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

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

[Out]

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

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Rubi [A]  time = 0.14, 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 = {} \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 11.33, size = 0, normalized size = 0.00 \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b x + a\right ) \csc \left (b x + a\right )^{3}}{d x + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x +a \right ) \left (\csc ^{3}\left (b x +a \right )\right )}{d x +c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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