3.20 \(\int \frac{1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx\)

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{1}{(c+d x)^2 (a \sec (e+f x)+a)^2},x\right ) \]

[Out]

Unintegrable[1/((c + d*x)^2*(a + a*Sec[e + f*x])^2), x]

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Rubi [A]  time = 0.0518937, 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 = {} \[ \int \frac{1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c + d*x)^2*(a + a*Sec[e + f*x])^2),x]

[Out]

Defer[Int][1/((c + d*x)^2*(a + a*Sec[e + f*x])^2), x]

Rubi steps

\begin{align*} \int \frac{1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx &=\int \frac{1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx\\ \end{align*}

Mathematica [A]  time = 18.0891, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((c + d*x)^2*(a + a*Sec[e + f*x])^2),x]

[Out]

Integrate[1/((c + d*x)^2*(a + a*Sec[e + f*x])^2), x]

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Maple [A]  time = 2.818, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) ^{2} \left ( a+a\sec \left ( fx+e \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^2/(a+a*sec(f*x+e))^2,x)

[Out]

int(1/(d*x+c)^2/(a+a*sec(f*x+e))^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(1/(d*x+c)^2/(a+a*sec(f*x+e))^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{1}{a^{2} d^{2} x^{2} + 2 \, a^{2} c d x + a^{2} c^{2} +{\left (a^{2} d^{2} x^{2} + 2 \, a^{2} c d x + a^{2} c^{2}\right )} \sec \left (f x + e\right )^{2} + 2 \,{\left (a^{2} d^{2} x^{2} + 2 \, a^{2} c d x + a^{2} c^{2}\right )} \sec \left (f x + e\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+a*sec(f*x+e))^2,x, algorithm="fricas")

[Out]

integral(1/(a^2*d^2*x^2 + 2*a^2*c*d*x + a^2*c^2 + (a^2*d^2*x^2 + 2*a^2*c*d*x + a^2*c^2)*sec(f*x + e)^2 + 2*(a^
2*d^2*x^2 + 2*a^2*c*d*x + a^2*c^2)*sec(f*x + e)), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{c^{2} \sec ^{2}{\left (e + f x \right )} + 2 c^{2} \sec{\left (e + f x \right )} + c^{2} + 2 c d x \sec ^{2}{\left (e + f x \right )} + 4 c d x \sec{\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \sec ^{2}{\left (e + f x \right )} + 2 d^{2} x^{2} \sec{\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**2/(a+a*sec(f*x+e))**2,x)

[Out]

Integral(1/(c**2*sec(e + f*x)**2 + 2*c**2*sec(e + f*x) + c**2 + 2*c*d*x*sec(e + f*x)**2 + 4*c*d*x*sec(e + f*x)
+ 2*c*d*x + d**2*x**2*sec(e + f*x)**2 + 2*d**2*x**2*sec(e + f*x) + d**2*x**2), x)/a**2

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x + c\right )}^{2}{\left (a \sec \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+a*sec(f*x+e))^2,x, algorithm="giac")

[Out]

integrate(1/((d*x + c)^2*(a*sec(f*x + e) + a)^2), x)