Integrand size = 10, antiderivative size = 48 \[ \int \arccos \left (\frac {c}{a+b x}\right ) \, dx=\frac {(a+b x) \sec ^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {c \text {arctanh}\left (\sqrt {1-\frac {c^2}{(a+b x)^2}}\right )}{b} \]
Leaf count is larger than twice the leaf count of optimal. \(528\) vs. \(2(48)=96\).
Time = 0.93 (sec) , antiderivative size = 528, normalized size of antiderivative = 11.00 \[ \int \arccos \left (\frac {c}{a+b x}\right ) \, dx=x \arccos \left (\frac {c}{a+b x}\right )+\frac {(a+b x) \sqrt {\frac {a^2-c^2+2 a b x+b^2 x^2}{(a+b x)^2}} \left (\left (-c+\sqrt {-a^2+c^2}\right ) \sqrt {-a^2+2 c \left (c+\sqrt {-a^2+c^2}\right )} \arctan \left (\frac {b \sqrt {-a^2+2 c \left (c+\sqrt {-a^2+c^2}\right )} x}{a \left (\sqrt {a^2-c^2}-\sqrt {a^2-c^2+2 a b x+b^2 x^2}\right )}\right )+\left (c+\sqrt {-a^2+c^2}\right ) \sqrt {a^2+2 c \left (-c+\sqrt {-a^2+c^2}\right )} \text {arctanh}\left (\frac {b \sqrt {a^2-2 c^2+2 c \sqrt {-a^2+c^2}} x}{a \sqrt {a^2-c^2}-a \sqrt {a^2-c^2+2 a b x+b^2 x^2}}\right )+a \left (a \arctan \left (\frac {b^2 c \sqrt {a^2-c^2} x^2}{a^4+a^3 b x+b^2 c^2 x^2-a^2 \left (c^2+\sqrt {a^2-c^2} \sqrt {a^2-c^2+2 a b x+b^2 x^2}\right )}\right )+c \left (-\log \left (\sqrt {a^2-c^2}-b x-\sqrt {a^2-c^2+2 a b x+b^2 x^2}\right )+\log \left (b^2 \left (\sqrt {a^2-c^2}+b x-\sqrt {a^2-c^2+2 a b x+b^2 x^2}\right )\right )\right )\right )\right )}{a b \sqrt {a^2-c^2+2 a b x+b^2 x^2}} \]
x*ArcCos[c/(a + b*x)] + ((a + b*x)*Sqrt[(a^2 - c^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*((-c + Sqrt[-a^2 + c^2])*Sqrt[-a^2 + 2*c*(c + Sqrt[-a^2 + c^2]) ]*ArcTan[(b*Sqrt[-a^2 + 2*c*(c + Sqrt[-a^2 + c^2])]*x)/(a*(Sqrt[a^2 - c^2] - Sqrt[a^2 - c^2 + 2*a*b*x + b^2*x^2]))] + (c + Sqrt[-a^2 + c^2])*Sqrt[a^ 2 + 2*c*(-c + Sqrt[-a^2 + c^2])]*ArcTanh[(b*Sqrt[a^2 - 2*c^2 + 2*c*Sqrt[-a ^2 + c^2]]*x)/(a*Sqrt[a^2 - c^2] - a*Sqrt[a^2 - c^2 + 2*a*b*x + b^2*x^2])] + a*(a*ArcTan[(b^2*c*Sqrt[a^2 - c^2]*x^2)/(a^4 + a^3*b*x + b^2*c^2*x^2 - a^2*(c^2 + Sqrt[a^2 - c^2]*Sqrt[a^2 - c^2 + 2*a*b*x + b^2*x^2]))] + c*(-Lo g[Sqrt[a^2 - c^2] - b*x - Sqrt[a^2 - c^2 + 2*a*b*x + b^2*x^2]] + Log[b^2*( Sqrt[a^2 - c^2] + b*x - Sqrt[a^2 - c^2 + 2*a*b*x + b^2*x^2])]))))/(a*b*Sqr t[a^2 - c^2 + 2*a*b*x + b^2*x^2])
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5332, 5773, 895, 798, 73, 219}
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.
