\(\int \frac {(c+d x)^m}{(a+b \sin (e+f x))^2} \, dx\) [178]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int \frac {(c+d x)^m}{(a+b \sin (e+f x))^2} \, dx=\int \frac {(c+d x)^m}{(a+b \sin (e+f x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 8.98 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^m}{(a+b \sin (e+f x))^2} \, dx=\int \frac {(c+d x)^m}{(a+b \sin (e+f x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.30 \[ \int \frac {(c+d x)^m}{(a+b \sin (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 10.86 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {(c+d x)^m}{(a+b \sin (e+f x))^2} \, dx=\int \frac {\left (c + d x\right )^{m}}{\left (a + b \sin {\left (e + f x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^m}{(a+b \sin (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.72 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^m}{(a+b \sin (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^m}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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