Integrand size = 19, antiderivative size = 66 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx=-\frac {b e \sqrt {1-c^2 x^2}}{c}-\frac {d (a+b \arccos (c x))}{x}+e x (a+b \arccos (c x))+b c d \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \]
-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
Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.21 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx=-\frac {a d}{x}+a e x-\frac {b e \sqrt {1-c^2 x^2}}{c}-\frac {b d \arccos (c x)}{x}+b e x \arccos (c x)-b c d \log (x)+b c d \log \left (1+\sqrt {1-c^2 x^2}\right ) \]
-((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]]
Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5231, 25, 354, 90, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx\) |
\(\Big \downarrow \) 5231 |
\(\displaystyle b c \int -\frac {d-e x^2}{x \sqrt {1-c^2 x^2}}dx-\frac {d (a+b \arccos (c x))}{x}+e x (a+b \arccos (c x))\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -b c \int \frac {d-e x^2}{x \sqrt {1-c^2 x^2}}dx-\frac {d (a+b \arccos (c x))}{x}+e x (a+b \arccos (c x))\) |
\(\Big \downarrow \) 354 |
\(\displaystyle -\frac {1}{2} b c \int \frac {d-e x^2}{x^2 \sqrt {1-c^2 x^2}}dx^2-\frac {d (a+b \arccos (c x))}{x}+e x (a+b \arccos (c x))\) |
\(\Big \downarrow \) 90 |
\(\displaystyle -\frac {1}{2} b c \left (d \int \frac {1}{x^2 \sqrt {1-c^2 x^2}}dx^2+\frac {2 e \sqrt {1-c^2 x^2}}{c^2}\right )-\frac {d (a+b \arccos (c x))}{x}+e x (a+b \arccos (c x))\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {1}{2} b c \left (\frac {2 e \sqrt {1-c^2 x^2}}{c^2}-\frac {2 d \int \frac {1}{\frac {1}{c^2}-\frac {x^4}{c^2}}d\sqrt {1-c^2 x^2}}{c^2}\right )-\frac {d (a+b \arccos (c x))}{x}+e x (a+b \arccos (c x))\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {d (a+b \arccos (c x))}{x}+e x (a+b \arccos (c x))-\frac {1}{2} b c \left (\frac {2 e \sqrt {1-c^2 x^2}}{c^2}-2 d \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )\) |
-((d*(a + b*ArcCos[c*x]))/x) + e*x*(a + b*ArcCos[c*x]) - (b*c*((2*e*Sqrt[1 - c^2*x^2])/c^2 - 2*d*ArcTanh[Sqrt[1 - c^2*x^2]]))/2
3.1.21.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(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]
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
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]}, Simp [(a + b*ArcCos[c*x]) u, x] + Simp[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]))
Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(c \left (\frac {a \left (c e x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (\arccos \left (c x \right ) e c x -\frac {\arccos \left (c x \right ) d c}{x}-e \sqrt {-c^{2} x^{2}+1}+d \,c^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{2}}\right )\) | \(79\) |
default | \(c \left (\frac {a \left (c e x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (\arccos \left (c x \right ) e c x -\frac {\arccos \left (c x \right ) d c}{x}-e \sqrt {-c^{2} x^{2}+1}+d \,c^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{2}}\right )\) | \(79\) |
parts | \(a \left (e x -\frac {d}{x}\right )+b c \left (\frac {\arccos \left (c x \right ) e x}{c}-\frac {\arccos \left (c x \right ) d}{c x}+\frac {-e \sqrt {-c^{2} x^{2}+1}+d \,c^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}\right )\) | \(79\) |
c*(a/c^2*(c*e*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))))
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (62) = 124\).
Time = 0.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.35 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx=\frac {b c^{2} d x \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - b c^{2} d x \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) + 2 \, a c e x^{2} - 2 \, \sqrt {-c^{2} x^{2} + 1} b e x - 2 \, a c d - 2 \, {\left (b c d - b c e\right )} x \arctan \left (\frac {\sqrt {-c^{2} x^{2} + 1} c x}{c^{2} x^{2} - 1}\right ) + 2 \, {\left (b c e x^{2} - b c d + {\left (b c d - b c e\right )} x\right )} \arccos \left (c x\right )}{2 \, c x} \]
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)
Time = 2.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.18 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx=- \frac {a d}{x} + a e x - b c d \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) - \frac {b d \operatorname {acos}{\left (c x \right )}}{x} + b e \left (\begin {cases} \frac {\pi x}{2} & \text {for}\: c = 0 \\x \operatorname {acos}{\left (c x \right )} - \frac {\sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right ) \]
-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))
Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.23 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx={\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {\arccos \left (c x\right )}{x}\right )} b d + a e x + \frac {{\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} b e}{c} - \frac {a d}{x} \]
(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
Leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (62) = 124\).
Time = 0.57 (sec) , antiderivative size = 859, normalized size of antiderivative = 13.02 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx=\text {Too large to display} \]
-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*ar ccos(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))
Time = 0.43 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.08 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx=b\,c\,d\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-c^2\,x^2}}\right )-\frac {b\,d\,\mathrm {acos}\left (c\,x\right )}{x}-\frac {a\,\left (d-e\,x^2\right )}{x}-\frac {b\,e\,\left (\sqrt {1-c^2\,x^2}-c\,x\,\mathrm {acos}\left (c\,x\right )\right )}{c} \]