3.83 \(\int (a+b \cos ^{-1}(-1+d x^2)) \, dx\)

Optimal. Leaf size=43 \[ a x-\frac {2 b \sqrt {2 d x^2-d^2 x^4}}{d x}+b x \cos ^{-1}\left (d x^2-1\right ) \]

[Out]

a*x+b*x*arccos(d*x^2-1)-2*b*(-d^2*x^4+2*d*x^2)^(1/2)/d/x

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4841, 12, 1588} \[ a x-\frac {2 b \sqrt {2 d x^2-d^2 x^4}}{d x}+b x \cos ^{-1}\left (d x^2-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcCos[-1 + d*x^2],x]

[Out]

a*x - (2*b*Sqrt[2*d*x^2 - d^2*x^4])/(d*x) + b*x*ArcCos[-1 + d*x^2]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rule 4841

Int[ArcCos[u_], x_Symbol] :> Simp[x*ArcCos[u], x] + Int[SimplifyIntegrand[(x*D[u, x])/Sqrt[1 - u^2], x], x] /;
InverseFunctionFreeQ[u, x] &&  !FunctionOfExponentialQ[u, x]

Rubi steps

\begin {align*} \int \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right ) \, dx &=a x+b \int \cos ^{-1}\left (-1+d x^2\right ) \, dx\\ &=a x+b x \cos ^{-1}\left (-1+d x^2\right )+b \int \frac {2 d x^2}{\sqrt {2 d x^2-d^2 x^4}} \, dx\\ &=a x+b x \cos ^{-1}\left (-1+d x^2\right )+(2 b d) \int \frac {x^2}{\sqrt {2 d x^2-d^2 x^4}} \, dx\\ &=a x-\frac {2 b \sqrt {2 d x^2-d^2 x^4}}{d x}+b x \cos ^{-1}\left (-1+d x^2\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 41, normalized size = 0.95 \[ a x-\frac {2 b \sqrt {-d x^2 \left (d x^2-2\right )}}{d x}+b x \cos ^{-1}\left (d x^2-1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcCos[-1 + d*x^2],x]

[Out]

a*x - (2*b*Sqrt[-(d*x^2*(-2 + d*x^2))])/(d*x) + b*x*ArcCos[-1 + d*x^2]

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 48, normalized size = 1.12 \[ \frac {b d x^{2} \arccos \left (d x^{2} - 1\right ) + a d x^{2} - 2 \, \sqrt {-d^{2} x^{4} + 2 \, d x^{2}} b}{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arccos(d*x^2-1),x, algorithm="fricas")

[Out]

(b*d*x^2*arccos(d*x^2 - 1) + a*d*x^2 - 2*sqrt(-d^2*x^4 + 2*d*x^2)*b)/(d*x)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 50, normalized size = 1.16 \[ {\left (x \arccos \left (d x^{2} - 1\right ) + \frac {2 \, \sqrt {2} \mathrm {sgn}\relax (x)}{\sqrt {d}} - \frac {2 \, \sqrt {-d^{2} x^{2} + 2 \, d}}{d \mathrm {sgn}\relax (x)}\right )} b + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arccos(d*x^2-1),x, algorithm="giac")

[Out]

(x*arccos(d*x^2 - 1) + 2*sqrt(2)*sgn(x)/sqrt(d) - 2*sqrt(-d^2*x^2 + 2*d)/(d*sgn(x)))*b + a*x

________________________________________________________________________________________

maple [A]  time = 0.01, size = 45, normalized size = 1.05 \[ a x +b \left (x \arccos \left (d \,x^{2}-1\right )+\frac {2 x \left (d \,x^{2}-2\right )}{\sqrt {-d^{2} x^{4}+2 d \,x^{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*arccos(d*x^2-1),x)

[Out]

a*x+b*(x*arccos(d*x^2-1)+2/(-d^2*x^4+2*d*x^2)^(1/2)*x*(d*x^2-2))

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 45, normalized size = 1.05 \[ {\left (x \arccos \left (d x^{2} - 1\right ) + \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} - 2 \, \sqrt {d}\right )}}{\sqrt {-d x^{2} + 2} d}\right )} b + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arccos(d*x^2-1),x, algorithm="maxima")

[Out]

(x*arccos(d*x^2 - 1) + 2*(d^(3/2)*x^2 - 2*sqrt(d))/(sqrt(-d*x^2 + 2)*d))*b + a*x

________________________________________________________________________________________

mupad [B]  time = 0.44, size = 39, normalized size = 0.91 \[ a\,x+b\,x\,\mathrm {acos}\left (d\,x^2-1\right )-\frac {2\,b\,\sqrt {1-{\left (d\,x^2-1\right )}^2}}{d\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*acos(d*x^2 - 1),x)

[Out]

a*x + b*x*acos(d*x^2 - 1) - (2*b*(1 - (d*x^2 - 1)^2)^(1/2))/(d*x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {acos}{\left (d x^{2} - 1 \right )}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*acos(d*x**2-1),x)

[Out]

Integral(a + b*acos(d*x**2 - 1), x)

________________________________________________________________________________________