Optimal. Leaf size=78 \[ -\frac{b \left (3 a^2-3 a b+b^2\right ) \cot (c+d x)}{d}-\frac{b^2 (3 a-b) \cot ^3(c+d x)}{3 d}+x (a-b)^3-\frac{b^3 \cot ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0472866, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3661, 390, 203} \[ -\frac{b \left (3 a^2-3 a b+b^2\right ) \cot (c+d x)}{d}-\frac{b^2 (3 a-b) \cot ^3(c+d x)}{3 d}+x (a-b)^3-\frac{b^3 \cot ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3661
Rule 390
Rule 203
Rubi steps
\begin{align*} \int \left (a+b \cot ^2(c+d x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^3}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (b \left (3 a^2-3 a b+b^2\right )+(3 a-b) b^2 x^2+b^3 x^4+\frac{(a-b)^3}{1+x^2}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{b \left (3 a^2-3 a b+b^2\right ) \cot (c+d x)}{d}-\frac{(3 a-b) b^2 \cot ^3(c+d x)}{3 d}-\frac{b^3 \cot ^5(c+d x)}{5 d}-\frac{(a-b)^3 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=(a-b)^3 x-\frac{b \left (3 a^2-3 a b+b^2\right ) \cot (c+d x)}{d}-\frac{(3 a-b) b^2 \cot ^3(c+d x)}{3 d}-\frac{b^3 \cot ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 2.75154, size = 111, normalized size = 1.42 \[ -\frac{\cot ^5(c+d x) \left (b \left (15 \left (3 a^2-3 a b+b^2\right ) \tan ^4(c+d x)+5 b (3 a-b) \tan ^2(c+d x)+3 b^2\right )+\frac{15 (a-b)^3 \tan ^8(c+d x) \tanh ^{-1}\left (\sqrt{-\tan ^2(c+d x)}\right )}{\left (-\tan ^2(c+d x)\right )^{3/2}}\right )}{15 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 116, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ( -{\frac{{b}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5}}- \left ( \cot \left ( dx+c \right ) \right ) ^{3}a{b}^{2}+{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{3}{b}^{3}}{3}}-3\,\cot \left ( dx+c \right ){a}^{2}b+3\,a{b}^{2}\cot \left ( dx+c \right ) -{b}^{3}\cot \left ( dx+c \right ) + \left ( -{a}^{3}+3\,{a}^{2}b-3\,a{b}^{2}+{b}^{3} \right ) \left ({\frac{\pi }{2}}-{\rm arccot} \left (\cot \left ( dx+c \right ) \right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46974, size = 151, normalized size = 1.94 \begin{align*} a^{3} x - \frac{3 \,{\left (d x + c + \frac{1}{\tan \left (d x + c\right )}\right )} a^{2} b}{d} + \frac{{\left (3 \, d x + 3 \, c + \frac{3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a b^{2}}{d} - \frac{{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} b^{3}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.6678, size = 583, normalized size = 7.47 \begin{align*} -\frac{{\left (45 \, a^{2} b - 60 \, a b^{2} + 23 \, b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )^{3} + 45 \, a^{2} b - 30 \, a b^{2} + 13 \, b^{3} -{\left (45 \, a^{2} b - 30 \, a b^{2} + b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )^{2} -{\left (45 \, a^{2} b - 60 \, a b^{2} + 11 \, b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right ) - 15 \,{\left ({\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} d x \cos \left (2 \, d x + 2 \, c\right )^{2} - 2 \,{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} d x \cos \left (2 \, d x + 2 \, c\right ) +{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} d x\right )} \sin \left (2 \, d x + 2 \, c\right )}{15 \,{\left (d \cos \left (2 \, d x + 2 \, c\right )^{2} - 2 \, d \cos \left (2 \, d x + 2 \, c\right ) + d\right )} \sin \left (2 \, d x + 2 \, c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.698127, size = 126, normalized size = 1.62 \begin{align*} \begin{cases} a^{3} x - 3 a^{2} b x - \frac{3 a^{2} b \cot{\left (c + d x \right )}}{d} + 3 a b^{2} x - \frac{a b^{2} \cot ^{3}{\left (c + d x \right )}}{d} + \frac{3 a b^{2} \cot{\left (c + d x \right )}}{d} - b^{3} x - \frac{b^{3} \cot ^{5}{\left (c + d x \right )}}{5 d} + \frac{b^{3} \cot ^{3}{\left (c + d x \right )}}{3 d} - \frac{b^{3} \cot{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \cot ^{2}{\left (c \right )}\right )^{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24844, size = 309, normalized size = 3.96 \begin{align*} \frac{3 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 60 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 35 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 720 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 900 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 330 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 480 \,{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}{\left (d x + c\right )} - \frac{720 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 900 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 330 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 60 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 35 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3 \, b^{3}}{\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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