3.255 \(\int \frac {\cos (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\)

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {\cos (c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right ) \]

[Out]

Unintegrable(cos(d*x+c)/(f*x+e)/(a+a*sin(d*x+c)),x)

________________________________________________________________________________________

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.

[In]

Int[Cos[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])),x]

[Out]

Defer[Int][Cos[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])), x]

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*}

________________________________________________________________________________________

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.

[In]

Integrate[Cos[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])),x]

[Out]

Integrate[Cos[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])), x]

________________________________________________________________________________________

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.

[In]

integrate(cos(d*x+c)/(f*x+e)/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

integral(cos(d*x + c)/(a*f*x + a*e + (a*f*x + a*e)*sin(d*x + c)), x)

________________________________________________________________________________________

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.

[In]

integrate(cos(d*x+c)/(f*x+e)/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate(cos(d*x + c)/((f*x + e)*(a*sin(d*x + c) + a)), x)

________________________________________________________________________________________

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.

[In]

int(cos(d*x+c)/(f*x+e)/(a+a*sin(d*x+c)),x)

[Out]

int(cos(d*x+c)/(f*x+e)/(a+a*sin(d*x+c)),x)

________________________________________________________________________________________

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.

[In]

integrate(cos(d*x+c)/(f*x+e)/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)/((f*x + e)*(a*sin(d*x + c) + a)), x)

________________________________________________________________________________________

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.

[In]

int(cos(c + d*x)/((e + f*x)*(a + a*sin(c + d*x))),x)

[Out]

int(cos(c + d*x)/((e + f*x)*(a + a*sin(c + d*x))), x)

________________________________________________________________________________________

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.

[In]

integrate(cos(d*x+c)/(f*x+e)/(a+a*sin(d*x+c)),x)

[Out]

Integral(cos(c + d*x)/(e*sin(c + d*x) + e + f*x*sin(c + d*x) + f*x), x)/a

________________________________________________________________________________________