Optimal. Leaf size=23 \[ \text {Int}\left (\frac {\cot (a+b x) \csc (a+b x)}{c+d x},x\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cot (a+b x) \csc (a+b x)}{c+d x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\cot (a+b x) \csc (a+b x)}{c+d x} \, dx &=\int \frac {\cot (a+b x) \csc (a+b x)}{c+d x} \, dx\\ \end {align*}
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Mathematica [A] time = 16.76, size = 0, normalized size = 0.00 \[ \int \frac {\cot (a+b x) \csc (a+b x)}{c+d x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b x + a\right ) \csc \left (b x + a\right )^{2}}{d x + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x + a\right ) \csc \left (b x + a\right )^{2}}{d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x +a \right ) \left (\csc ^{2}\left (b x +a \right )\right )}{d x +c}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, \cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 2 \, \cos \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + \frac {{\left (b d^{2} x + b c d + {\left (b d^{2} x + b c d\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (b d^{2} x + b c d\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, {\left (b d^{2} x + b c d\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \int \frac {\sin \left (b x + a\right )}{{\left (d x + c\right )}^{2} {\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right )}}\,{d x}}{b} + \frac {{\left (b d^{2} x + b c d + {\left (b d^{2} x + b c d\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (b d^{2} x + b c d\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, {\left (b d^{2} x + b c d\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \int \frac {\sin \left (b x + a\right )}{{\left (d x + c\right )}^{2} {\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right )}}\,{d x}}{b} + 2 \, \sin \left (b x + a\right )}{b d x + {\left (b d x + b c\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (b d x + b c\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} + b c - 2 \, {\left (b d x + b c\right )} \cos \left (2 \, b x + 2 \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\cos \left (a+b\,x\right )}{{\sin \left (a+b\,x\right )}^2\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (a + b x \right )} \csc ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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