\(\int \frac {(c+a^2 c x^2)^{3/2} \arctan (a x)}{x^4} \, dx\) [215]

Optimal result

Integrand size = 22, antiderivative size = 310 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^4} \, dx=-\frac {a c \sqrt {c+a^2 c x^2}}{6 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7}{6} a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}} \] Output:

-1/6*a*c*(a^2*c*x^2+c)^(1/2)/x^2-a^2*c*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/x-1 
/3*(a^2*c*x^2+c)^(3/2)*arctan(a*x)/x^3-2*I*a^3*c^2*(a^2*x^2+1)^(1/2)*arcta 
n(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2*c*x^2+c)^(1/2)-7/6*a^3 
*c^(3/2)*arctanh((a^2*c*x^2+c)^(1/2)/c^(1/2))+I*a^3*c^2*(a^2*x^2+1)^(1/2)* 
polylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2*c*x^2+c)^(1/2)-I*a^3*c^ 
2*(a^2*x^2+1)^(1/2)*polylog(2,I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2*c*x^ 
2+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.85 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^4} \, dx=-\frac {c \sqrt {c+a^2 c x^2} \left (a x \sqrt {1+a^2 x^2}+2 \sqrt {1+a^2 x^2} \arctan (a x)+8 a^2 x^2 \sqrt {1+a^2 x^2} \arctan (a x)+a^3 x^3 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right )-6 a^3 x^3 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )+6 a^3 x^3 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )+6 a^3 x^3 \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )\right )-6 a^3 x^3 \log \left (\sin \left (\frac {1}{2} \arctan (a x)\right )\right )-6 i a^3 x^3 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+6 i a^3 x^3 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )}{6 x^3 \sqrt {1+a^2 x^2}} \] Input:

Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x^4,x]
 

Output:

-1/6*(c*Sqrt[c + a^2*c*x^2]*(a*x*Sqrt[1 + a^2*x^2] + 2*Sqrt[1 + a^2*x^2]*A 
rcTan[a*x] + 8*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + a^3*x^3*ArcTanh[Sqr 
t[1 + a^2*x^2]] - 6*a^3*x^3*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] + 6*a 
^3*x^3*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] + 6*a^3*x^3*Log[Cos[ArcTan 
[a*x]/2]] - 6*a^3*x^3*Log[Sin[ArcTan[a*x]/2]] - (6*I)*a^3*x^3*PolyLog[2, ( 
-I)*E^(I*ArcTan[a*x])] + (6*I)*a^3*x^3*PolyLog[2, I*E^(I*ArcTan[a*x])]))/( 
x^3*Sqrt[1 + a^2*x^2])
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.95, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {5485, 5479, 243, 51, 73, 221, 5485, 5425, 5421, 5479, 243, 73, 221}

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.

\(\displaystyle \int \frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{x^4} \, dx\)

\(\Big \downarrow \) 5485

\(\displaystyle a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^4}dx\)

\(\Big \downarrow \) 5479

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

\(\Big \downarrow \) 243

\(\displaystyle a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+c \left (\frac {1}{6} a \int \frac {\sqrt {a^2 c x^2+c}}{x^4}dx^2-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\)

\(\Big \downarrow \) 51

\(\displaystyle a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+c \left (\frac {1}{6} a \left (\frac {1}{2} a^2 c \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx^2-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+c \left (\frac {1}{6} a \left (\int \frac {1}{\frac {x^4}{a^2 c}-\frac {1}{a^2}}d\sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\)

\(\Big \downarrow \) 5485

\(\displaystyle a^2 c \left (a^2 c \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)}{x^2 \sqrt {a^2 c x^2+c}}dx\right )+c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\)

\(\Big \downarrow \) 5425

\(\displaystyle a^2 c \left (\frac {a^2 c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)}{x^2 \sqrt {a^2 c x^2+c}}dx\right )+c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\)

\(\Big \downarrow \) 5421

\(\displaystyle c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )+a^2 c \left (c \int \frac {\arctan (a x)}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )\)

\(\Big \downarrow \) 5479

\(\displaystyle c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )+a^2 c \left (c \left (a \int \frac {1}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )\)

\(\Big \downarrow \) 243

\(\displaystyle c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )+a^2 c \left (c \left (\frac {1}{2} a \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx^2-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )+a^2 c \left (c \left (\frac {\int \frac {1}{\frac {x^4}{a^2 c}-\frac {1}{a^2}}d\sqrt {a^2 c x^2+c}}{a c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )+a^2 c \left (c \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )\)

Input:

Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x^4,x]
 

Output:

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

Defintions of rubi rules used

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 5421

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]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && 
GtQ[d, 0]
 

rule 5425

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[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 5479

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] - Simp[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 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]))
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2334 vs. \(2 (257 ) = 514\).

