3.71 \(\int \frac{a+b \tan ^{-1}(c x^2)}{x^2} \, dx\)

Optimal. Leaf size=143 \[ -\frac{a+b \tan ^{-1}\left (c x^2\right )}{x}-\frac{b \sqrt{c} \log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2}}+\frac{b \sqrt{c} \log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2}}-\frac{b \sqrt{c} \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2}}+\frac{b \sqrt{c} \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )}{\sqrt{2}} \]

[Out]

-((a + b*ArcTan[c*x^2])/x) - (b*Sqrt[c]*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/Sqrt[2] + (b*Sqrt[c]*ArcTan[1 + Sqrt[2]
*Sqrt[c]*x])/Sqrt[2] - (b*Sqrt[c]*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2]) + (b*Sqrt[c]*Log[1 + Sqrt[2]
*Sqrt[c]*x + c*x^2])/(2*Sqrt[2])

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Rubi [A]  time = 0.0861341, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5033, 211, 1165, 628, 1162, 617, 204} \[ -\frac{a+b \tan ^{-1}\left (c x^2\right )}{x}-\frac{b \sqrt{c} \log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2}}+\frac{b \sqrt{c} \log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2}}-\frac{b \sqrt{c} \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2}}+\frac{b \sqrt{c} \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x^2])/x^2,x]

[Out]

-((a + b*ArcTan[c*x^2])/x) - (b*Sqrt[c]*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/Sqrt[2] + (b*Sqrt[c]*ArcTan[1 + Sqrt[2]
*Sqrt[c]*x])/Sqrt[2] - (b*Sqrt[c]*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2]) + (b*Sqrt[c]*Log[1 + Sqrt[2]
*Sqrt[c]*x + c*x^2])/(2*Sqrt[2])

Rule 5033

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

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{a+b \tan ^{-1}\left (c x^2\right )}{x^2} \, dx &=-\frac{a+b \tan ^{-1}\left (c x^2\right )}{x}+(2 b c) \int \frac{1}{1+c^2 x^4} \, dx\\ &=-\frac{a+b \tan ^{-1}\left (c x^2\right )}{x}+(b c) \int \frac{1-c x^2}{1+c^2 x^4} \, dx+(b c) \int \frac{1+c x^2}{1+c^2 x^4} \, dx\\ &=-\frac{a+b \tan ^{-1}\left (c x^2\right )}{x}+\frac{1}{2} b \int \frac{1}{\frac{1}{c}-\frac{\sqrt{2} x}{\sqrt{c}}+x^2} \, dx+\frac{1}{2} b \int \frac{1}{\frac{1}{c}+\frac{\sqrt{2} x}{\sqrt{c}}+x^2} \, dx-\frac{\left (b \sqrt{c}\right ) \int \frac{\frac{\sqrt{2}}{\sqrt{c}}+2 x}{-\frac{1}{c}-\frac{\sqrt{2} x}{\sqrt{c}}-x^2} \, dx}{2 \sqrt{2}}-\frac{\left (b \sqrt{c}\right ) \int \frac{\frac{\sqrt{2}}{\sqrt{c}}-2 x}{-\frac{1}{c}+\frac{\sqrt{2} x}{\sqrt{c}}-x^2} \, dx}{2 \sqrt{2}}\\ &=-\frac{a+b \tan ^{-1}\left (c x^2\right )}{x}-\frac{b \sqrt{c} \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{2 \sqrt{2}}+\frac{b \sqrt{c} \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{2 \sqrt{2}}+\frac{\left (b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2}}-\frac{\left (b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{2}}\\ &=-\frac{a+b \tan ^{-1}\left (c x^2\right )}{x}-\frac{b \sqrt{c} \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2}}+\frac{b \sqrt{c} \tan ^{-1}\left (1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{2}}-\frac{b \sqrt{c} \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{2 \sqrt{2}}+\frac{b \sqrt{c} \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{2 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.042499, size = 158, normalized size = 1.1 \[ -\frac{a}{x}-\frac{b \sqrt{c} \log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2}}+\frac{b \sqrt{c} \log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2}}-\frac{b \tan ^{-1}\left (c x^2\right )}{x}+\frac{b \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} x-\sqrt{2}}{\sqrt{2}}\right )}{\sqrt{2}}+\frac{b \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} x+\sqrt{2}}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a/x) - (b*ArcTan[c*x^2])/x + (b*Sqrt[c]*ArcTan[(-Sqrt[2] + 2*Sqrt[c]*x)/Sqrt[2]])/Sqrt[2] + (b*Sqrt[c]*ArcTa
n[(Sqrt[2] + 2*Sqrt[c]*x)/Sqrt[2]])/Sqrt[2] - (b*Sqrt[c]*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2]) + (b*
Sqrt[c]*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2])

