3.29 \(\int \frac {a+b \tan (c+d \sqrt {x})}{x} \, dx\)

Optimal. Leaf size=24 \[ b \text {Int}\left (\frac {\tan \left (c+d \sqrt {x}\right )}{x},x\right )+a \log (x) \]

[Out]

a*ln(x)+b*Unintegrable(tan(c+d*x^(1/2))/x,x)

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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 = {} \[ \int \frac {a+b \tan \left (c+d \sqrt {x}\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Tan[c + d*Sqrt[x]])/x,x]

[Out]

a*Log[x] + b*Defer[Int][Tan[c + d*Sqrt[x]]/x, x]

Rubi steps

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

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Mathematica [A]  time = 2.62, size = 0, normalized size = 0.00 \[ \int \frac {a+b \tan \left (c+d \sqrt {x}\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Tan[c + d*Sqrt[x]])/x,x]

[Out]

Integrate[(a + b*Tan[c + d*Sqrt[x]])/x, x]

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fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \tan \left (d \sqrt {x} + c\right ) + a}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(c+d*x^(1/2)))/x,x, algorithm="fricas")

[Out]

integral((b*tan(d*sqrt(x) + c) + a)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(c+d*x^(1/2)))/x,x, algorithm="giac")

[Out]

integrate((b*tan(d*sqrt(x) + c) + a)/x, x)

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maple [A]  time = 0.93, size = 0, normalized size = 0.00 \[ \int \frac {a +b \tan \left (c +d \sqrt {x}\right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(c+d*x^(1/2)))/x,x)

[Out]

int((a+b*tan(c+d*x^(1/2)))/x,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, b \int \frac {\sin \left (2 \, d \sqrt {x} + 2 \, c\right )}{{\left (\cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + 1\right )} x}\,{d x} + a \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(c+d*x^(1/2)))/x,x, algorithm="maxima")

[Out]

2*b*integrate(sin(2*d*sqrt(x) + 2*c)/((cos(2*d*sqrt(x) + 2*c)^2 + sin(2*d*sqrt(x) + 2*c)^2 + 2*cos(2*d*sqrt(x)
+ 2*c) + 1)*x), x) + a*log(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tan(c + d*x^(1/2)))/x,x)

[Out]

int((a + b*tan(c + d*x^(1/2)))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(c+d*x**(1/2)))/x,x)

[Out]

Integral((a + b*tan(c + d*sqrt(x)))/x, x)

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