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

Optimal. Leaf size=23 \[ b \text{Unintegrable}\left (\frac{\tan \left (c+d \sqrt{x}\right )}{x},x\right )+a \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0185292, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 2.43751, size = 0, normalized size = 0. \[ \int \frac{a+b \tan \left (c+d \sqrt{x}\right )}{x} \, dx \]

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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Maple [A]  time = 0.14, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( a+b\tan \left ( c+d\sqrt{x} \right ) \right ) }\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} 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 \left (x\right ) \end{align*}

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \tan \left (d \sqrt{x} + c\right ) + a}{x}, x\right ) \end{align*}

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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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \tan{\left (c + d \sqrt{x} \right )}}{x}\, dx \end{align*}

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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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \tan \left (d \sqrt{x} + c\right ) + a}{x}\,{d x} \end{align*}

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)