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

Optimal. Leaf size=23 \[ 4 \text{Unintegrable}\left (\frac{\csc ^2(2 a+2 b x)}{(c+d x)^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0857983, 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 = {} \[ \int \frac{\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx &=4 \int \frac{\csc ^2(2 a+2 b x)}{(c+d x)^2} \, dx\\ \end{align*}

Mathematica [A]  time = 7.58574, size = 0, normalized size = 0. \[ \int \frac{\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.811, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \csc \left ( bx+a \right ) \right ) ^{2} \left ( \sec \left ( bx+a \right ) \right ) ^{2}}{ \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)^2/(d*x+c)^2,x)

[Out]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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