Optimal. Leaf size=72 \[ -\frac {\sin \left (c-\frac {d e}{f}\right ) \text {Ci}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{a f}+\frac {\log (e+f x)}{a f} \]
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Rubi [A] time = 0.20, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {4523, 31, 3303, 3299, 3302} \[ -\frac {\sin \left (c-\frac {d e}{f}\right ) \text {CosIntegral}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{a f}+\frac {\log (e+f x)}{a f} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3299
Rule 3302
Rule 3303
Rule 4523
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx &=\frac {\int \frac {1}{e+f x} \, dx}{a}-\frac {\int \frac {\sin (c+d x)}{e+f x} \, dx}{a}\\ &=\frac {\log (e+f x)}{a f}-\frac {\cos \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {d e}{f}+d x\right )}{e+f x} \, dx}{a}-\frac {\sin \left (c-\frac {d e}{f}\right ) \int \frac {\cos \left (\frac {d e}{f}+d x\right )}{e+f x} \, dx}{a}\\ &=\frac {\log (e+f x)}{a f}-\frac {\text {Ci}\left (\frac {d e}{f}+d x\right ) \sin \left (c-\frac {d e}{f}\right )}{a f}-\frac {\cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{a f}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 58, normalized size = 0.81 \[ \frac {-\sin \left (c-\frac {d e}{f}\right ) \text {Ci}\left (d \left (\frac {e}{f}+x\right )\right )-\cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (d \left (\frac {e}{f}+x\right )\right )+\log (e+f x)}{a f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 89, normalized size = 1.24 \[ -\frac {{\left (\operatorname {Ci}\left (\frac {d f x + d e}{f}\right ) + \operatorname {Ci}\left (-\frac {d f x + d e}{f}\right )\right )} \sin \left (-\frac {d e - c f}{f}\right ) + 2 \, \cos \left (-\frac {d e - c f}{f}\right ) \operatorname {Si}\left (\frac {d f x + d e}{f}\right ) - 2 \, \log \left (f x + e\right )}{2 \, a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.31, size = 716, normalized size = 9.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 102, normalized size = 1.42 \[ -\frac {\Si \left (d x +c +\frac {-c f +d e}{f}\right ) \cos \left (\frac {-c f +d e}{f}\right )}{a f}+\frac {\Ci \left (d x +c +\frac {-c f +d e}{f}\right ) \sin \left (\frac {-c f +d e}{f}\right )}{a f}+\frac {\ln \left (\left (d x +c \right ) f -c f +d e \right )}{a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.41, size = 163, normalized size = 2.26 \[ \frac {d {\left (i \, E_{1}\left (\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right ) - i \, E_{1}\left (-\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \cos \left (-\frac {d e - c f}{f}\right ) + d {\left (E_{1}\left (\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right ) + E_{1}\left (-\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \sin \left (-\frac {d e - c f}{f}\right ) + 2 \, d \log \left (d e + {\left (d x + c\right )} f - c f\right )}{2 \, a d f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^2}{\left (e+f\,x\right )\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]
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 \[ \frac {\int \frac {\cos ^{2}{\left (c + d x \right )}}{e \sin {\left (c + d x \right )} + e + f x \sin {\left (c + d x \right )} + f x}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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