3.34 \(\int \frac{(a+b \tan (c+d \sqrt{x}))^2}{x} \, dx\)

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

[Out]

Unintegrable[(a + b*Tan[c + d*Sqrt[x]])^2/x, x]

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Rubi [A]  time = 0.0220278, 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 = {} \[ \int \frac{\left (a+b \tan \left (c+d \sqrt{x}\right )\right )^2}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Tan[c + d*Sqrt[x]])^2/x,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 21.5438, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \tan \left (c+d \sqrt{x}\right )\right )^2}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.247, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( a+b\tan \left ( c+d\sqrt{x} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{4 \, b^{2} \sqrt{x} \sin \left (2 \, d \sqrt{x} + 2 \, c\right ) + 2 \,{\left (d \cos \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt{x} + 2 \, c\right ) + d\right )} x \int \frac{2 \, a b d x \sin \left (2 \, d \sqrt{x} + 2 \, c\right ) + b^{2} \sqrt{x} \sin \left (2 \, d \sqrt{x} + 2 \, c\right )}{{\left (d \cos \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt{x} + 2 \, c\right ) + d\right )} x^{2}}\,{d x} +{\left ({\left (a^{2} - b^{2}\right )} d \cos \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} +{\left (a^{2} - b^{2}\right )} d \sin \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} + 2 \,{\left (a^{2} - b^{2}\right )} d \cos \left (2 \, d \sqrt{x} + 2 \, c\right ) +{\left (a^{2} - b^{2}\right )} d\right )} x \log \left (x\right )}{{\left (d \cos \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt{x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt{x} + 2 \, c\right ) + d\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \tan \left (d \sqrt{x} + c\right )^{2} + 2 \, a b \tan \left (d \sqrt{x} + c\right ) + a^{2}}{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)))^2/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \tan{\left (c + d \sqrt{x} \right )}\right )^{2}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \tan \left (d \sqrt{x} + c\right ) + a\right )}^{2}}{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)))^2/x,x, algorithm="giac")

[Out]

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