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

3.3.14.1 Optimal result

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

output

-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
 

3.3.14.2 Mathematica [A] (verified)

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

input

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

output

(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*Arc 
Tan[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[ArcTan[a*x]/2]] - 8*Cot[ArcTan[a*x]/2]*Lo 
g[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] + (12*I)*Cot[ArcTan[a*x]/2]*Pol 
yLog[2, -E^(I*ArcTan[a*x])] - (12*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, E^(I*Ar 
cTan[a*x])] + ArcTan[a*x]*Csc[ArcTan[a*x]/2]*Sec[ArcTan[a*x]/2])*Tan[ArcTa 
n[a*x]/2])/(8*Sqrt[1 + a^2*x^2])
 

3.3.14.3 Rubi [A] (verified)

Time = 1.56 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.36, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5485, 5481, 224, 219, 242, 5493, 5489, 5497, 242, 5493, 5489}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5485

\(\displaystyle a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^3}dx\)

\(\Big \downarrow \) 5481

\(\displaystyle a^2 c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-a c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\arctan (a x) \sqrt {a^2 c x^2+c}\right )+c \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx+a c \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}\right )\)

\(\Big \downarrow \) 224

\(\displaystyle a^2 c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-a c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}\right )+c \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx+a c \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle a^2 c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )+c \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx+a c \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}\right )\)

\(\Big \downarrow \) 242

\(\displaystyle a^2 c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )+c \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )\)

\(\Big \downarrow \) 5493

\(\displaystyle a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )+c \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )\)

\(\Big \downarrow \) 5489

\(\displaystyle c \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )\)

\(\Big \downarrow \) 5497

\(\displaystyle c \left (-c \left (-\frac {1}{2} a^2 \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx+\frac {1}{2} a \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}\right )-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )\)

\(\Big \downarrow \) 242

\(\displaystyle c \left (-c \left (-\frac {1}{2} a^2 \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x}\right )-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )\)

\(\Big \downarrow \) 5493

\(\displaystyle c \left (-c \left (-\frac {a^2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x}\right )-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )\)

\(\Big \downarrow \) 5489

\(\displaystyle a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )+c \left (-c \left (-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x}\right )-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )\)

input

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

output

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

3.3.14.3.1 Defintions of rubi rules used

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

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 242

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

rule 5481

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] + (Simp[d/(m + 2)   Int[(f*x)^m*((a + b*ArcTan[c*x])/Sq 
rt[d + e*x^2]), x], x] - Simp[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 5485

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(q_.), x_Symbol] :> Simp[d   Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + 
 b*ArcTan[c*x])^p, x], x] + Simp[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 5489

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]/Sq 
rt[1 - I*c*x]], x] + (Simp[I*(b/Sqrt[d])*PolyLog[2, -Sqrt[1 + I*c*x]/Sqrt[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 5493

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2 
]), x_Symbol] :> Simp[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 5497

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*Ar 
cTan[c*x])^p/(d*f*(m + 1))), x] + (-Simp[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] - Simp[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] && GtQ[p, 0] 
 && LtQ[m, -1] && NeQ[m, -2]
 

3.3.14.4 Maple [A] (verified)

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

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

input

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

output

-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
 

3.3.14.5 Fricas [F]

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

input

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

output

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

3.3.14.6 Sympy [F]

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

input

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

output

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

3.3.14.7 Maxima [F]

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

input

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

output

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

3.3.14.8 Giac [F(-2)]

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

input

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

output

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

3.3.14.9 Mupad [F(-1)]

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

input

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

output

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