3.51 \(\int \frac {\cot (a+b x) \csc ^2(a+b x)}{(c+d x)^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.17, 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 = {} \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 10.64, size = 0, normalized size = 0.00 \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b x + a\right ) \csc \left (b x + a\right )^{3}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*csc(b*x+a)^3/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral(cos(b*x + a)*csc(b*x + a)^3/(d^2*x^2 + 2*c*d*x + c^2), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*csc(b*x+a)^3/(d*x+c)^2,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.54, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x +a \right ) \left (\csc ^{3}\left (b x +a \right )\right )}{\left (d x +c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)*csc(b*x+a)^3/(d*x+c)^2,x)

[Out]

int(cos(b*x+a)*csc(b*x+a)^3/(d*x+c)^2,x)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*csc(b*x+a)^3/(d*x+c)^2,x, algorithm="maxima")

[Out]

Timed out

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\cos \left (a+b\,x\right )}{{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)/(sin(a + b*x)^3*(c + d*x)^2),x)

[Out]

int(cos(a + b*x)/(sin(a + b*x)^3*(c + d*x)^2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (a + b x \right )} \csc ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*csc(b*x+a)**3/(d*x+c)**2,x)

[Out]

Integral(cos(a + b*x)*csc(a + b*x)**3/(c + d*x)**2, x)

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