3.1.37 \(\int \frac {1}{(c+d x) (a+b \sec (e+f x))} \, dx\) [37]

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

[Out]

Unintegrable(1/(d*x+c)/(a+b*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+b \sec (e+f x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 1.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c+d x) (a+b \sec (e+f x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d x +c \right ) \left (a +b \sec \left (f x +e \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

________________________________________________________________________________________

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+b*sec(f*x+e)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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+b*sec(f*x+e)),x, algorithm="giac")

[Out]

integrate(1/((d*x + c)*(b*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 {b}{\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 + b/cos(e + f*x))*(c + d*x)),x)

[Out]

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

________________________________________________________________________________________