\(\int \frac {(c+d x)^m}{(a+a \sin (e+f x))^2} \, dx\) [150]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {(c+d x)^m}{(a+a \sin (e+f x))^2} \, dx=\text {Int}\left (\frac {(c+d x)^m}{(a+a \sin (e+f x))^2},x\right ) \]

[Out]

Unintegrable((d*x+c)^m/(a+a*sin(f*x+e))^2,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 = {} \[ \int \frac {(c+d x)^m}{(a+a \sin (e+f x))^2} \, dx=\int \frac {(c+d x)^m}{(a+a \sin (e+f x))^2} \, dx \]

[In]

Int[(c + d*x)^m/(a + a*Sin[e + f*x])^2,x]

[Out]

Defer[Int][(c + d*x)^m/(a + a*Sin[e + f*x])^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(c+d x)^m}{(a+a \sin (e+f x))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 6.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^m}{(a+a \sin (e+f x))^2} \, dx=\int \frac {(c+d x)^m}{(a+a \sin (e+f x))^2} \, dx \]

[In]

Integrate[(c + d*x)^m/(a + a*Sin[e + f*x])^2,x]

[Out]

Integrate[(c + d*x)^m/(a + a*Sin[e + f*x])^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.16 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {\left (d x +c \right )^{m}}{\left (a +a \sin \left (f x +e \right )\right )^{2}}d x\]

[In]

int((d*x+c)^m/(a+a*sin(f*x+e))^2,x)

[Out]

int((d*x+c)^m/(a+a*sin(f*x+e))^2,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.10 \[ \int \frac {(c+d x)^m}{(a+a \sin (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((d*x+c)^m/(a+a*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 10.39 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {(c+d x)^m}{(a+a \sin (e+f x))^2} \, dx=\frac {\int \frac {\left (c + d x\right )^{m}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]

[In]

integrate((d*x+c)**m/(a+a*sin(f*x+e))**2,x)

[Out]

Integral((c + d*x)**m/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1), x)/a**2

Maxima [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^m}{(a+a \sin (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((d*x+c)^m/(a+a*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^m}{(a+a \sin (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((d*x+c)^m/(a+a*sin(f*x+e))^2,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.83 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^m}{(a+a \sin (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^m}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]

[In]

int((c + d*x)^m/(a + a*sin(e + f*x))^2,x)

[Out]

int((c + d*x)^m/(a + a*sin(e + f*x))^2, x)