3.1.21 \(\int \frac {(d+e x^2) (a+b \text {ArcCos}(c x))}{x^2} \, dx\) [21]

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

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, 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, 4816, 457, 81, 65, 214} \begin {gather*} -\frac {d (a+b \text {ArcCos}(c x))}{x}+e x (a+b \text {ArcCos}(c x))+b c d \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )-\frac {b e \sqrt {1-c^2 x^2}}{c} \end {gather*}

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 65

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 81

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 214

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

Rule 457

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 4816

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]))

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) \text {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) \text {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) \text {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*}

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Mathematica [A]
time = 0.05, size = 80, normalized size = 1.21 \begin {gather*} -\frac {a d}{x}+a e x-\frac {b e \sqrt {1-c^2 x^2}}{c}-\frac {b d \text {ArcCos}(c x)}{x}+b e x \text {ArcCos}(c x)-b c d \log (x)+b c d \log \left (1+\sqrt {1-c^2 x^2}\right ) \end {gather*}

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 = 0.01, size = 79, normalized size = 1.20

method	result	size
derivativedivides	\(c \left (\frac {a \left (e c x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (\arccos \left (c x \right ) e c x -\frac {\arccos \left (c x \right ) d c}{x}-e \sqrt {-c^{2} x^{2}+1}+d \,c^{2} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{2}}\right )\)	\(79\)
default	\(c \left (\frac {a \left (e c x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (\arccos \left (c x \right ) e c x -\frac {\arccos \left (c x \right ) d c}{x}-e \sqrt {-c^{2} x^{2}+1}+d \,c^{2} \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{2}}\right )\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 83, normalized size = 1.26 \begin {gather*} {\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 x e + \frac {{\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} b e}{c} - \frac {a d}{x} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (64) = 128\).
time = 2.99, size = 160, normalized size = 2.42 \begin {gather*} \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 x^{2} e - 2 \, \sqrt {-c^{2} x^{2} + 1} b x e - 2 \, a c d + 2 \, {\left (b c d x - b c d + {\left (b c x^{2} - b c x\right )} e\right )} \arccos \left (c x\right ) - 2 \, {\left (b c d x - b c x e\right )} \arctan \left (\frac {\sqrt {-c^{2} x^{2} + 1} c x}{c^{2} x^{2} - 1}\right )}{2 \, c x} \end {gather*}

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

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Sympy [A]
time = 2.36, size = 78, normalized size = 1.18 \begin {gather*} - \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (62) = 124\).
time = 0.73, size = 859, normalized size = 13.02 \begin {gather*} -\frac {b c^{2} d \arccos \left (c x\right )}{c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}} + \frac {b c^{2} d \log \left ({\left | c x + \sqrt {-c^{2} x^{2} + 1} + 1 \right |}\right )}{c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}} - \frac {b c^{2} d \log \left ({\left | -c x + \sqrt {-c^{2} x^{2} + 1} - 1 \right |}\right )}{c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}} - \frac {a c^{2} d}{c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b c^{2} d \arccos \left (c x\right )}{{\left (c x + 1\right )}^{2} {\left (c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}\right )}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a c^{2} d}{{\left (c x + 1\right )}^{2} {\left (c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}\right )}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b c^{2} d \arccos \left (c x\right )}{{\left (c x + 1\right )}^{4} {\left (c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}\right )}} + \frac {b e \arccos \left (c x\right )}{c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b c^{2} d \log \left ({\left | c x + \sqrt {-c^{2} x^{2} + 1} + 1 \right |}\right )}{{\left (c x + 1\right )}^{4} {\left (c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}\right )}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b c^{2} d \log \left ({\left | -c x + \sqrt {-c^{2} x^{2} + 1} - 1 \right |}\right )}{{\left (c x + 1\right )}^{4} {\left (c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}\right )}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} a c^{2} d}{{\left (c x + 1\right )}^{4} {\left (c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}\right )}} + \frac {a e}{c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b e \arccos \left (c x\right )}{{\left (c x + 1\right )}^{2} {\left (c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}\right )}} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b e}{{\left (c x + 1\right )} {\left (c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}\right )}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a e}{{\left (c x + 1\right )}^{2} {\left (c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}\right )}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b e \arccos \left (c x\right )}{{\left (c x + 1\right )}^{4} {\left (c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}\right )}} + \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e}{{\left (c x + 1\right )}^{3} {\left (c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}\right )}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} a e}{{\left (c x + 1\right )}^{4} {\left (c - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} c}{{\left (c x + 1\right )}^{4}}\right )}} \end {gather*}

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*e*arccos(c*x)/(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*e*arccos(c*x)/((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*e*arccos(c*x)/((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))

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Mupad [B]
time = 0.38, size = 71, normalized size = 1.08 \begin {gather*} b\,c\,d\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-c^2\,x^2}}\right )-\frac {b\,d\,\mathrm {acos}\left (c\,x\right )}{x}-\frac {a\,\left (d-e\,x^2\right )}{x}-\frac {b\,e\,\left (\sqrt {1-c^2\,x^2}-c\,x\,\mathrm {acos}\left (c\,x\right )\right )}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*c*d*atanh(1/(1 - c^2*x^2)^(1/2)) - (b*d*acos(c*x))/x - (a*(d - e*x^2))/x - (b*e*((1 - c^2*x^2)^(1/2) - c*x*a
cos(c*x)))/c

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