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 = 0.0813487, antiderivative size = 60, normalized size of antiderivative = 1., 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 3529
Rule 3531
Rule 3475
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*}
Mathematica [C] time = 0.291407, 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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Maple [A] time = 0.037, size = 63, normalized size = 1.1 \begin{align*} -{\frac{b \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{b\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{\cot \left ( dx+c \right ) a}{d}}+ax+{\frac{ac}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50713, size = 96, normalized size = 1.6 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67818, size = 224, normalized size = 3.73 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.06712, size = 97, normalized size = 1.62 \begin{align*} \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{\left (c \right )}\right ) \cot ^{4}{\left (c \right )} & \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.3623, size = 189, normalized size = 3.15 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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