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

Optimal. Leaf size=25 \[ \text {Int}\left (\cot (a+b x) \csc ^2(a+b x) (c+d x)^m,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.22, 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 = {} \[ \int (c+d x)^m \cot (a+b x) \csc ^2(a+b x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 6.31, size = 0, normalized size = 0.00 \[ \int (c+d x)^m \cot (a+b x) \csc ^2(a+b x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \csc \left (b x + a\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*cos(b*x+a)*csc(b*x+a)^3,x, algorithm="fricas")

[Out]

integral((d*x + c)^m*cos(b*x + a)*csc(b*x + a)^3, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \csc \left (b x + a\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*cos(b*x+a)*csc(b*x+a)^3,x, algorithm="giac")

[Out]

integrate((d*x + c)^m*cos(b*x + a)*csc(b*x + a)^3, x)

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maple [A]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \cos \left (b x +a \right ) \left (\csc ^{3}\left (b x +a \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^m*cos(b*x+a)*csc(b*x+a)^3,x)

[Out]

int((d*x+c)^m*cos(b*x+a)*csc(b*x+a)^3,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \csc \left (b x + a\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*cos(b*x+a)*csc(b*x+a)^3,x, algorithm="maxima")

[Out]

integrate((d*x + c)^m*cos(b*x + a)*csc(b*x + a)^3, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^m}{{\sin \left (a+b\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(a + b*x)*(c + d*x)^m)/sin(a + b*x)^3,x)

[Out]

int((cos(a + b*x)*(c + d*x)^m)/sin(a + b*x)^3, x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**m*cos(b*x+a)*csc(b*x+a)**3,x)

[Out]

Exception raised: HeuristicGCDFailed

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