3.21 \(\int \frac{(d+e x^2) (a+b \cos ^{-1}(c x))}{x^2} \, dx\)

Optimal. Leaf size=66 \[ -\frac{d \left (a+b \cos ^{-1}(c x)\right )}{x}+e x \left (a+b \cos ^{-1}(c x)\right )+b c d \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )-\frac{b e \sqrt{1-c^2 x^2}}{c} \]

[Out]

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

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Rubi [A]  time = 0.074847, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {14, 4732, 446, 80, 63, 208} \[ -\frac{d \left (a+b \cos ^{-1}(c x)\right )}{x}+e x \left (a+b \cos ^{-1}(c x)\right )+b c d \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )-\frac{b e \sqrt{1-c^2 x^2}}{c} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)*(a + b*ArcCos[c*x]))/x^2,x]

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 4732

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 -
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] ||
(IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 63

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 208

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

Rubi steps

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

Mathematica [A]  time = 0.0615048, size = 80, normalized size = 1.21 \[ -\frac{a d}{x}+a e x+b c d \log \left (\sqrt{1-c^2 x^2}+1\right )-\frac{b e \sqrt{1-c^2 x^2}}{c}-b c d \log (x)-\frac{b d \cos ^{-1}(c x)}{x}+b e x \cos ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)*(a + b*ArcCos[c*x]))/x^2,x]

[Out]

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

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Maple [A]  time = 0.007, size = 79, normalized size = 1.2 \begin{align*} c \left ({\frac{a}{{c}^{2}} \left ( ecx-{\frac{dc}{x}} \right ) }+{\frac{b}{{c}^{2}} \left ( \arccos \left ( cx \right ) ecx-{\frac{\arccos \left ( cx \right ) cd}{x}}-e\sqrt{-{c}^{2}{x}^{2}+1}+{c}^{2}d{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*(a+b*arccos(c*x))/x^2,x)

[Out]

c*(a/c^2*(e*c*x-c*d/x)+b/c^2*(arccos(c*x)*e*c*x-arccos(c*x)*c*d/x-e*(-c^2*x^2+1)^(1/2)+c^2*d*arctanh(1/(-c^2*x
^2+1)^(1/2))))

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Maxima [A]  time = 1.50752, size = 109, normalized size = 1.65 \begin{align*}{\left (c \log \left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{\arccos \left (c x\right )}{x}\right )} b d + a e x + \frac{{\left (c x \arccos \left (c x\right ) - \sqrt{-c^{2} x^{2} + 1}\right )} b e}{c} - \frac{a d}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) - arccos(c*x)/x)*b*d + a*e*x + (c*x*arccos(c*x) - sqrt(-c^2*x^2
+ 1))*b*e/c - a*d/x

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Fricas [B]  time = 2.46413, size = 360, normalized size = 5.45 \begin{align*} \frac{b c^{2} d x \log \left (\sqrt{-c^{2} x^{2} + 1} + 1\right ) - b c^{2} d x \log \left (\sqrt{-c^{2} x^{2} + 1} - 1\right ) + 2 \, a c e x^{2} - 2 \, \sqrt{-c^{2} x^{2} + 1} b e x - 2 \, a c d - 2 \,{\left (b c d - b c e\right )} x \arctan \left (\frac{\sqrt{-c^{2} x^{2} + 1} c x}{c^{2} x^{2} - 1}\right ) + 2 \,{\left (b c e x^{2} - b c d +{\left (b c d - b c e\right )} x\right )} \arccos \left (c x\right )}{2 \, c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 4.0338, size = 78, normalized size = 1.18 \begin{align*} - \frac{a d}{x} + a e x - b c d \left (\begin{cases} - \operatorname{acosh}{\left (\frac{1}{c x} \right )} & \text{for}\: \frac{1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname{asin}{\left (\frac{1}{c x} \right )} & \text{otherwise} \end{cases}\right ) - \frac{b d \operatorname{acos}{\left (c x \right )}}{x} + b e \left (\begin{cases} \frac{\pi x}{2} & \text{for}\: c = 0 \\x \operatorname{acos}{\left (c x \right )} - \frac{\sqrt{- c^{2} x^{2} + 1}}{c} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)*(a+b*acos(c*x))/x**2,x)

