\(\int \frac {(c+a^2 c x^2)^{3/2} \arctan (a x)}{x^3} \, dx\) [214]

Optimal result

Integrand size = 22, antiderivative size = 304 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^3} \, dx=-\frac {a c \sqrt {c+a^2 c x^2}}{2 x}+a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)}{2 x^2}-\frac {3 a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-a^2 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}} \]

[Out]

-a^2*c^(3/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))-3*a^2*c^2*arctan(a*x)*arctanh((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)+3/2*I*a^2*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)-3/2*I*a^2*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)-1/2*a*c*(a^2*c*x^2+c)^(1/2)/x+a^2*c*arctan(a*x)*(a^2*c*x^2+c)^(1/2)-1/2*c*arctan(a*x)*(a^2*c*x
^2+c)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5070, 5066, 5082, 270, 5078, 5074, 223, 212} \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^3} \, dx=-\frac {3 a^2 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}+a^2 c \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {c \arctan (a x) \sqrt {a^2 c x^2+c}}{2 x^2}-a^2 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )+\frac {3 i a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {3 i a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {a c \sqrt {a^2 c x^2+c}}{2 x} \]

[In]

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

[Out]

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

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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -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]

Rule 5082

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((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])^p/(d*f*(m + 1))), x] + (-Dist[b*c*(p/(f*(m + 1))), Int[(f*x
)^(m + 1)*((a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Dist[c^2*((m + 2)/(f^2*(m + 1))), Int[(f*x)^
(m + 2)*((a + b*ArcTan[c*x])^p/Sqrt[d + e*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && G
tQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]

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

Mathematica [A] (verified)

Time = 1.87 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.99 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^3} \, dx=\frac {a^2 c \sqrt {c+a^2 c x^2} \left (-2-2 \cot ^2\left (\frac {1}{2} \arctan (a x)\right )+4 a x \arctan (a x) \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-\arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+12 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1-e^{i \arctan (a x)}\right )-12 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1+e^{i \arctan (a x)}\right )+8 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )-\sin \left (\frac {1}{2} \arctan (a x)\right )\right )-8 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )+\sin \left (\frac {1}{2} \arctan (a x)\right )\right )+12 i \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-12 i \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+\arctan (a x) \csc \left (\frac {1}{2} \arctan (a x)\right ) \sec \left (\frac {1}{2} \arctan (a x)\right )\right ) \tan \left (\frac {1}{2} \arctan (a x)\right )}{8 \sqrt {1+a^2 x^2}} \]

[In]

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

[Out]

(a^2*c*Sqrt[c + a^2*c*x^2]*(-2 - 2*Cot[ArcTan[a*x]/2]^2 + 4*a*x*ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 - ArcTan[a*x]
*Cot[ArcTan[a*x]/2]*Csc[ArcTan[a*x]/2]^2 + 12*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 - E^(I*ArcTan[a*x])] - 12*A
rcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 + E^(I*ArcTan[a*x])] + 8*Cot[ArcTan[a*x]/2]*Log[Cos[ArcTan[a*x]/2] - Sin[A
rcTan[a*x]/2]] - 8*Cot[ArcTan[a*x]/2]*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] + (12*I)*Cot[ArcTan[a*x]/2]
*PolyLog[2, -E^(I*ArcTan[a*x])] - (12*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, E^(I*ArcTan[a*x])] + ArcTan[a*x]*Csc[Ar
cTan[a*x]/2]*Sec[ArcTan[a*x]/2])*Tan[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2])

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.69

method	result	size
default	\(-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (3 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right ) \arctan \left (a x \right ) a^{2} x^{2}-3 i \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) a^{2} x^{2}-3 i \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right ) a^{2} x^{2}-4 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) a^{2} x^{2}-2 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+\sqrt {a^{2} x^{2}+1}\, a x +\arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\right ) c}{2 \sqrt {a^{2} x^{2}+1}\, x^{2}}\)	\(211\)

[In]

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

[Out]

-1/2/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)*(3*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)*arctan(a*x)*a^2*x^2-3*I*
dilog((1+I*a*x)/(a^2*x^2+1)^(1/2))*a^2*x^2-3*I*dilog((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)*a^2*x^2-4*I*arctan((1+I*a*
x)/(a^2*x^2+1)^(1/2))*a^2*x^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+arctan(a*x)*(a^2*x
^2+1)^(1/2))*c/x^2

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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