\(\int \frac {(c+a^2 c x^2)^{3/2} \arctan (a x)}{x} \, dx\) [212]

Optimal result

Integrand size = 22, antiderivative size = 281 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=-\frac {1}{6} a c x \sqrt {c+a^2 c x^2}+c \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7}{6} c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}} \]

[Out]

1/3*(a^2*c*x^2+c)^(3/2)*arctan(a*x)-7/6*c^(3/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))-2*c^2*arctan(a*x)*arc
tanh((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*c^2*polylog(2,-(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*c^2*polylog(2,(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)-1/6*a*c*x*(a^2*c*x^2+c)^(1/2)+c*arctan(a*x)*(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5070, 5066, 5078, 5074, 223, 212, 5050, 201} \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=-\frac {2 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}+c \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {1}{3} \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}-\frac {7}{6} c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )+\frac {i c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {i c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {1}{6} a c x \sqrt {a^2 c x^2+c} \]

[In]

Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x,x]

[Out]

-1/6*(a*c*x*Sqrt[c + a^2*c*x^2]) + c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/3 -
(2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (7*c^(3/
2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/6 + (I*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[
1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (I*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt
[c + a^2*c*x^2]

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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]

Rule 5066

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

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]))

Rule 5074

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

Rule 5078

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + c^2*
x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/(x*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]

Rubi steps \begin{align*} \text {integral}& = c \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x} \, dx+\left (a^2 c\right ) \int x \sqrt {c+a^2 c x^2} \arctan (a x) \, dx \\ & = c \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {1}{3} (a c) \int \sqrt {c+a^2 c x^2} \, dx+c^2 \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx-\left (a c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {1}{6} a c x \sqrt {c+a^2 c x^2}+c \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {1}{6} \left (a c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx-\left (a c^2\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{6} a c x \sqrt {c+a^2 c x^2}+c \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {1}{6} \left (a c^2\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right ) \\ & = -\frac {1}{6} a c x \sqrt {c+a^2 c x^2}+c \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{3} \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7}{6} c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.83 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=\frac {c \sqrt {c+a^2 c x^2} \left (-a x \sqrt {1+a^2 x^2}+8 \sqrt {1+a^2 x^2} \arctan (a x)+2 a^2 x^2 \sqrt {1+a^2 x^2} \arctan (a x)+6 \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-6 \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )+\log \left (-a x+\sqrt {1+a^2 x^2}\right )+6 \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )-\sin \left (\frac {1}{2} \arctan (a x)\right )\right )-6 \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )+\sin \left (\frac {1}{2} \arctan (a x)\right )\right )+6 i \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-6 i \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )\right )}{6 \sqrt {1+a^2 x^2}} \]

[In]

Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x,x]

[Out]

(c*Sqrt[c + a^2*c*x^2]*(-(a*x*Sqrt[1 + a^2*x^2]) + 8*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 2*a^2*x^2*Sqrt[1 + a^2*x^
2]*ArcTan[a*x] + 6*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 6*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + Log[-(a
*x) + Sqrt[1 + a^2*x^2]] + 6*Log[Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]] - 6*Log[Cos[ArcTan[a*x]/2] + Sin[Arc
Tan[a*x]/2]] + (6*I)*PolyLog[2, -E^(I*ArcTan[a*x])] - (6*I)*PolyLog[2, E^(I*ArcTan[a*x])]))/(6*Sqrt[1 + a^2*x^
2])

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.66

method	result	size
default	\(-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (-2 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+\sqrt {a^{2} x^{2}+1}\, a x +6 \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-8 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}-14 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-6 i \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) c}{6 \sqrt {a^{2} x^{2}+1}}\)	\(185\)

[In]

int((a^2*c*x^2+c)^(3/2)*arctan(a*x)/x,x,method=_RETURNVERBOSE)

[Out]

-1/6/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)*(-2*arctan(a*x)*(a^2*x^2+1)^(1/2)*a^2*x^2+(a^2*x^2+1)^(1/2)*a
*x+6*arctan(a*x)*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-8*arctan(a*x)*(a^2*x^2+1)^(1/2)-14*I*arctan((1+I*a*x)/(a^2*
x^2+1)^(1/2))-6*I*dilog((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-6*I*dilog((1+I*a*x)/(a^2*x^2+1)^(1/2)))*c

Fricas [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )}{x} \,d x } \]

[In]

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)/x,x, algorithm="fricas")

[Out]

integral((a^2*c*x^2 + c)^(3/2)*arctan(a*x)/x, x)

Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}{\left (a x \right )}}{x}\, dx \]

[In]

integrate((a**2*c*x**2+c)**(3/2)*atan(a*x)/x,x)

[Out]

Integral((c*(a**2*x**2 + 1))**(3/2)*atan(a*x)/x, x)

Maxima [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )}{x} \,d x } \]

[In]

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)/x,x, algorithm="maxima")

[Out]

1/3*(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1)*sqrt(c)*arctan(a*x) - 1/6*(a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*(a*c*x*cos(1/
2*arctan2(4*a*x, -a^2*x^2 + 3)) + 2*c*sin(1/2*arctan2(4*a*x, -a^2*x^2 + 3)))*sqrt(c) + 1/12*(c*arctan2((a^4*x^
4 + 10*a^2*x^2 + 9)^(1/4)*sin(1/2*arctan2(4*a*x, a^2*x^2 - 3)) + 2, a*x + (a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*cos
(1/2*arctan2(4*a*x, a^2*x^2 - 3))) + c*arctan2((a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*sin(1/2*arctan2(4*a*x, a^2*x^2
- 3)) - 2, -a*x + (a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*cos(1/2*arctan2(4*a*x, a^2*x^2 - 3))) + 12*c*integrate(sqr
t(a^2*x^2 + 1)*arctan(a*x)/x, x))*sqrt(c)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)/x,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. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{x} \,d x \]

[In]

int((atan(a*x)*(c + a^2*c*x^2)^(3/2))/x,x)

[Out]

int((atan(a*x)*(c + a^2*c*x^2)^(3/2))/x, x)