Optimal. Leaf size=29 \[ \text {Int}\left (\frac {\cos (c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right ) \]
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Rubi [A] time = 0.05, 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 = {} \[ \int \frac {\cos (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx &=\int \frac {\cos (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\\ \end {align*}
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Mathematica [A] time = 3.25, size = 0, normalized size = 0.00 \[ \int \frac {\cos (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (d x + c\right )}{a f x + a e + {\left (a f x + a e\right )} \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )}{{\left (f x + e\right )} {\left (a \sin \left (d x + c\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x +c \right )}{\left (f x +e \right ) \left (a +a \sin \left (d x +c \right )\right )}\, dx \]
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 \[ \int \frac {\cos \left (d x + c\right )}{{\left (f x + e\right )} {\left (a \sin \left (d x + c\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\cos \left (c+d\,x\right )}{\left (e+f\,x\right )\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cos {\left (c + d x \right )}}{e \sin {\left (c + d x \right )} + e + f x \sin {\left (c + d x \right )} + f x}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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