\(\int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x^2} \, dx\) [205]

Optimal result

Integrand size = 22, antiderivative size = 242 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x^2} \, dx=-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {2 i a c \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {i a c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}} \]

[Out]

-a*arctanh((a^2*c*x^2+c)^(1/2)/c^(1/2))*c^(1/2)-2*I*a*c*arctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a
^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+I*a*c*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2
*c*x^2+c)^(1/2)-I*a*c*polylog(2,I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-arcta
n(a*x)*(a^2*c*x^2+c)^(1/2)/x

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5070, 5064, 272, 65, 214, 5010, 5006} \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x^2} \, dx=-\frac {2 i a c \sqrt {a^2 x^2+1} \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \arctan (a x)}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x}-a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )+\frac {i a c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {i a c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}} \]

[In]

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

[Out]

-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x) - ((2*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[
1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - a*Sqrt[c]*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + (I*a*c*Sqrt[1 + a^2*x^2]*P
olyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (I*a*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (
I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5006

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[-2*I*(a + b*ArcTan[c*x])*(
ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d])), x] + (Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 + I*c*x]/Sqrt[1
- I*c*x])]/(c*Sqrt[d])), x] - Simp[I*b*(PolyLog[2, I*(Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x])]/(c*Sqrt[d])), x]) /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]

Rule 5010

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

Rule 5064

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Dist[b*c*(p/(f*(m + 1))), Int[(
f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,
c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]

Rule 5070

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

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

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x^2} \, dx=-\frac {a \sqrt {c \left (1+a^2 x^2\right )} \left (\frac {\sqrt {1+a^2 x^2} \arctan (a x)}{a x}-\arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )+\arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )+\log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )\right )-\log \left (\sin \left (\frac {1}{2} \arctan (a x)\right )\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}} \]

[In]

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

[Out]

-((a*Sqrt[c*(1 + a^2*x^2)]*((Sqrt[1 + a^2*x^2]*ArcTan[a*x])/(a*x) - ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] +
ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] + Log[Cos[ArcTan[a*x]/2]] - Log[Sin[ArcTan[a*x]/2]] - I*PolyLog[2, (
-I)*E^(I*ArcTan[a*x])] + I*PolyLog[2, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 929 vs. \(2 (199 ) = 398\).

Time = 0.60 (sec) , antiderivative size = 930, normalized size of antiderivative = 3.84

method	result	size
default	\(\frac {\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right ) \left (i a^{2} x^{2}+2 a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{4 \sqrt {a^{2} x^{2}+1}\, x}-\frac {\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right ) \left (i a^{2} x^{2}+2 a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{4 \sqrt {a^{2} x^{2}+1}\, x}+\frac {\operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \left (a^{2} x^{2}-2 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{4 \sqrt {a^{2} x^{2}+1}\, x}-\frac {\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \left (i a^{2} x^{2}+2 a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{4 \sqrt {a^{2} x^{2}+1}\, x}+\frac {\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \left (i a^{2} x^{2}+2 a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{4 \sqrt {a^{2} x^{2}+1}\, x}-\frac {\operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \left (a^{2} x^{2}-2 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{4 \sqrt {a^{2} x^{2}+1}\, x}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a^{2} x^{2}+2 i a x -1\right ) \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}{4 \sqrt {a^{2} x^{2}+1}\, x}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a^{2} x^{2}-2 a x -i\right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \arctan \left (a x \right )}{4 \sqrt {a^{2} x^{2}+1}\, x}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a^{2} x^{2}-2 a x -i\right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \arctan \left (a x \right )}{4 \sqrt {a^{2} x^{2}+1}\, x}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a^{2} x^{2}+2 i a x -1\right ) \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}{4 \sqrt {a^{2} x^{2}+1}\, x}-\frac {\arctan \left (a x \right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 x}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a^{2} x^{2}-2 a x -i\right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right )}{4 \sqrt {a^{2} x^{2}+1}\, x}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a^{2} x^{2}-2 a x -i\right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{4 \sqrt {a^{2} x^{2}+1}\, x}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \arctan \left (a x \right )}{2 x}\)	\(930\)

[In]

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

[Out]

1/4*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)*(I*a^2*x^2+2*a*x-I)/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)/x-1/4*ln
((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)*(I*a^2*x^2+2*a*x-I)/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)/x+1/4*dilog(1-
I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2-2*I*a*x-1)/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)/x-1/4*arctan(a*
x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(I*a^2*x^2+2*a*x-I)/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)/x+1/4*a
rctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(I*a^2*x^2+2*a*x-I)/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)
/x-1/4*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2-2*I*a*x-1)/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)/
x+1/4/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)*(a^2*x^2+2*I*a*x-1)*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/x
+1/4/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)*(I*a^2*x^2-2*a*x-I)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*arcta
n(a*x)/x-1/4/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)*(I*a^2*x^2-2*a*x-I)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2
))*arctan(a*x)/x-1/4/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)*(a^2*x^2+2*I*a*x-1)*dilog(1-I*(1+I*a*x)/(a^2*
x^2+1)^(1/2))/x-1/2*arctan(a*x)*(1+I*a*x)*(c*(a*x-I)*(I+a*x))^(1/2)/x-1/4/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x)
)^(1/2)*(I*a^2*x^2-2*a*x-I)*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)/x+1/4/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2
)*(I*a^2*x^2-2*a*x-I)*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)/x+1/2*(c*(a*x-I)*(I+a*x))^(1/2)*(I*a*x-1)*arctan(a*x)/
x

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x^2} \, dx=\int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}{\left (a x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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