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

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

[Out]

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

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Rubi [A]  time = 0.0512894, 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 = {} \[ \int \frac{(a+a \sec (e+f x))^2}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [F]  time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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Maple [A]  time = 1.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+a\sec \left ( fx+e \right ) \right ) ^{2}}{dx+c}}\, 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),x)

[Out]

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

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

[Out]

((a^2*d*f*x + a^2*c*f)*cos(2*f*x + 2*e)^2*log(d*x + c) + 2*a^2*d*sin(2*f*x + 2*e) + (a^2*d*f*x + a^2*c*f)*log(
d*x + c)*sin(2*f*x + 2*e)^2 + 2*(a^2*d*f*x + a^2*c*f)*cos(2*f*x + 2*e)*log(d*x + c) + (d^2*f*x + c*d*f + (d^2*
f*x + c*d*f)*cos(2*f*x + 2*e)^2 + (d^2*f*x + c*d*f)*sin(2*f*x + 2*e)^2 + 2*(d^2*f*x + c*d*f)*cos(2*f*x + 2*e))
*integrate(2*(2*(a^2*d*f*x + a^2*c*f)*cos(2*f*x + 2*e)*cos(f*x + e) + 2*(a^2*d*f*x + a^2*c*f)*cos(f*x + e) + (
a^2*d + 2*(a^2*d*f*x + a^2*c*f)*sin(f*x + e))*sin(2*f*x + 2*e))/(d^2*f*x^2 + 2*c*d*f*x + c^2*f + (d^2*f*x^2 +
2*c*d*f*x + c^2*f)*cos(2*f*x + 2*e)^2 + (d^2*f*x^2 + 2*c*d*f*x + c^2*f)*sin(2*f*x + 2*e)^2 + 2*(d^2*f*x^2 + 2*
c*d*f*x + c^2*f)*cos(2*f*x + 2*e)), x) + (a^2*d*f*x + a^2*c*f)*log(d*x + c))/(d^2*f*x + c*d*f + (d^2*f*x + c*d
*f)*cos(2*f*x + 2*e)^2 + (d^2*f*x + c*d*f)*sin(2*f*x + 2*e)^2 + 2*(d^2*f*x + c*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 x + c}, 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),x, algorithm="fricas")

[Out]

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

[Out]

a**2*(Integral(2*sec(e + f*x)/(c + d*x), x) + Integral(sec(e + f*x)**2/(c + d*x), x) + Integral(1/(c + d*x), 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}}{d x + c}\,{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),x, algorithm="giac")

[Out]

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