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

Optimal. Leaf size=39 \[ \text {Int}\left (\csc ^3(a+b x) (c+d x)^m,x\right )-\text {Int}\left (\csc (a+b x) (c+d x)^m,x\right ) \]

[Out]

-Unintegrable((d*x+c)^m*csc(b*x+a),x)+Unintegrable((d*x+c)^m*csc(b*x+a)^3,x)

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Rubi [A]  time = 0.08, 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 = {} \[ \int (c+d x)^m \cot ^2(a+b x) \csc (a+b x) \, dx \]

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*}

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Mathematica [A]  time = 38.57, size = 0, normalized size = 0.00 \[ \int (c+d x)^m \cot ^2(a+b x) \csc (a+b x) \, dx \]

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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fricas [A]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d x + c\right )}^{m} \cot \left (b x + a\right )^{2} \csc \left (b x + a\right ), x\right ) \]

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

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)

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maple [A]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \left (\cot ^{2}\left (b x +a \right )\right ) \csc \left (b x +a \right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{m} \cot \left (b x + a\right )^{2} \csc \left (b x + a\right )\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\mathrm {cot}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^m}{\sin \left (a+b\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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