\(\int \frac {1}{(c+d x) (a+a \sin (e+f x))} \, dx\) [110]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{(c+d x) (a+a \sin (e+f x))} \, dx=\text {Int}\left (\frac {1}{(c+d x) (a+a \sin (e+f x))},x\right ) \]

[Out]

Unintegrable(1/(d*x+c)/(a+a*sin(f*x+e)),x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{(c+d x) (a+a \sin (e+f x))} \, dx=\int \frac {1}{(c+d x) (a+a \sin (e+f x))} \, dx \]

[In]

Int[1/((c + d*x)*(a + a*Sin[e + f*x])),x]

[Out]

Defer[Int][1/((c + d*x)*(a + a*Sin[e + f*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(c+d x) (a+a \sin (e+f x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 5.35 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+a \sin (e+f x))} \, dx=\int \frac {1}{(c+d x) (a+a \sin (e+f x))} \, dx \]

[In]

Integrate[1/((c + d*x)*(a + a*Sin[e + f*x])),x]

[Out]

Integrate[1/((c + d*x)*(a + a*Sin[e + f*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (d x +c \right ) \left (a +a \sin \left (f x +e \right )\right )}d x\]

[In]

int(1/(d*x+c)/(a+a*sin(f*x+e)),x)

[Out]

int(1/(d*x+c)/(a+a*sin(f*x+e)),x)

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(c+d x) (a+a \sin (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (a \sin \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)/(a+a*sin(f*x+e)),x, algorithm="fricas")

[Out]

integral(1/(a*d*x + a*c + (a*d*x + a*c)*sin(f*x + e)), x)

Sympy [N/A]

Not integrable

Time = 1.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(c+d x) (a+a \sin (e+f x))} \, dx=\frac {\int \frac {1}{c \sin {\left (e + f x \right )} + c + d x \sin {\left (e + f x \right )} + d x}\, dx}{a} \]

[In]

integrate(1/(d*x+c)/(a+a*sin(f*x+e)),x)

[Out]

Integral(1/(c*sin(e + f*x) + c + d*x*sin(e + f*x) + d*x), x)/a

Maxima [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 285, normalized size of antiderivative = 14.25 \[ \int \frac {1}{(c+d x) (a+a \sin (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (a \sin \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)/(a+a*sin(f*x+e)),x, algorithm="maxima")

[Out]

-2*((a*d^2*f*x + a*c*d*f + (a*d^2*f*x + a*c*d*f)*cos(f*x + e)^2 + (a*d^2*f*x + a*c*d*f)*sin(f*x + e)^2 + 2*(a*
d^2*f*x + a*c*d*f)*sin(f*x + e))*integrate(cos(f*x + e)/(a*d^2*f*x^2 + 2*a*c*d*f*x + a*c^2*f + (a*d^2*f*x^2 +
2*a*c*d*f*x + a*c^2*f)*cos(f*x + e)^2 + (a*d^2*f*x^2 + 2*a*c*d*f*x + a*c^2*f)*sin(f*x + e)^2 + 2*(a*d^2*f*x^2
+ 2*a*c*d*f*x + a*c^2*f)*sin(f*x + e)), x) + cos(f*x + e))/(a*d*f*x + a*c*f + (a*d*f*x + a*c*f)*cos(f*x + e)^2
+ (a*d*f*x + a*c*f)*sin(f*x + e)^2 + 2*(a*d*f*x + a*c*f)*sin(f*x + e))

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+a \sin (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (a \sin \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)/(a+a*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate(1/((d*x + c)*(a*sin(f*x + e) + a)), x)

Mupad [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+a \sin (e+f x))} \, dx=\int \frac {1}{\left (a+a\,\sin \left (e+f\,x\right )\right )\,\left (c+d\,x\right )} \,d x \]

[In]

int(1/((a + a*sin(e + f*x))*(c + d*x)),x)

[Out]

int(1/((a + a*sin(e + f*x))*(c + d*x)), x)