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

Optimal. Leaf size=24 \[ \text{CannotIntegrate}\left (\frac{\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.189883, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 11.9084, size = 0, normalized size = 0. \[ \int \frac{\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 2.649, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \csc \left ( bx+a \right ) \right ) ^{2}\sec \left ( bx+a \right ) }{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\csc \left (b x + a\right )^{2} \sec \left (b x + a\right )}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{2}{\left (a + b x \right )} \sec{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x