3.111 \(\int (c+d x)^m \cot ^2(a+b x) \csc (a+b x) \, dx\)

Optimal. Leaf size=37 \[ \text{Unintegrable}\left (\csc ^3(a+b x) (c+d x)^m,x\right )-\text{Unintegrable}\left (\csc (a+b x) (c+d x)^m,x\right ) \]

[Out]

-Unintegrable[(c + d*x)^m*Csc[a + b*x], x] + Unintegrable[(c + d*x)^m*Csc[a + b*x]^3, x]

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Rubi [A]  time = 0.0752131, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (c+d x)^m \cot ^2(a+b x) \csc (a+b x) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c + d*x)^m*Cot[a + b*x]^2*Csc[a + b*x],x]

[Out]

-Defer[Int][(c + d*x)^m*Csc[a + b*x], x] + Defer[Int][(c + d*x)^m*Csc[a + b*x]^3, x]

Rubi steps

\begin{align*} \int (c+d x)^m \cot ^2(a+b x) \csc (a+b x) \, dx &=-\int (c+d x)^m \csc (a+b x) \, dx+\int (c+d x)^m \csc ^3(a+b x) \, dx\\ \end{align*}

Mathematica [A]  time = 10.2968, size = 0, normalized size = 0. \[ \int (c+d x)^m \cot ^2(a+b x) \csc (a+b x) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c + d*x)^m*Cot[a + b*x]^2*Csc[a + b*x],x]

[Out]

Integrate[(c + d*x)^m*Cot[a + b*x]^2*Csc[a + b*x], x]

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Maple [A]  time = 0.18, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( \cot \left ( bx+a \right ) \right ) ^{2}\csc \left ( bx+a \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^m*cot(b*x+a)^2*csc(b*x+a),x)

[Out]

int((d*x+c)^m*cot(b*x+a)^2*csc(b*x+a),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} \cot \left (b x + a\right )^{2} \csc \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*cot(b*x+a)^2*csc(b*x+a),x, algorithm="maxima")

[Out]

integrate((d*x + c)^m*cot(b*x + a)^2*csc(b*x + a), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d x + c\right )}^{m} \cot \left (b x + a\right )^{2} \csc \left (b x + a\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*cot(b*x+a)^2*csc(b*x+a),x, algorithm="fricas")

[Out]

integral((d*x + c)^m*cot(b*x + a)^2*csc(b*x + a), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right )^{m} \cot ^{2}{\left (a + b x \right )} \csc{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**m*cot(b*x+a)**2*csc(b*x+a),x)

[Out]

Integral((c + d*x)**m*cot(a + b*x)**2*csc(a + b*x), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} \cot \left (b x + a\right )^{2} \csc \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*cot(b*x+a)^2*csc(b*x+a),x, algorithm="giac")

[Out]

integrate((d*x + c)^m*cot(b*x + a)^2*csc(b*x + a), x)