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

Optimal result

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

[Out]

-1/3*(a^2*c*x^2+c)^(3/2)*arctan(a*x)/x^3-7/6*a^3*c^(3/2)*arctanh((a^2*c*x^2+c)^(1/2)/c^(1/2))-2*I*a^3*c^2*arct
an(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^3*c^2*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^3*c^2*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)-1/6*a*c*(a^2*c*x^2+c)^(1/2)/x^2-a^2*c*arctan(a*x)*(a^2*c*
x^2+c)^(1/2)/x

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5070, 5064, 272, 43, 65, 214, 5010, 5006} \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^4} \, dx=-\frac {a^2 c \arctan (a x) \sqrt {a^2 c x^2+c}}{x}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 x^3}-\frac {a c \sqrt {a^2 c x^2+c}}{6 x^2}-\frac {2 i a^3 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {7}{6} a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )+\frac {i a^3 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {i a^3 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}} \]

[In]

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

[Out]

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

Rule 43

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] - Dist[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] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

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 {\sqrt {c+a^2 c x^2} \arctan (a x)}{x^4} \, dx+\left (a^2 c\right ) \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x^2} \, dx \\ & = -\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{3 x^3}+\frac {1}{3} (a c) \int \frac {\sqrt {c+a^2 c x^2}}{x^3} \, dx+\left (a^2 c^2\right ) \int \frac {\arctan (a x)}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^2\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\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 {1}{6} (a c) \text {Subst}\left (\int \frac {\sqrt {c+a^2 c x}}{x^2} \, dx,x,x^2\right )+\left (a^3 c^2\right ) \int \frac {1}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^4 c^2 \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 {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 {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}}+\frac {1}{12} \left (a^3 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {1}{2} \left (a^3 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+a^2 c x}} \, dx,x,x^2\right ) \\ & = -\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 {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}}+\frac {1}{6} (a c) \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 c) \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 ) \\ & = -\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.85 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{x^4} \, dx=-\frac {c \sqrt {c+a^2 c x^2} \left (a x \sqrt {1+a^2 x^2}+2 \sqrt {1+a^2 x^2} \arctan (a x)+8 a^2 x^2 \sqrt {1+a^2 x^2} \arctan (a x)+a^3 x^3 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right )-6 a^3 x^3 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )+6 a^3 x^3 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )+6 a^3 x^3 \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )\right )-6 a^3 x^3 \log \left (\sin \left (\frac {1}{2} \arctan (a x)\right )\right )-6 i a^3 x^3 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+6 i a^3 x^3 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )}{6 x^3 \sqrt {1+a^2 x^2}} \]

[In]

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

[Out]

-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]*ArcTan[a*x] + 8*a^2*x^2*Sqrt[1 + a^2*
x^2]*ArcTan[a*x] + a^3*x^3*ArcTanh[Sqrt[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[ArcT
an[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])

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 = 0.79 (sec) , antiderivative size = 2335, normalized size of antiderivative = 7.53

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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