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

Optimal. Leaf size=24 \[ b \text {Int}\left (\frac {\sec \left (c+d x^2\right )}{x^2},x\right )-\frac {a}{x} \]

[Out]

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

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Rubi [A]  time = 0.01, 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 = {} \[ \int \frac {a+b \sec \left (c+d x^2\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sec[c + d*x^2])/x^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {a+b \sec \left (c+d x^2\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \sec \left (d x^{2} + c\right ) + a}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {a +b \sec \left (d \,x^{2}+c \right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x^2+c))/x^2,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^2*co
s(2*d*x^2 + 2*c)^2 + x^2*sin(2*d*x^2 + 2*c)^2 + 2*x^2*cos(2*d*x^2 + 2*c) + x^2), x) - a/x

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {a+\frac {b}{\cos \left (d\,x^2+c\right )}}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cos(c + d*x^2))/x^2,x)

[Out]

int((a + b/cos(c + d*x^2))/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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