[Out]

-a*d/x + a*e*x - b*c*d*Piecewise((-acosh(1/(c*x)), 1/Abs(c**2*x**2) > 1), (I*asin(1/(c*x)), True)) - b*d*acos(
c*x)/x + b*e*Piecewise((pi*x/2, Eq(c, 0)), (x*acos(c*x) - sqrt(-c**2*x**2 + 1)/c, True))

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Giac [B]  time = 1.86946, size = 1170, normalized size = 17.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*c^2*d*arccos(c*x)/(c - (c^2*x^2 - 1)^2*c/(c*x + 1)^4) + b*c^2*d*log(abs(c*x + sqrt(-c^2*x^2 + 1) + 1))/(c -
(c^2*x^2 - 1)^2*c/(c*x + 1)^4) - b*c^2*d*log(abs(-c*x + sqrt(-c^2*x^2 + 1) - 1))/(c - (c^2*x^2 - 1)^2*c/(c*x
+ 1)^4) - a*c^2*d/(c - (c^2*x^2 - 1)^2*c/(c*x + 1)^4) + 2*(c^2*x^2 - 1)*b*c^2*d*arccos(c*x)/((c*x + 1)^2*(c -
(c^2*x^2 - 1)^2*c/(c*x + 1)^4)) + 2*(c^2*x^2 - 1)*a*c^2*d/((c*x + 1)^2*(c - (c^2*x^2 - 1)^2*c/(c*x + 1)^4)) -
(c^2*x^2 - 1)^2*b*c^2*d*arccos(c*x)/((c*x + 1)^4*(c - (c^2*x^2 - 1)^2*c/(c*x + 1)^4)) + b*arccos(c*x)*e/(c - (
c^2*x^2 - 1)^2*c/(c*x + 1)^4) - (c^2*x^2 - 1)^2*b*c^2*d*log(abs(c*x + sqrt(-c^2*x^2 + 1) + 1))/((c*x + 1)^4*(c
- (c^2*x^2 - 1)^2*c/(c*x + 1)^4)) + (c^2*x^2 - 1)^2*b*c^2*d*log(abs(-c*x + sqrt(-c^2*x^2 + 1) - 1))/((c*x + 1
)^4*(c - (c^2*x^2 - 1)^2*c/(c*x + 1)^4)) - (c^2*x^2 - 1)^2*a*c^2*d/((c*x + 1)^4*(c - (c^2*x^2 - 1)^2*c/(c*x +
1)^4)) + a*e/(c - (c^2*x^2 - 1)^2*c/(c*x + 1)^4) + 2*(c^2*x^2 - 1)*b*arccos(c*x)*e/((c*x + 1)^2*(c - (c^2*x^2
- 1)^2*c/(c*x + 1)^4)) - 2*sqrt(-c^2*x^2 + 1)*b*e/((c*x + 1)*(c - (c^2*x^2 - 1)^2*c/(c*x + 1)^4)) + 2*(c^2*x^2
- 1)*a*e/((c*x + 1)^2*(c - (c^2*x^2 - 1)^2*c/(c*x + 1)^4)) + (c^2*x^2 - 1)^2*b*arccos(c*x)*e/((c*x + 1)^4*(c
- (c^2*x^2 - 1)^2*c/(c*x + 1)^4)) + 2*(-c^2*x^2 + 1)^(3/2)*b*e/((c*x + 1)^3*(c - (c^2*x^2 - 1)^2*c/(c*x + 1)^4
)) + (c^2*x^2 - 1)^2*a*e/((c*x + 1)^4*(c - (c^2*x^2 - 1)^2*c/(c*x + 1)^4))