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

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {\cot (a+b x) \csc (a+b x)}{c+d x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.11, 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 = {} \[ \int \frac {\cot (a+b x) \csc (a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 16.76, size = 0, normalized size = 0.00 \[ \int \frac {\cot (a+b x) \csc (a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b x + a\right ) \csc \left (b x + a\right )^{2}}{d x + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x +a \right ) \left (\csc ^{2}\left (b x +a \right )\right )}{d x +c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, \cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 2 \, \cos \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + \frac {{\left (b d^{2} x + b c d + {\left (b d^{2} x + b c d\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (b d^{2} x + b c d\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, {\left (b d^{2} x + b c d\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \int \frac {\sin \left (b x + a\right )}{{\left (d x + c\right )}^{2} {\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right )}}\,{d x}}{b} + \frac {{\left (b d^{2} x + b c d + {\left (b d^{2} x + b c d\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (b d^{2} x + b c d\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, {\left (b d^{2} x + b c d\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \int \frac {\sin \left (b x + a\right )}{{\left (d x + c\right )}^{2} {\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right )}}\,{d x}}{b} + 2 \, \sin \left (b x + a\right )}{b d x + {\left (b d x + b c\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (b d x + b c\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} + b c - 2 \, {\left (b d x + b c\right )} \cos \left (2 \, b x + 2 \, a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b*d^2*x + b*c*d + (b*d^2*x + b*c*d)*cos(2*b*x + 2*a)^2 + (b*d^2*x + b*c*d)*sin(2*b*x + 2*a)^2 - 2*(b*d^2*x
+ b*c*d)*cos(2*b*x + 2*a))*integrate(sin(b*x + a)/(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + (b*d^2*x^2 + 2*b*c*d*x + b*
c^2)*cos(b*x + a)^2 + (b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sin(b*x + a)^2 + 2*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*cos(b
*x + a)), x) + (b*d^2*x + b*c*d + (b*d^2*x + b*c*d)*cos(2*b*x + 2*a)^2 + (b*d^2*x + b*c*d)*sin(2*b*x + 2*a)^2
- 2*(b*d^2*x + b*c*d)*cos(2*b*x + 2*a))*integrate(sin(b*x + a)/(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + (b*d^2*x^2 + 2
*b*c*d*x + b*c^2)*cos(b*x + a)^2 + (b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sin(b*x + a)^2 - 2*(b*d^2*x^2 + 2*b*c*d*x +
b*c^2)*cos(b*x + a)), x) + 2*cos(b*x + a)*sin(2*b*x + 2*a) - 2*cos(2*b*x + 2*a)*sin(b*x + a) + 2*sin(b*x + a)
)/(b*d*x + (b*d*x + b*c)*cos(2*b*x + 2*a)^2 + (b*d*x + b*c)*sin(2*b*x + 2*a)^2 + b*c - 2*(b*d*x + b*c)*cos(2*b
*x + 2*a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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