Optimal. Leaf size=120 \[ \frac{\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{\left (a^2-b^2\right ) \cot (c+d x)}{d}-x \left (a^2-b^2\right )-\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a b \cot ^4(c+d x)}{2 d}+\frac{a b \cot ^2(c+d x)}{d}+\frac{2 a b \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.190071, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3542, 3529, 3531, 3475} \[ \frac{\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{\left (a^2-b^2\right ) \cot (c+d x)}{d}-x \left (a^2-b^2\right )-\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a b \cot ^4(c+d x)}{2 d}+\frac{a b \cot ^2(c+d x)}{d}+\frac{2 a b \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3542
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx &=-\frac{a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^5(c+d x) \left (2 a b-\left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-\frac{a b \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^4(c+d x) \left (-a^2+b^2-2 a b \tan (c+d x)\right ) \, dx\\ &=\frac{\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{a b \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^3(c+d x) \left (-2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{a b \cot ^2(c+d x)}{d}+\frac{\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{a b \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^2(c+d x) \left (a^2-b^2+2 a b \tan (c+d x)\right ) \, dx\\ &=-\frac{\left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a b \cot ^2(c+d x)}{d}+\frac{\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{a b \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\int \cot (c+d x) \left (2 a b-\left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-\left (a^2-b^2\right ) x-\frac{\left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a b \cot ^2(c+d x)}{d}+\frac{\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{a b \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+(2 a b) \int \cot (c+d x) \, dx\\ &=-\left (a^2-b^2\right ) x-\frac{\left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a b \cot ^2(c+d x)}{d}+\frac{\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{a b \cot ^4(c+d x)}{2 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{2 a b \log (\sin (c+d x))}{d}\\ \end{align*}
Mathematica [C] time = 1.00411, size = 121, normalized size = 1.01 \[ -\frac{a^2 \cot ^5(c+d x) \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(c+d x)\right )}{5 d}+\frac{a b \left (-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right )}{2 d}-\frac{b^2 \cot ^3(c+d x) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 148, normalized size = 1.2 \begin{align*} -{\frac{{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{b}^{2}\cot \left ( dx+c \right ) }{d}}+{b}^{2}x+{\frac{{b}^{2}c}{d}}-{\frac{ab \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{2\,d}}+{\frac{ab \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}+2\,{\frac{ab\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{a}^{2}\cot \left ( dx+c \right ) }{d}}-{a}^{2}x-{\frac{{a}^{2}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.69103, size = 167, normalized size = 1.39 \begin{align*} -\frac{30 \, a b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \, a b \log \left (\tan \left (d x + c\right )\right ) + 30 \,{\left (a^{2} - b^{2}\right )}{\left (d x + c\right )} - \frac{30 \, a b \tan \left (d x + c\right )^{3} - 30 \,{\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{4} - 15 \, a b \tan \left (d x + c\right ) + 10 \,{\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} - 6 \, a^{2}}{\tan \left (d x + c\right )^{5}}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78693, size = 346, normalized size = 2.88 \begin{align*} \frac{30 \, a 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 )^{5} - 15 \,{\left (2 \,{\left (a^{2} - b^{2}\right )} d x - 3 \, a b\right )} \tan \left (d x + c\right )^{5} + 30 \, a b \tan \left (d x + c\right )^{3} - 30 \,{\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{4} - 15 \, a b \tan \left (d x + c\right ) + 10 \,{\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} - 6 \, a^{2}}{30 \, d \tan \left (d x + c\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 73.7135, size = 172, normalized size = 1.43 \begin{align*} \begin{cases} \tilde{\infty } a^{2} 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 )^{2} \cot ^{6}{\left (c \right )} & \text{for}\: d = 0 \\- a^{2} x - \frac{a^{2}}{d \tan{\left (c + d x \right )}} + \frac{a^{2}}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac{a^{2}}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac{a b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac{2 a b \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + \frac{a b}{d \tan ^{2}{\left (c + d x \right )}} - \frac{a b}{2 d \tan ^{4}{\left (c + d x \right )}} + b^{2} x + \frac{b^{2}}{d \tan{\left (c + d x \right )}} - \frac{b^{2}}{3 d \tan ^{3}{\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 = 1.59888, size = 387, normalized size = 3.22 \begin{align*} \frac{3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 15 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 35 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 20 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 180 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 960 \, a b \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) + 960 \, a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 330 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 300 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 480 \,{\left (a^{2} - b^{2}\right )}{\left (d x + c\right )} - \frac{2192 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 330 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 300 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 180 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 35 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 20 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 15 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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