3.6 \(\int \frac {a+b \sec (c+d x^2)}{x} \, dx\)

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

[Out]

a*ln(x)+b*Unintegrable(sec(d*x^2+c)/x,x)

________________________________________________________________________________________

Rubi [A]  time = 0.02, 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 = {} \[ \int \frac {a+b \sec \left (c+d x^2\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sec[c + d*x^2])/x,x]

[Out]

a*Log[x] + b*Defer[Int][Sec[c + d*x^2]/x, x]

Rubi steps

\begin {align*} \int \frac {a+b \sec \left (c+d x^2\right )}{x} \, dx &=\int \left (\frac {a}{x}+\frac {b \sec \left (c+d x^2\right )}{x}\right ) \, dx\\ &=a \log (x)+b \int \frac {\sec \left (c+d x^2\right )}{x} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.77, size = 0, normalized size = 0.00 \[ \int \frac {a+b \sec \left (c+d x^2\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Sec[c + d*x^2])/x,x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \sec \left (d x^{2} + c\right ) + a}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*sec(d*x^2 + c) + a)/x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.76, size = 0, normalized size = 0.00 \[ \int \frac {a +b \sec \left (d \,x^{2}+c \right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, b \int \frac {\cos \left (2 \, d x^{2} + 2 \, c\right ) \cos \left (d x^{2} + c\right ) + \sin \left (2 \, d x^{2} + 2 \, c\right ) \sin \left (d x^{2} + c\right ) + \cos \left (d x^{2} + c\right )}{x \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + x \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, x \cos \left (2 \, d x^{2} + 2 \, c\right ) + x}\,{d x} + a \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*integrate((cos(2*d*x^2 + 2*c)*cos(d*x^2 + c) + sin(2*d*x^2 + 2*c)*sin(d*x^2 + c) + cos(d*x^2 + c))/(x*cos(
2*d*x^2 + 2*c)^2 + x*sin(2*d*x^2 + 2*c)^2 + 2*x*cos(2*d*x^2 + 2*c) + x), x) + a*log(x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cos(c + d*x^2))/x,x)

[Out]

int((a + b/cos(c + d*x^2))/x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*sec(c + d*x**2))/x, x)

________________________________________________________________________________________