3.1178 \(\int \frac{\sqrt{d+e x^2} (a+b \tan ^{-1}(c x))}{x^2} \, dx\)

Optimal. Leaf size=67 \[ b \text{Unintegrable}\left (\frac{\tan ^{-1}(c x) \sqrt{d+e x^2}}{x^2},x\right )-\frac{a \sqrt{d+e x^2}}{x}+a \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \]

[Out]

-((a*Sqrt[d + e*x^2])/x) + a*Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + b*Unintegrable[(Sqrt[d + e*x^2]*Ar
cTan[c*x])/x^2, x]

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Rubi [A]  time = 0.146594, 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 = {} \[ \int \frac{\sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-((a*Sqrt[d + e*x^2])/x) + a*Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + b*Defer[Int][(Sqrt[d + e*x^2]*ArcT
an[c*x])/x^2, x]

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx &=a \int \frac{\sqrt{d+e x^2}}{x^2} \, dx+b \int \frac{\sqrt{d+e x^2} \tan ^{-1}(c x)}{x^2} \, dx\\ &=-\frac{a \sqrt{d+e x^2}}{x}+b \int \frac{\sqrt{d+e x^2} \tan ^{-1}(c x)}{x^2} \, dx+(a e) \int \frac{1}{\sqrt{d+e x^2}} \, dx\\ &=-\frac{a \sqrt{d+e x^2}}{x}+b \int \frac{\sqrt{d+e x^2} \tan ^{-1}(c x)}{x^2} \, dx+(a e) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )\\ &=-\frac{a \sqrt{d+e x^2}}{x}+a \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )+b \int \frac{\sqrt{d+e x^2} \tan ^{-1}(c x)}{x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 8.54584, size = 0, normalized size = 0. \[ \int \frac{\sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.758, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arctan \left ( cx \right ) }{{x}^{2}}\sqrt{e{x}^{2}+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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