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Maple [A]  time = 0.025, size = 125, normalized size = 0.9 \begin{align*} -{\frac{a}{x}}-{\frac{b\arctan \left ( c{x}^{2} \right ) }{x}}+{\frac{bc\sqrt{2}}{2}\sqrt [4]{{c}^{-2}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{c}^{-2}}}}}-1 \right ) }+{\frac{bc\sqrt{2}}{4}\sqrt [4]{{c}^{-2}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{c}^{-2}}x\sqrt{2}+\sqrt{{c}^{-2}} \right ) \left ({x}^{2}-\sqrt [4]{{c}^{-2}}x\sqrt{2}+\sqrt{{c}^{-2}} \right ) ^{-1}} \right ) }+{\frac{bc\sqrt{2}}{2}\sqrt [4]{{c}^{-2}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{c}^{-2}}}}}+1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c*x^2))/x^2,x)

[Out]

-a/x-b/x*arctan(c*x^2)+1/2*b*c*(1/c^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c^2)^(1/4)*x-1)+1/4*b*c*(1/c^2)^(1/4)*2
^(1/2)*ln((x^2+(1/c^2)^(1/4)*x*2^(1/2)+(1/c^2)^(1/2))/(x^2-(1/c^2)^(1/4)*x*2^(1/2)+(1/c^2)^(1/2)))+1/2*b*c*(1/
c^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c^2)^(1/4)*x+1)

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Maxima [B]  time = 1.52878, size = 339, normalized size = 2.37 \begin{align*} \frac{1}{4} \,{\left ({\left (\frac{\sqrt{2} \log \left (\sqrt{c^{2}} x^{2} + \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}} x + 1\right )}{{\left (c^{2}\right )}^{\frac{1}{4}}} - \frac{\sqrt{2} \log \left (\sqrt{c^{2}} x^{2} - \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}} x + 1\right )}{{\left (c^{2}\right )}^{\frac{1}{4}}} + \frac{\sqrt{2} \log \left (\frac{2 \, \sqrt{c^{2}} x - \sqrt{2} \sqrt{-\sqrt{c^{2}}} + \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}{2 \, \sqrt{c^{2}} x + \sqrt{2} \sqrt{-\sqrt{c^{2}}} + \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{-\sqrt{c^{2}}}} + \frac{\sqrt{2} \log \left (\frac{2 \, \sqrt{c^{2}} x - \sqrt{2} \sqrt{-\sqrt{c^{2}}} - \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}{2 \, \sqrt{c^{2}} x + \sqrt{2} \sqrt{-\sqrt{c^{2}}} - \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{-\sqrt{c^{2}}}}\right )} c - \frac{4 \, \arctan \left (c x^{2}\right )}{x}\right )} b - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((sqrt(2)*log(sqrt(c^2)*x^2 + sqrt(2)*(c^2)^(1/4)*x + 1)/(c^2)^(1/4) - sqrt(2)*log(sqrt(c^2)*x^2 - sqrt(2)
*(c^2)^(1/4)*x + 1)/(c^2)^(1/4) + sqrt(2)*log((2*sqrt(c^2)*x - sqrt(2)*sqrt(-sqrt(c^2)) + sqrt(2)*(c^2)^(1/4))
/(2*sqrt(c^2)*x + sqrt(2)*sqrt(-sqrt(c^2)) + sqrt(2)*(c^2)^(1/4)))/sqrt(-sqrt(c^2)) + sqrt(2)*log((2*sqrt(c^2)
*x - sqrt(2)*sqrt(-sqrt(c^2)) - sqrt(2)*(c^2)^(1/4))/(2*sqrt(c^2)*x + sqrt(2)*sqrt(-sqrt(c^2)) - sqrt(2)*(c^2)
^(1/4)))/sqrt(-sqrt(c^2)))*c - 4*arctan(c*x^2)/x)*b - a/x

