\frac {4 \cot \left (d x +c \right ) a \,b^{3}}{d}-\frac {4 a \,b^{3} c}{d}+\frac {b^{4} \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 149, normalized size = 1.06 \[ \frac {48 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )} - 6 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac {16 \, a^{3} b \tan \left (d x + c\right ) + 3 \, a^{4} - 48 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} - 6 \, {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.97, size = 150, normalized size = 1.06 \[ \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^4-6\,a^2\,b^2+b^4\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^4}{2\,d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^4\,\left ({\mathrm {tan}\left (c+d\,x\right )}^3\,\left (4\,a\,b^3-4\,a^3\,b\right )-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {a^4}{2}-3\,a^2\,b^2\right )+\frac {a^4}{4}+\frac {4\,a^3\,b\,\mathrm {tan}\left (c+d\,x\right )}{3}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^4}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.83, size = 252, normalized size = 1.79 \[ \begin {cases} \tilde {\infty } a^{4} x & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (a + b \tan {\relax (c )}\right )^{4} \cot ^{5}{\relax (c )} & \text {for}\: d = 0 \\- \frac {a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {a^{4}}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {a^{4}}{4 d \tan ^{4}{\left (c + d x \right )}} + 4 a^{3} b x + \frac {4 a^{3} b}{d \tan {\left (c + d x \right )}} - \frac {4 a^{3} b}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {3 a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac {6 a^{2} b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {3 a^{2} b^{2}}{d \tan ^{2}{\left (c + d x \right )}} - 4 a b^{3} x - \frac {4 a b^{3}}{d \tan {\left (c + d x \right )}} - \frac {b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {b^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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