\(\int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx\) [59]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx=\text {Int}\left ((a+b \csc (e+f x))^m \sin ^2(e+f x),x\right ) \]

[Out]

Unintegrable((a+b*csc(f*x+e))^m*sin(f*x+e)^2,x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 21, 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 = {} \[ \int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 6.53 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

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

[In]

int((a+b*csc(f*x+e))^m*sin(f*x+e)^2,x)

[Out]

int((a+b*csc(f*x+e))^m*sin(f*x+e)^2,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int { {\left (b \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{2} \,d x } \]

[In]

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

[Out]

integral(-(cos(f*x + e)^2 - 1)*(b*csc(f*x + e) + a)^m, x)

Sympy [N/A]

Not integrable

Time = 21.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int \left (a + b \csc {\left (e + f x \right )}\right )^{m} \sin ^{2}{\left (e + f x \right )}\, dx \]

[In]

integrate((a+b*csc(f*x+e))**m*sin(f*x+e)**2,x)

[Out]

Integral((a + b*csc(e + f*x))**m*sin(e + f*x)**2, x)

Maxima [N/A]

Not integrable

Time = 2.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int { {\left (b \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{2} \,d x } \]

[In]

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

[Out]

integrate((b*csc(f*x + e) + a)^m*sin(f*x + e)^2, x)

Giac [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int { {\left (b \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{2} \,d x } \]

[In]

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

[Out]

integrate((b*csc(f*x + e) + a)^m*sin(f*x + e)^2, x)

Mupad [N/A]

Not integrable

Time = 19.49 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^2\,{\left (a+\frac {b}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]

[In]

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

[Out]

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