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

Optimal. Leaf size=24 \[ \text {Int}\left (\sin ^2(e+f x) (a+b \csc (e+f x))^m,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx \]

Verification is Not applicable to the result.

[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*} \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\\ \end {align*}

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Mathematica [A]  time = 6.24, size = 0, normalized size = 0.00 \[ \int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx \]

Verification is Not applicable to the result.

[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]

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fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (f x + e\right )^{2} - 1\right )} {\left (b \csc \left (f x + e\right ) + a\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 4.45, size = 0, normalized size = 0.00 \[ \int \left (a +b \csc \left (f x +e \right )\right )^{m} \left (\sin ^{2}\left (f x +e \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \csc \left (f x + e\right ) + a\right )}^{m} \sin \left (f x + e\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\sin \left (e+f\,x\right )}^2\,{\left (a+\frac {b}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \csc {\left (e + f x \right )}\right )^{m} \sin ^{2}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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