\(\int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx\) [226]

Optimal result

Integrand size = 20, antiderivative size = 59 \[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^2 \sqrt {c}} \]

[Out]

-arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))/a^2/c^(1/2)+arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^2/c

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5050, 223, 212} \[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}} \]

[In]

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

[Out]

(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^2*c) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^2*Sqrt[c])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 5050

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(d + e*x^2)^(
q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Dist[b*(p/(2*c*(q + 1))), Int[(d + e*x^2)^q*(a + b*ArcTan[c
*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2 c}-\frac {\int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2 c}-\frac {\text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{a} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^2 \sqrt {c}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.02 \[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \arctan (a x)-\sqrt {c} \log \left (a c x+\sqrt {c} \sqrt {c+a^2 c x^2}\right )}{a^2 c} \]

[In]

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

[Out]

(Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - Sqrt[c]*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/(a^2*c)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.40 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.69

method	result	size
default	\(\frac {\left (\arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}-\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right )+\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{2} c}\)	\(100\)

[In]

int(x*arctan(a*x)/(a^2*c*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(arctan(a*x)*(a^2*x^2+1)^(1/2)-ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+I)+ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-I))/(a^2*x^2+1
)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)/a^2/c

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08 \[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\frac {2 \, \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right ) + \sqrt {c} \log \left (-2 \, a^{2} c x^{2} + 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right )}{2 \, a^{2} c} \]

[In]

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

[Out]

1/2*(2*sqrt(a^2*c*x^2 + c)*arctan(a*x) + sqrt(c)*log(-2*a^2*c*x^2 + 2*sqrt(a^2*c*x^2 + c)*a*sqrt(c)*x - c))/(a
^2*c)

Sympy [F(-2)]

Exception generated. \[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.03 \[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\frac {2 \, \sqrt {a^{2} x^{2} + 1} \arctan \left (a x\right ) - \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, a^{2} \sqrt {c}} \]

[In]

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

[Out]

1/2*(2*sqrt(a^2*x^2 + 1)*arctan(a*x) - log(a*x + sqrt(a^2*x^2 + 1)) + log(-a*x + sqrt(a^2*x^2 + 1)))/(a^2*sqrt
(c))

Giac [F]

\[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x \arctan \left (a x\right )}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x\,\mathrm {atan}\left (a\,x\right )}{\sqrt {c\,a^2\,x^2+c}} \,d x \]

[In]

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

[Out]

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