Time = 1.78 (sec) , antiderivative size = 2335, normalized size of antiderivative = 7.53

method	result	size
default	\(\text {Expression too large to display}\)	\(2335\)

Input:

int((a^2*c*x^2+c)^(3/2)*arctan(a*x)/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/48*c*(a^3*x^3-I*a^2*x^2+a*x-I)*(c*(a*x-I)*(a*x+I))^(1/2)/x^3-7/96*c*ln( 
1+(1+I*a*x)/(a^2*x^2+1)^(1/2))*(I*a^4*x^4+4*a^3*x^3-6*I*a^2*x^2-4*a*x+I)/( 
a^2*x^2+1)^(1/2)*(c*(a*x-I)*(a*x+I))^(1/2)/x^3+1/8*c*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 
)*(a*x+I))^(1/2)/x^3-1/8*c*arctan(a*x)*(I*a^3*x^3+3*a^2*x^2-3*I*a*x-1)*(c* 
(a*x-I)*(a*x+I))^(1/2)/x^3-1/16*c*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*( 
a^4*x^4-4*I*a^3*x^3-6*a^2*x^2+4*I*a*x+1)/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(a*x 
+I))^(1/2)/x^3-1/8*c*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)*(a*x+I))^(1/2)/x^3+1/16/(a^2*x^2+1)^(1/ 
2)*(c*(a*x-I)*(a*x+I))^(1/2)*(a^4*x^4+4*I*a^3*x^3-6*a^2*x^2-4*I*a*x+1)*dil 
og(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*c/x^3+1/48*(c*(a*x-I)*(a*x+I))^(1/2)*( 
a^3*x^3+3*I*a^2*x^2-3*a*x-I)*c/x^3-1/16*c*arctan(a*x)*ln(1+I*(1+I*a*x)/(a^ 
2*x^2+1)^(1/2))*(I*a^4*x^4+4*a^3*x^3-6*I*a^2*x^2-4*a*x+I)/(a^2*x^2+1)^(1/2 
)*(c*(a*x-I)*(a*x+I))^(1/2)/x^3-1/8*c*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)*(a*x+I))^(1/2)/x^3+1/8 
*c*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)*(a*x+I))^(1/2)/x^3-7/48*c*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)*(a*x+I))^(1/2)/x^3-7/9 
6/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(a*x+I))^(1/2)*(I*a^4*x^4-4*a^3*x^3-6*I*a^2 
*x^2+4*a*x+I)*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)*c/x^3-7/24*c*arctan(a*x...
 

Fricas [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )}{x^{4}} \,d x } \] Input:

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)/x^4,x, algorithm="fricas")
 

Output:

integral((a^2*c*x^2 + c)^(3/2)*arctan(a*x)/x^4, x)
 

Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^4} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}{\left (a x \right )}}{x^{4}}\, dx \] Input:

integrate((a**2*c*x**2+c)**(3/2)*atan(a*x)/x**4,x)
 

Output:

Integral((c*(a**2*x**2 + 1))**(3/2)*atan(a*x)/x**4, x)
 

Maxima [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )}{x^{4}} \,d x } \] Input:

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)/x^4,x, algorithm="maxima")
 

Output:

integrate((a^2*c*x^2 + c)^(3/2)*arctan(a*x)/x^4, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)/x^4,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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^4} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{x^4} \,d x \] Input:

int((atan(a*x)*(c + a^2*c*x^2)^(3/2))/x^4,x)
 

Output:

int((atan(a*x)*(c + a^2*c*x^2)^(3/2))/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^4} \, dx=\frac {\sqrt {c}\, c \left (4 \mathit {atan} \left (\sqrt {a^{2} x^{2}+1}+a x \right ) a^{3} x^{3}-2 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right ) a^{2} x^{2}-2 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )-2 \mathit {atan} \left (a x \right ) a^{3} x^{3}-\sqrt {a^{2} x^{2}+1}\, a x +6 \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )}{x^{2}}d x \right ) a^{2} x^{3}+\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+a x -1\right ) a^{3} x^{3}-\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+a x +1\right ) a^{3} x^{3}\right )}{6 x^{3}} \] Input:

int((a^2*c*x^2+c)^(3/2)*atan(a*x)/x^4,x)
 

Output:

(sqrt(c)*c*(4*atan(sqrt(a**2*x**2 + 1) + a*x)*a**3*x**3 - 2*sqrt(a**2*x**2 
 + 1)*atan(a*x)*a**2*x**2 - 2*sqrt(a**2*x**2 + 1)*atan(a*x) - 2*atan(a*x)* 
a**3*x**3 - sqrt(a**2*x**2 + 1)*a*x + 6*int((sqrt(a**2*x**2 + 1)*atan(a*x) 
)/x**2,x)*a**2*x**3 + log(sqrt(a**2*x**2 + 1) + a*x - 1)*a**3*x**3 - log(s 
qrt(a**2*x**2 + 1) + a*x + 1)*a**3*x**3))/(6*x**3)