\(\int (a+b \arctan (c x^2)) \, dx\) [70]

Optimal result

Integrand size = 10, antiderivative size = 140 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=a x+b x \arctan \left (c x^2\right )+\frac {b \arctan \left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}-\frac {b \arctan \left (1+\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}-\frac {b \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2} \sqrt {c}}+\frac {b \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2} \sqrt {c}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4930, 303, 1176, 631, 210, 1179, 642} \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=a x+b x \arctan \left (c x^2\right )+\frac {b \arctan \left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}-\frac {b \arctan \left (\sqrt {2} \sqrt {c} x+1\right )}{\sqrt {2} \sqrt {c}}-\frac {b \log \left (c x^2-\sqrt {2} \sqrt {c} x+1\right )}{2 \sqrt {2} \sqrt {c}}+\frac {b \log \left (c x^2+\sqrt {2} \sqrt {c} x+1\right )}{2 \sqrt {2} \sqrt {c}} \]

[In]

Int[a + b*ArcTan[c*x^2],x]

[Out]

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

Rule 210

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

Rule 303

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

Rule 631

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 642

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 1176

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 1179

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 4930

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Dist[b*c
*n*p, Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0
] && (EqQ[n, 1] || EqQ[p, 1])

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.76 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=a x+b x \arctan \left (c x^2\right )-\frac {b \left (-2 \arctan \left (1-\sqrt {2} \sqrt {c} x\right )+2 \arctan \left (1+\sqrt {2} \sqrt {c} x\right )+\log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )-\log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )\right )}{2 \sqrt {2} \sqrt {c}} \]

[In]

Integrate[a + b*ArcTan[c*x^2],x]

[Out]

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

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.74

method	result	size
default	\(a x +b \left (x \arctan \left (c \,x^{2}\right )-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}\right )\)	\(103\)
parts	\(a x +b \left (x \arctan \left (c \,x^{2}\right )-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}\right )\)	\(103\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=b x \arctan \left (c x^{2}\right ) + a x - \frac {1}{2} \, \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {1}{4}} \log \left (b^{3} x + \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {3}{4}} c\right ) + \frac {1}{2} i \, \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {1}{4}} \log \left (b^{3} x + i \, \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {3}{4}} c\right ) - \frac {1}{2} i \, \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {1}{4}} \log \left (b^{3} x - i \, \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {3}{4}} c\right ) + \frac {1}{2} \, \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {1}{4}} \log \left (b^{3} x - \left (-\frac {b^{4}}{c^{2}}\right )^{\frac {3}{4}} c\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 4.34 (sec) , antiderivative size = 617, normalized size of antiderivative = 4.41 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=a x + b \left (\begin {cases} 0 & \text {for}\: c = 0 \\- \infty i x & \text {for}\: c = - \frac {i}{x^{2}} \\\infty i x & \text {for}\: c = \frac {i}{x^{2}} \\\frac {2 c^{5} x^{5} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} \operatorname {atan}{\left (c x^{2} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} - \frac {2 c^{4} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{2}} \log {\left (x - \sqrt [4]{- \frac {1}{c^{2}}} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} + \frac {c^{4} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{2}} \log {\left (x^{2} + \sqrt {- \frac {1}{c^{2}}} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} - \frac {2 c^{4} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{c^{2}}}} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} + \frac {2 c^{3} x \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} \operatorname {atan}{\left (c x^{2} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} - \frac {2 c^{2} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{2}} \log {\left (x - \sqrt [4]{- \frac {1}{c^{2}}} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} + \frac {c^{2} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{2}} \log {\left (x^{2} + \sqrt {- \frac {1}{c^{2}}} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} - \frac {2 c^{2} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{c^{2}}}} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} + \frac {2 c x^{4} \operatorname {atan}{\left (c x^{2} \right )}}{2 c^{5} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} + \frac {2 \operatorname {atan}{\left (c x^{2} \right )}}{2 c^{6} x^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}} + 2 c^{4} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{4}}} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

