Optimal. Leaf size=29 \[ \frac {d \log (\sin (a+b x))}{b^2}-\frac {(c+d x) \cot (a+b x)}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4184, 3475} \[ \frac {d \log (\sin (a+b x))}{b^2}-\frac {(c+d x) \cot (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 4184
Rubi steps
\begin {align*} \int (c+d x) \csc ^2(a+b x) \, dx &=-\frac {(c+d x) \cot (a+b x)}{b}+\frac {d \int \cot (a+b x) \, dx}{b}\\ &=-\frac {(c+d x) \cot (a+b x)}{b}+\frac {d \log (\sin (a+b x))}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 52, normalized size = 1.79 \[ \frac {d \log (\sin (a+b x))}{b^2}-\frac {c \cot (a+b x)}{b}-\frac {d x \cot (a)}{b}+\frac {d x \csc (a) \sin (b x) \csc (a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 46, normalized size = 1.59 \[ \frac {d \log \left (\frac {1}{2} \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) - {\left (b d x + b c\right )} \cos \left (b x + a\right )}{b^{2} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.37, size = 1251, normalized size = 43.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 39, normalized size = 1.34 \[ -\frac {d \cot \left (b x +a \right ) x}{b}+\frac {d \ln \left (\sin \left (b x +a \right )\right )}{b^{2}}-\frac {c \cot \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 217, normalized size = 7.48 \[ \frac {\frac {{\left ({\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) + {\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) - 4 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d}{{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} b} - \frac {2 \, c}{\tan \left (b x + a\right )} + \frac {2 \, a d}{b \tan \left (b x + a\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 55, normalized size = 1.90 \[ \frac {d\,\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}-1\right )}{b^2}-\frac {\left (c+d\,x\right )\,2{}\mathrm {i}}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}-\frac {d\,x\,2{}\mathrm {i}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right ) \csc ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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