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

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

[Out]

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

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Rubi [A]  time = 0.0480409, 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 = {} \[ \int \frac{(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 28.806, size = 0, normalized size = 0. \[ \int \frac{(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.433, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+a\sec \left ( fx+e \right ) \right ) ^{2}}{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{2} d f x + a^{2} c f - 2 \, a^{2} d \sin \left (2 \, f x + 2 \, e\right ) +{\left (a^{2} d f x + a^{2} c f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} +{\left (a^{2} d f x + a^{2} c f\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \,{\left (a^{2} d f x + a^{2} c f\right )} \cos \left (2 \, f x + 2 \, e\right ) - 4 \,{\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f +{\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} +{\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \,{\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \cos \left (2 \, f x + 2 \, e\right )\right )} \int \frac{{\left (a^{2} d f x + a^{2} c f\right )} \cos \left (2 \, f x + 2 \, e\right ) \cos \left (f x + e\right ) +{\left (a^{2} d f x + a^{2} c f\right )} \cos \left (f x + e\right ) +{\left (a^{2} d +{\left (a^{2} d f x + a^{2} c f\right )} \sin \left (f x + e\right )\right )} \sin \left (2 \, f x + 2 \, e\right )}{d^{3} f x^{3} + 3 \, c d^{2} f x^{2} + 3 \, c^{2} d f x + c^{3} f +{\left (d^{3} f x^{3} + 3 \, c d^{2} f x^{2} + 3 \, c^{2} d f x + c^{3} f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} +{\left (d^{3} f x^{3} + 3 \, c d^{2} f x^{2} + 3 \, c^{2} d f x + c^{3} f\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \,{\left (d^{3} f x^{3} + 3 \, c d^{2} f x^{2} + 3 \, c^{2} d f x + c^{3} f\right )} \cos \left (2 \, f x + 2 \, e\right )}\,{d x}}{d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f +{\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} +{\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \,{\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \cos \left (2 \, f x + 2 \, e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^2*d*f*x + a^2*c*f - 2*a^2*d*sin(2*f*x + 2*e) + (a^2*d*f*x + a^2*c*f)*cos(2*f*x + 2*e)^2 + (a^2*d*f*x + a^2
*c*f)*sin(2*f*x + 2*e)^2 + 2*(a^2*d*f*x + a^2*c*f)*cos(2*f*x + 2*e) - (d^3*f*x^2 + 2*c*d^2*f*x + c^2*d*f + (d^
3*f*x^2 + 2*c*d^2*f*x + c^2*d*f)*cos(2*f*x + 2*e)^2 + (d^3*f*x^2 + 2*c*d^2*f*x + c^2*d*f)*sin(2*f*x + 2*e)^2 +
2*(d^3*f*x^2 + 2*c*d^2*f*x + c^2*d*f)*cos(2*f*x + 2*e))*integrate(4*((a^2*d*f*x + a^2*c*f)*cos(2*f*x + 2*e)*c
os(f*x + e) + (a^2*d*f*x + a^2*c*f)*cos(f*x + e) + (a^2*d + (a^2*d*f*x + a^2*c*f)*sin(f*x + e))*sin(2*f*x + 2*
e))/(d^3*f*x^3 + 3*c*d^2*f*x^2 + 3*c^2*d*f*x + c^3*f + (d^3*f*x^3 + 3*c*d^2*f*x^2 + 3*c^2*d*f*x + c^3*f)*cos(2
*f*x + 2*e)^2 + (d^3*f*x^3 + 3*c*d^2*f*x^2 + 3*c^2*d*f*x + c^3*f)*sin(2*f*x + 2*e)^2 + 2*(d^3*f*x^3 + 3*c*d^2*
f*x^2 + 3*c^2*d*f*x + c^3*f)*cos(2*f*x + 2*e)), x))/(d^3*f*x^2 + 2*c*d^2*f*x + c^2*d*f + (d^3*f*x^2 + 2*c*d^2*
f*x + c^2*d*f)*cos(2*f*x + 2*e)^2 + (d^3*f*x^2 + 2*c*d^2*f*x + c^2*d*f)*sin(2*f*x + 2*e)^2 + 2*(d^3*f*x^2 + 2*
c*d^2*f*x + c^2*d*f)*cos(2*f*x + 2*e))

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{2} \sec \left (f x + e\right )^{2} + 2 \, a^{2} \sec \left (f x + e\right ) + a^{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((a+a*sec(f*x+e))^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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