\(\int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx\) [806]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\text {Int}\left (\frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\int \frac {(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 5.95 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx \]

[In]

Integrate[(3 + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^2,x]

[Out]

Integrate[(3 + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^2, x]

Maple [N/A] (verified)

Not integrable

Time = 2.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04 \[ \int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 24.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 13.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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