\(\int \frac {(c+d x)^m}{a+a \sin (e+f x)} \, dx\) [149]

Optimal result

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

[Out]

Unintegrable((d*x+c)^m/(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 = {} \[ \int \frac {(c+d x)^m}{a+a \sin (e+f x)} \, dx=\int \frac {(c+d x)^m}{a+a \sin (e+f x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.67 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^m}{a+a \sin (e+f x)} \, dx=\int \frac {(c+d x)^m}{a+a \sin (e+f x)} \, dx \]

[In]

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

[Out]

Integrate[(c + d*x)^m/(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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^m}{a+a \sin (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{a \sin \left (f x + e\right ) + a} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {(c+d x)^m}{a+a \sin (e+f x)} \, dx=\frac {\int \frac {\left (c + d x\right )^{m}}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^m}{a+a \sin (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{a \sin \left (f x + e\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^m}{a+a \sin (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{a \sin \left (f x + e\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^m}{a+a \sin (e+f x)} \, dx=\int \frac {{\left (c+d\,x\right )}^m}{a+a\,\sin \left (e+f\,x\right )} \,d x \]

[In]

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

[Out]

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