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

Optimal. Leaf size=68 \[ b \text {Int}\left (\frac {\tan ^{-1}(c x) \sqrt {d+e x^2}}{x^2},x\right )-\frac {a \sqrt {d+e x^2}}{x}+a \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.15, 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 = {} \[ \int \frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx \]

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*}

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Mathematica [A]  time = 8.95, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx \]

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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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{2}}, x\right ) \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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maple [A]  time = 1.16, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e \,x^{2}+d}\, \left (a +b \arctan \left (c x \right )\right )}{x^{2}}\, dx \]

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 [A]  time = 0.00, size = 0, normalized size = 0.00 \[ {\left (\sqrt {e} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right ) - \frac {\sqrt {e x^{2} + d}}{x}\right )} a + b \int \frac {\sqrt {e x^{2} + d} \arctan \left (c x\right )}{x^{2}}\,{d x} \]

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]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\sqrt {e\,x^2+d}}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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