3.1.14 \(\int \frac {1}{(c+d x) (a+a \sec (e+f x))} \, dx\) [14]

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{(c+d x) (a+a \sec (e+f x))},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, 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 = {} \begin {gather*} \int \frac {1}{(c+d x) (a+a \sec (e+f x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 7.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c+d x) (a+a \sec (e+f x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.30, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d x +c \right ) \left (a +a \sec \left (f x +e \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{c \sec {\left (e + f x \right )} + c + d x \sec {\left (e + f x \right )} + d x}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )\,\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + a/cos(e + f*x))*(c + d*x)),x)

[Out]

int(1/((a + a/cos(e + f*x))*(c + d*x)), x)

________________________________________________________________________________________