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Fricas [B]  time = 2.75745, size = 790, normalized size = 5.52 \begin{align*} -\frac{4 \, \sqrt{2} \left (b^{4} c^{2}\right )^{\frac{1}{4}} x \arctan \left (-\frac{b^{4} c^{2} + \sqrt{2} \left (b^{4} c^{2}\right )^{\frac{3}{4}} b c x - \sqrt{2} \left (b^{4} c^{2}\right )^{\frac{3}{4}} \sqrt{b^{2} c^{2} x^{2} + \sqrt{2} \left (b^{4} c^{2}\right )^{\frac{1}{4}} b c x + \sqrt{b^{4} c^{2}}}}{b^{4} c^{2}}\right ) + 4 \, \sqrt{2} \left (b^{4} c^{2}\right )^{\frac{1}{4}} x \arctan \left (\frac{b^{4} c^{2} - \sqrt{2} \left (b^{4} c^{2}\right )^{\frac{3}{4}} b c x + \sqrt{2} \left (b^{4} c^{2}\right )^{\frac{3}{4}} \sqrt{b^{2} c^{2} x^{2} - \sqrt{2} \left (b^{4} c^{2}\right )^{\frac{1}{4}} b c x + \sqrt{b^{4} c^{2}}}}{b^{4} c^{2}}\right ) - \sqrt{2} \left (b^{4} c^{2}\right )^{\frac{1}{4}} x \log \left (b^{2} c^{2} x^{2} + \sqrt{2} \left (b^{4} c^{2}\right )^{\frac{1}{4}} b c x + \sqrt{b^{4} c^{2}}\right ) + \sqrt{2} \left (b^{4} c^{2}\right )^{\frac{1}{4}} x \log \left (b^{2} c^{2} x^{2} - \sqrt{2} \left (b^{4} c^{2}\right )^{\frac{1}{4}} b c x + \sqrt{b^{4} c^{2}}\right ) + 4 \, b \arctan \left (c x^{2}\right ) + 4 \, a}{4 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*sqrt(2)*(b^4*c^2)^(1/4)*x*arctan(-(b^4*c^2 + sqrt(2)*(b^4*c^2)^(3/4)*b*c*x - sqrt(2)*(b^4*c^2)^(3/4)*s
qrt(b^2*c^2*x^2 + sqrt(2)*(b^4*c^2)^(1/4)*b*c*x + sqrt(b^4*c^2)))/(b^4*c^2)) + 4*sqrt(2)*(b^4*c^2)^(1/4)*x*arc
tan((b^4*c^2 - sqrt(2)*(b^4*c^2)^(3/4)*b*c*x + sqrt(2)*(b^4*c^2)^(3/4)*sqrt(b^2*c^2*x^2 - sqrt(2)*(b^4*c^2)^(1
/4)*b*c*x + sqrt(b^4*c^2)))/(b^4*c^2)) - sqrt(2)*(b^4*c^2)^(1/4)*x*log(b^2*c^2*x^2 + sqrt(2)*(b^4*c^2)^(1/4)*b
*c*x + sqrt(b^4*c^2)) + sqrt(2)*(b^4*c^2)^(1/4)*x*log(b^2*c^2*x^2 - sqrt(2)*(b^4*c^2)^(1/4)*b*c*x + sqrt(b^4*c
^2)) + 4*b*arctan(c*x^2) + 4*a)/x

