Optimal. Leaf size=60 \[ -\frac {a \cot ^3(c+d x)}{3 d}+\frac {a \cot (c+d x)}{d}+a x-\frac {b \cot ^2(c+d x)}{2 d}-\frac {b \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.08, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3529, 3531, 3475} \[ -\frac {a \cot ^3(c+d x)}{3 d}+\frac {a \cot (c+d x)}{d}+a x-\frac {b \cot ^2(c+d x)}{2 d}-\frac {b \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3529
Rule 3531
Rubi steps
\begin {align*} \int \cot ^4(c+d x) (a+b \tan (c+d x)) \, dx &=-\frac {a \cot ^3(c+d x)}{3 d}+\int \cot ^3(c+d x) (b-a \tan (c+d x)) \, dx\\ &=-\frac {b \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}+\int \cot ^2(c+d x) (-a-b \tan (c+d x)) \, dx\\ &=\frac {a \cot (c+d x)}{d}-\frac {b \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}+\int \cot (c+d x) (-b+a \tan (c+d x)) \, dx\\ &=a x+\frac {a \cot (c+d x)}{d}-\frac {b \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}-b \int \cot (c+d x) \, dx\\ &=a x+\frac {a \cot (c+d x)}{d}-\frac {b \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}-\frac {b \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 70, normalized size = 1.17 \[ -\frac {a \cot ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(c+d x)\right )}{3 d}-\frac {b \left (\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 89, normalized size = 1.48 \[ -\frac {3 \, b \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{3} - 3 \, {\left (2 \, a d x - b\right )} \tan \left (d x + c\right )^{3} - 6 \, a \tan \left (d x + c\right )^{2} + 3 \, b \tan \left (d x + c\right ) + 2 \, a}{6 \, d \tan \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.08, size = 140, normalized size = 2.33 \[ \frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, {\left (d x + c\right )} a + 24 \, b \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 24 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {44 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 63, normalized size = 1.05 \[ -\frac {a \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a \cot \left (d x +c \right )}{d}+a x +\frac {c a}{d}-\frac {b \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {b \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 71, normalized size = 1.18 \[ \frac {6 \, {\left (d x + c\right )} a + 3 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, b \log \left (\tan \left (d x + c\right )\right ) + \frac {6 \, a \tan \left (d x + c\right )^{2} - 3 \, b \tan \left (d x + c\right ) - 2 \, a}{\tan \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.98, size = 96, normalized size = 1.60 \[ -\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-\frac {b}{2}+\frac {a\,1{}\mathrm {i}}{2}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (\frac {b}{2}+\frac {a\,1{}\mathrm {i}}{2}\right )}{d}-\frac {b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left (-a\,{\mathrm {tan}\left (c+d\,x\right )}^2+\frac {b\,\mathrm {tan}\left (c+d\,x\right )}{2}+\frac {a}{3}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.53, size = 97, normalized size = 1.62 \[ \begin {cases} \tilde {\infty } a x & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (a + b \tan {\relax (c )}\right ) \cot ^{4}{\relax (c )} & \text {for}\: d = 0 \\a x + \frac {a}{d \tan {\left (c + d x \right )}} - \frac {a}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {b}{2 d \tan ^{2}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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