a*x + b*Piecewise((0, Eq(c, 0)), (-oo*I*x, Eq(c, -I/x**2)), (oo*I*x, Eq(c, I/x**2)), (2*c**5*x**5*(-1/c**2)**(
7/4)*atan(c*x**2)/(2*c**5*x**4*(-1/c**2)**(7/4) + 2*c**3*(-1/c**2)**(7/4)) - 2*c**4*x**4*(-1/c**2)**(3/2)*log(
x - (-1/c**2)**(1/4))/(2*c**5*x**4*(-1/c**2)**(7/4) + 2*c**3*(-1/c**2)**(7/4)) + c**4*x**4*(-1/c**2)**(3/2)*lo
g(x**2 + sqrt(-1/c**2))/(2*c**5*x**4*(-1/c**2)**(7/4) + 2*c**3*(-1/c**2)**(7/4)) - 2*c**4*x**4*(-1/c**2)**(3/2
)*atan(x/(-1/c**2)**(1/4))/(2*c**5*x**4*(-1/c**2)**(7/4) + 2*c**3*(-1/c**2)**(7/4)) + 2*c**3*x*(-1/c**2)**(7/4
)*atan(c*x**2)/(2*c**5*x**4*(-1/c**2)**(7/4) + 2*c**3*(-1/c**2)**(7/4)) - 2*c**2*(-1/c**2)**(3/2)*log(x - (-1/
c**2)**(1/4))/(2*c**5*x**4*(-1/c**2)**(7/4) + 2*c**3*(-1/c**2)**(7/4)) + c**2*(-1/c**2)**(3/2)*log(x**2 + sqrt
(-1/c**2))/(2*c**5*x**4*(-1/c**2)**(7/4) + 2*c**3*(-1/c**2)**(7/4)) - 2*c**2*(-1/c**2)**(3/2)*atan(x/(-1/c**2)
**(1/4))/(2*c**5*x**4*(-1/c**2)**(7/4) + 2*c**3*(-1/c**2)**(7/4)) + 2*c*x**4*atan(c*x**2)/(2*c**5*x**4*(-1/c**
2)**(7/4) + 2*c**3*(-1/c**2)**(7/4)) + 2*atan(c*x**2)/(2*c**6*x**4*(-1/c**2)**(7/4) + 2*c**4*(-1/c**2)**(7/4))
, True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=-\frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x + \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x - \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (c x^{2} + \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (c x^{2} - \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}}\right )} - 4 \, x \arctan \left (c x^{2}\right )\right )} b + a x \]

[In]

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

[Out]

-1/4*(c*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*c*x + sqrt(2)*sqrt(c))/sqrt(c))/c^(3/2) + 2*sqrt(2)*arctan(1/2*sqrt(2
)*(2*c*x - sqrt(2)*sqrt(c))/sqrt(c))/c^(3/2) - sqrt(2)*log(c*x^2 + sqrt(2)*sqrt(c)*x + 1)/c^(3/2) + sqrt(2)*lo
g(c*x^2 - sqrt(2)*sqrt(c)*x + 1)/c^(3/2)) - 4*x*arctan(c*x^2))*b + a*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.06 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=-\frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {2} \sqrt {{\left | c \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{c^{2}} + \frac {2 \, \sqrt {2} \sqrt {{\left | c \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{c^{2}} - \frac {\sqrt {2} \sqrt {{\left | c \right |}} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{c^{2}} + \frac {\sqrt {2} \sqrt {{\left | c \right |}} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{c^{2}}\right )} - 4 \, x \arctan \left (c x^{2}\right )\right )} b + a x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.35 \[ \int \left (a+b \arctan \left (c x^2\right )\right ) \, dx=a\,x+b\,x\,\mathrm {atan}\left (c\,x^2\right )-\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {c}\,x\right )}{\sqrt {c}}-\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {c}\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{\sqrt {c}} \]

[In]

int(a + b*atan(c*x^2),x)

[Out]

a*x + b*x*atan(c*x^2) - ((-1)^(1/4)*b*atan((-1)^(1/4)*c^(1/2)*x))/c^(1/2) - ((-1)^(1/4)*b*atan((-1)^(1/4)*c^(1
/2)*x*1i)*1i)/c^(1/2)