Optimal. Leaf size=35 \[ -\frac{d \cot (a+b x)}{2 b^2}-\frac{(c+d x) \csc ^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.0314137, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4410, 3767, 8} \[ -\frac{d \cot (a+b x)}{2 b^2}-\frac{(c+d x) \csc ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 4410
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (c+d x) \cot (a+b x) \csc ^2(a+b x) \, dx &=-\frac{(c+d x) \csc ^2(a+b x)}{2 b}+\frac{d \int \csc ^2(a+b x) \, dx}{2 b}\\ &=-\frac{(c+d x) \csc ^2(a+b x)}{2 b}-\frac{d \operatorname{Subst}(\int 1 \, dx,x,\cot (a+b x))}{2 b^2}\\ &=-\frac{d \cot (a+b x)}{2 b^2}-\frac{(c+d x) \csc ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0738112, size = 48, normalized size = 1.37 \[ -\frac{d \cot (a+b x)}{2 b^2}-\frac{c \csc ^2(a+b x)}{2 b}-\frac{d x \csc ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 61, normalized size = 1.7 \begin{align*}{\frac{1}{b} \left ({\frac{d}{b} \left ( -{\frac{bx+a}{2\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}}}-{\frac{\cot \left ( bx+a \right ) }{2}} \right ) }+{\frac{ad}{2\,b \left ( \sin \left ( bx+a \right ) \right ) ^{2}}}-{\frac{c}{2\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16603, size = 387, normalized size = 11.06 \begin{align*} \frac{\frac{2 \,{\left (4 \,{\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + 4 \,{\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} -{\left (2 \,{\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )\right )} \cos \left (4 \, b x + 4 \, a\right ) - 2 \,{\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) -{\left (2 \,{\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \sin \left (4 \, b x + 4 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )\right )} d}{{\left (2 \,{\left (2 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )^{2} - 4 \, \cos \left (2 \, b x + 2 \, a\right )^{2} - \sin \left (4 \, b x + 4 \, a\right )^{2} + 4 \, \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 4 \, \sin \left (2 \, b x + 2 \, a\right )^{2} + 4 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} b} - \frac{c}{\sin \left (b x + a\right )^{2}} + \frac{a d}{b \sin \left (b x + a\right )^{2}}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.456576, size = 103, normalized size = 2.94 \begin{align*} \frac{b d x + d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + b c}{2 \,{\left (b^{2} \cos \left (b x + a\right )^{2} - b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20892, size = 710, normalized size = 20.29 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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