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Sympy [A]  time = 32.019, size = 1105, normalized size = 7.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a/x, Eq(c, 0)), (-(a - b*atan((-sqrt(2)/2 - sqrt(2)*I/2)**(-2)))/x, Eq(c, -1/(x**2*(-sqrt(2)/2 - s
qrt(2)*I/2)**2))), (-(a - b*atan((-sqrt(2)/2 + sqrt(2)*I/2)**(-2)))/x, Eq(c, -1/(x**2*(-sqrt(2)/2 + sqrt(2)*I/
2)**2))), (-(a - b*atan((sqrt(2)/2 - sqrt(2)*I/2)**(-2)))/x, Eq(c, -1/(x**2*(sqrt(2)/2 - sqrt(2)*I/2)**2))), (
-(a - b*atan((sqrt(2)/2 + sqrt(2)*I/2)**(-2)))/x, Eq(c, -1/(x**2*(sqrt(2)/2 + sqrt(2)*I/2)**2))), (-2*(-1)**(1
/4)*a*c**6*x**4*(c**(-2))**(9/4)/(2*(-1)**(1/4)*c**6*x**5*(c**(-2))**(9/4) + 2*(-1)**(1/4)*c**4*x*(c**(-2))**(
9/4)) - 2*(-1)**(1/4)*a*c**4*(c**(-2))**(9/4)/(2*(-1)**(1/4)*c**6*x**5*(c**(-2))**(9/4) + 2*(-1)**(1/4)*c**4*x
*(c**(-2))**(9/4)) - 2*I*b*c**7*x**5*(c**(-2))**(5/2)*log(x - (-1)**(1/4)*(c**(-2))**(1/4))/(2*(-1)**(1/4)*c**
6*x**5*(c**(-2))**(9/4) + 2*(-1)**(1/4)*c**4*x*(c**(-2))**(9/4)) + I*b*c**7*x**5*(c**(-2))**(5/2)*log(x**2 + I
*sqrt(c**(-2)))/(2*(-1)**(1/4)*c**6*x**5*(c**(-2))**(9/4) + 2*(-1)**(1/4)*c**4*x*(c**(-2))**(9/4)) - 2*I*b*c**
7*x**5*(c**(-2))**(5/2)*atan((-1)**(3/4)*x/(c**(-2))**(1/4))/(2*(-1)**(1/4)*c**6*x**5*(c**(-2))**(9/4) + 2*(-1
)**(1/4)*c**4*x*(c**(-2))**(9/4)) - 2*(-1)**(1/4)*b*c**6*x**4*(c**(-2))**(9/4)*atan(c*x**2)/(2*(-1)**(1/4)*c**
6*x**5*(c**(-2))**(9/4) + 2*(-1)**(1/4)*c**4*x*(c**(-2))**(9/4)) - 2*I*b*c**5*x*(c**(-2))**(5/2)*log(x - (-1)*
*(1/4)*(c**(-2))**(1/4))/(2*(-1)**(1/4)*c**6*x**5*(c**(-2))**(9/4) + 2*(-1)**(1/4)*c**4*x*(c**(-2))**(9/4)) +
I*b*c**5*x*(c**(-2))**(5/2)*log(x**2 + I*sqrt(c**(-2)))/(2*(-1)**(1/4)*c**6*x**5*(c**(-2))**(9/4) + 2*(-1)**(1
/4)*c**4*x*(c**(-2))**(9/4)) - 2*I*b*c**5*x*(c**(-2))**(5/2)*atan((-1)**(3/4)*x/(c**(-2))**(1/4))/(2*(-1)**(1/
4)*c**6*x**5*(c**(-2))**(9/4) + 2*(-1)**(1/4)*c**4*x*(c**(-2))**(9/4)) - 2*(-1)**(1/4)*b*c**4*(c**(-2))**(9/4)
*atan(c*x**2)/(2*(-1)**(1/4)*c**6*x**5*(c**(-2))**(9/4) + 2*(-1)**(1/4)*c**4*x*(c**(-2))**(9/4)) - 2*b*c**2*x*
*5*atan(c*x**2)/(2*(-1)**(1/4)*c**6*x**5*(c**(-2))**(9/4) + 2*(-1)**(1/4)*c**4*x*(c**(-2))**(9/4)) - 2*b*x*ata
n(c*x**2)/(2*(-1)**(1/4)*c**6*x**5*(c**(-2))**(9/4) + 2*(-1)**(1/4)*c**4*x*(c**(-2))**(9/4)), True))

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Giac [A]  time = 1.22791, size = 186, normalized size = 1.3 \begin{align*} \frac{1}{4} \, b c{\left (\frac{2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \frac{\sqrt{2}}{\sqrt{{\left | c \right |}}}\right )} \sqrt{{\left | c \right |}}\right )}{\sqrt{{\left | c \right |}}} + \frac{2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \frac{\sqrt{2}}{\sqrt{{\left | c \right |}}}\right )} \sqrt{{\left | c \right |}}\right )}{\sqrt{{\left | c \right |}}} + \frac{\sqrt{2} \log \left (x^{2} + \frac{\sqrt{2} x}{\sqrt{{\left | c \right |}}} + \frac{1}{{\left | c \right |}}\right )}{\sqrt{{\left | c \right |}}} - \frac{\sqrt{2} \log \left (x^{2} - \frac{\sqrt{2} x}{\sqrt{{\left | c \right |}}} + \frac{1}{{\left | c \right |}}\right )}{\sqrt{{\left | c \right |}}}\right )} - \frac{b \arctan \left (c x^{2}\right ) + a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*c*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)/sqrt(abs(c)))*sqrt(abs(c)))/sqrt(abs(c)) + 2*sqrt(2)*arct
an(1/2*sqrt(2)*(2*x - sqrt(2)/sqrt(abs(c)))*sqrt(abs(c)))/sqrt(abs(c)) + sqrt(2)*log(x^2 + sqrt(2)*x/sqrt(abs(
c)) + 1/abs(c))/sqrt(abs(c)) - sqrt(2)*log(x^2 - sqrt(2)*x/sqrt(abs(c)) + 1/abs(c))/sqrt(abs(c))) - (b*arctan(
c*x^2) + a)/x