3.1205 \(\int \frac {a+b \tan ^{-1}(c x)}{x \sqrt {d+e x^2}} \, dx\)

Optimal. Leaf size=51 \[ b \text {Int}\left (\frac {\tan ^{-1}(c x)}{x \sqrt {d+e x^2}},x\right )-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{\sqrt {d}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.16, 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 = {} \[ \int \frac {a+b \tan ^{-1}(c x)}{x \sqrt {d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 5.78, size = 0, normalized size = 0.00 \[ \int \frac {a+b \tan ^{-1}(c x)}{x \sqrt {d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )}}{e x^{3} + d x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(e*x^2 + d)*(b*arctan(c*x) + a)/(e*x^3 + d*x), x)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 1.14, size = 0, normalized size = 0.00 \[ \int \frac {a +b \arctan \left (c x \right )}{x \sqrt {e \,x^{2}+d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x\,\sqrt {e\,x^2+d}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________