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

Optimal. Leaf size=140 \[ a x-\frac{b \log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2} \sqrt{c}}+\frac{b \log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2} \sqrt{c}}+b x \tan ^{-1}\left (c x^2\right )+\frac{b \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}-\frac{b \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )}{\sqrt{2} \sqrt{c}} \]

[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])

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Rubi [A]  time = 0.104643, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5027, 297, 1162, 617, 204, 1165, 628} \[ a x-\frac{b \log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2} \sqrt{c}}+\frac{b \log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2} \sqrt{c}}+b x \tan ^{-1}\left (c x^2\right )+\frac{b \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}-\frac{b \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )}{\sqrt{2} \sqrt{c}} \]

Antiderivative was successfully verified.

[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 5027

Int[ArcTan[(c_.)*(x_)^(n_)], x_Symbol] :> Simp[x*ArcTan[c*x^n], x] - Dist[c*n, Int[x^n/(1 + c^2*x^(2*n)), x],
x] /; FreeQ[{c, n}, x]

Rule 297

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 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])

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]

Rubi steps

\begin{align*} \int \left (a+b \tan ^{-1}\left (c x^2\right )\right ) \, dx &=a x+b \int \tan ^{-1}\left (c x^2\right ) \, dx\\ &=a x+b x \tan ^{-1}\left (c x^2\right )-(2 b c) \int \frac{x^2}{1+c^2 x^4} \, dx\\ &=a x+b x \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}+\frac{b \operatorname{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 \tan ^{-1}\left (c x^2\right )+\frac{b \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}-\frac{b \tan ^{-1}\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]  time = 0.042094, size = 107, normalized size = 0.76 \[ a x+b x \tan ^{-1}\left (c x^2\right )-\frac{b \left (\log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )-\log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )-2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )+2 \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )\right )}{2 \sqrt{2} \sqrt{c}} \]

Antiderivative was successfully verified.

[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])

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Maple [A]  time = 0.023, size = 125, normalized size = 0.9 \begin{align*} ax+bx\arctan \left ( c{x}^{2} \right ) -{\frac{b\sqrt{2}}{4\,c}\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{1}{\sqrt [4]{{c}^{-2}}}}}-{\frac{b\sqrt{2}}{2\,c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{c}^{-2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{c}^{-2}}}}}-{\frac{b\sqrt{2}}{2\,c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{c}^{-2}}}}}-1 \right ){\frac{1}{\sqrt [4]{{c}^{-2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x+b*x*arctan(c*x^2)-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)-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 = 1.51946, size = 348, normalized size = 2.49 \begin{align*} \frac{1}{4} \,{\left (c{\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{3}{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{3}{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{c^{2}} \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{c^{2}} \sqrt{-\sqrt{c^{2}}}}\right )} + 4 \, x \arctan \left (c x^{2}\right )\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.82601, size = 768, normalized size = 5.49 \begin{align*} b x \arctan \left (c x^{2}\right ) + a x + \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} b^{3} c x + b^{4} - \sqrt{2} \sqrt{b^{6} x^{2} + \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{3}{4}} b^{3} c x + \sqrt{\frac{b^{4}}{c^{2}}} b^{4}} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} c}{b^{4}}\right ) + \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} b^{3} c x - b^{4} - \sqrt{2} \sqrt{b^{6} x^{2} - \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{3}{4}} b^{3} c x + \sqrt{\frac{b^{4}}{c^{2}}} b^{4}} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} c}{b^{4}}\right ) + \frac{1}{4} \, \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} \log \left (b^{6} x^{2} + \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{3}{4}} b^{3} c x + \sqrt{\frac{b^{4}}{c^{2}}} b^{4}\right ) - \frac{1}{4} \, \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} \log \left (b^{6} x^{2} - \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{3}{4}} b^{3} c x + \sqrt{\frac{b^{4}}{c^{2}}} b^{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x*arctan(c*x^2) + a*x + sqrt(2)*(b^4/c^2)^(1/4)*arctan(-(sqrt(2)*(b^4/c^2)^(1/4)*b^3*c*x + b^4 - sqrt(2)*sqr
t(b^6*x^2 + sqrt(2)*(b^4/c^2)^(3/4)*b^3*c*x + sqrt(b^4/c^2)*b^4)*(b^4/c^2)^(1/4)*c)/b^4) + sqrt(2)*(b^4/c^2)^(
1/4)*arctan(-(sqrt(2)*(b^4/c^2)^(1/4)*b^3*c*x - b^4 - sqrt(2)*sqrt(b^6*x^2 - sqrt(2)*(b^4/c^2)^(3/4)*b^3*c*x +
sqrt(b^4/c^2)*b^4)*(b^4/c^2)^(1/4)*c)/b^4) + 1/4*sqrt(2)*(b^4/c^2)^(1/4)*log(b^6*x^2 + sqrt(2)*(b^4/c^2)^(3/4
)*b^3*c*x + sqrt(b^4/c^2)*b^4) - 1/4*sqrt(2)*(b^4/c^2)^(1/4)*log(b^6*x^2 - sqrt(2)*(b^4/c^2)^(3/4)*b^3*c*x + s
qrt(b^4/c^2)*b^4)

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Sympy [A]  time = 15.0671, size = 146, normalized size = 1.04 \begin{align*} a x + b \left (\begin{cases} - \frac{\left (-1\right )^{\frac{3}{4}} c^{3} \left (\frac{1}{c^{2}}\right )^{\frac{7}{4}} \log{\left (x^{2} + i \sqrt{\frac{1}{c^{2}}} \right )}}{2} + \left (-1\right )^{\frac{3}{4}} c \left (\frac{1}{c^{2}}\right )^{\frac{3}{4}} \log{\left (x - \sqrt [4]{-1} \sqrt [4]{\frac{1}{c^{2}}} \right )} - \left (-1\right )^{\frac{3}{4}} c \left (\frac{1}{c^{2}}\right )^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} x}{\sqrt [4]{\frac{1}{c^{2}}}} \right )} + x \operatorname{atan}{\left (c x^{2} \right )} + \frac{\sqrt [4]{-1} \operatorname{atan}{\left (c x^{2} \right )}}{c^{4} \left (\frac{1}{c^{2}}\right )^{\frac{7}{4}}} & \text{for}\: c \neq 0 \\0 & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*Piecewise((-(-1)**(3/4)*c**3*(c**(-2))**(7/4)*log(x**2 + I*sqrt(c**(-2)))/2 + (-1)**(3/4)*c*(c**(-2))*
*(3/4)*log(x - (-1)**(1/4)*(c**(-2))**(1/4)) - (-1)**(3/4)*c*(c**(-2))**(3/4)*atan((-1)**(3/4)*x/(c**(-2))**(1
/4)) + x*atan(c*x**2) + (-1)**(1/4)*atan(c*x**2)/(c**4*(c**(-2))**(7/4)), Ne(c, 0)), (0, True))

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Giac [A]  time = 1.14414, size = 201, normalized size = 1.44 \begin{align*} -\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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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