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

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {a \sec (e+f x)+a}{(c+d x)^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {a+a \sec (e+f x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 6.47, size = 0, normalized size = 0.00 \[ \int \frac {a+a \sec (e+f x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a \sec \left (f x + e\right ) + a}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a \sec \left (f x + e\right ) + a}{{\left (d x + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.23, size = 0, normalized size = 0.00 \[ \int \frac {a +a \sec \left (f x +e \right )}{\left (d x +c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {a+\frac {a}{\cos \left (e+f\,x\right )}}{{\left (c+d\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \frac {\sec {\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {1}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(Integral(sec(e + f*x)/(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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