Optimal. Leaf size=82 \[ \text {Int}\left (\frac {\cot (a+b x)}{c+d x},x\right )-\frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}-\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \]
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Rubi [A] time = 0.14, 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 = {} \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx &=\int \frac {\cot (a+b x)}{c+d x} \, dx-\int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx\\ &=\int \frac {\cot (a+b x)}{c+d x} \, dx-\int \frac {\sin (2 a+2 b x)}{2 (c+d x)} \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sin (2 a+2 b x)}{c+d x} \, dx\right )+\int \frac {\cot (a+b x)}{c+d x} \, dx\\ &=-\left (\frac {1}{2} \cos \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx\right )-\frac {1}{2} \sin \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx+\int \frac {\cot (a+b x)}{c+d x} \, dx\\ &=-\frac {\text {Ci}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{2 d}-\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\int \frac {\cot (a+b x)}{c+d x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.81, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b x + a\right )^{2} \cot \left (b x + a\right )}{d x + c}, x\right ) \]
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 \[ \int \frac {\cos \left (b x + a\right )^{2} \cot \left (b x + a\right )}{d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{2}\left (b x +a \right )\right ) \cot \left (b x +a \right )}{d x +c}\, dx \]
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 \[ -\frac {{\left (-i \, E_{1}\left (\frac {2 i \, b d x + 2 i \, b c}{d}\right ) + i \, E_{1}\left (-\frac {2 i \, b d x + 2 i \, b c}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 4 \, d \int \frac {\sin \left (b x + a\right )}{{\left (d x + c\right )} {\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right )}}\,{d x} - 4 \, d \int \frac {\sin \left (b x + a\right )}{{\left (d x + c\right )} {\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right )}}\,{d x} - {\left (E_{1}\left (\frac {2 i \, b d x + 2 i \, b c}{d}\right ) + E_{1}\left (-\frac {2 i \, b d x + 2 i \, b c}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (a+b\,x\right )}^2\,\mathrm {cot}\left (a+b\,x\right )}{c+d\,x} \,d x \]
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 \[ \int \frac {\cos ^{2}{\left (a + b x \right )} \cot {\left (a + b x \right )}}{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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