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

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{\sin (c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0480519, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sin (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\sin (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx &=\int \frac{\sin (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\\ \end{align*}

Mathematica [A]  time = 8.57725, size = 0, normalized size = 0. \[ \int \frac{\sin (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.173, size = 0, normalized size = 0. \begin{align*} \int{\frac{\sin \left ( dx+c \right ) }{ \left ( fx+e \right ) \left ( a+a\sin \left ( dx+c \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sin \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sin{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (d x + c\right )}{{\left (f x + e\right )}{\left (a \sin \left (d x + c\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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