\(\displaystyle \int \arccos \left (\frac {c}{a+b x}\right ) \, dx\) |
\(\Big \downarrow \) 5332 |
\(\displaystyle \int \sec ^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )dx\) |
\(\Big \downarrow \) 5773 |
\(\displaystyle \frac {(a+b x) \sec ^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\int \frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right ) \sqrt {1-\frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right )^2}}}dx\) |
\(\Big \downarrow \) 895 |
\(\displaystyle \frac {(a+b x) \sec ^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {c \int \frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right ) \sqrt {1-\frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right )^2}}}d\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {c \int \frac {\left (\frac {a}{c}+\frac {b x}{c}\right )^2}{\sqrt {-\frac {a}{c}-\frac {b x}{c}+1}}d\frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right )^2}}{2 b}+\frac {(a+b x) \sec ^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(a+b x) \sec ^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {c \int \frac {1}{1-\frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right )^4}}d\sqrt {-\frac {a}{c}-\frac {b x}{c}+1}}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {(a+b x) \sec ^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {c \text {arctanh}\left (\sqrt {-\frac {a}{c}-\frac {b x}{c}+1}\right )}{b}\) |
3.2.14.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Simp[u^m/(Coeff icient[v, x, 1]*v^m) Subst[Int[x^m*(a + b*x^n)^p, x], x, v], x] /; FreeQ[ {a, b, m, n, p}, x] && LinearPairQ[u, v, x]
Int[ArcCos[(c_.)/((a_.) + (b_.)*(x_)^(n_.))]^(m_.)*(u_.), x_Symbol] :> Int[ u*ArcSec[a/c + b*(x^n/c)]^m, x] /; FreeQ[{a, b, c, n, m}, x]
Int[ArcSec[(c_) + (d_.)*(x_)], x_Symbol] :> Simp[(c + d*x)*(ArcSec[c + d*x] /d), x] - Int[1/((c + d*x)*Sqrt[1 - 1/(c + d*x)^2]), x] /; FreeQ[{c, d}, x]
Time = 0.63 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(-\frac {c \left (-\frac {\left (b x +a \right ) \arccos \left (\frac {c}{b x +a}\right )}{c}+\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\frac {c^{2}}{\left (b x +a \right )^{2}}}}\right )\right )}{b}\) | \(45\) |
default | \(-\frac {c \left (-\frac {\left (b x +a \right ) \arccos \left (\frac {c}{b x +a}\right )}{c}+\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\frac {c^{2}}{\left (b x +a \right )^{2}}}}\right )\right )}{b}\) | \(45\) |
parts | \(x \arccos \left (\frac {c}{b x +a}\right )-\frac {c \sqrt {b^{2} x^{2}+2 a b x +a^{2}-c^{2}}\, \left (\ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}-c^{2}}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) b \sqrt {-c^{2}}+a \ln \left (\frac {2 \left (\sqrt {-c^{2}}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-c^{2}}-c^{2}\right ) b}{b x +a}\right ) \sqrt {b^{2}}\right )}{b \sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}-c^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \sqrt {b^{2}}\, \sqrt {-c^{2}}}\) | \(203\) |
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (46) = 92\).
Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 2.92 \[ \int \arccos \left (\frac {c}{a+b x}\right ) \, dx=\frac {b x \arccos \left (\frac {c}{b x + a}\right ) + 2 \, a \arctan \left (-\frac {b x - {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + a}{c}\right ) + c \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a\right )}{b} \]
(b*x*arccos(c/(b*x + a)) + 2*a*arctan(-(b*x - (b*x + a)*sqrt((b^2*x^2 + 2* a*b*x + a^2 - c^2)/(b^2*x^2 + 2*a*b*x + a^2)) + a)/c) + c*log(-b*x + (b*x + a)*sqrt((b^2*x^2 + 2*a*b*x + a^2 - c^2)/(b^2*x^2 + 2*a*b*x + a^2)) - a)) /b
\[ \int \arccos \left (\frac {c}{a+b x}\right ) \, dx=\int \operatorname {acos}{\left (\frac {c}{a + b x} \right )}\, dx \]
\[ \int \arccos \left (\frac {c}{a+b x}\right ) \, dx=\int { \arccos \left (\frac {c}{b x + a}\right ) \,d x } \]
x*arctan(sqrt(b*x + a + c)*sqrt(b*x + a - c)/c) - integrate((b^2*c*x^2 + a *b*c*x)*e^(1/2*log(b*x + a + c) + 1/2*log(b*x + a - c))/(b^2*c^2*x^2 + 2*a *b*c^2*x + a^2*c^2 - c^4 + (b^2*x^2 + 2*a*b*x + a^2 - c^2)*e^(log(b*x + a + c) + log(b*x + a - c))), x)
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (46) = 92\).
Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.98 \[ \int \arccos \left (\frac {c}{a+b x}\right ) \, dx=-\frac {b {\left (\frac {c^{2} {\left (\log \left (\sqrt {-\frac {c^{2}}{{\left (b x + a\right )}^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {c^{2}}{{\left (b x + a\right )}^{2}} + 1} + 1\right )\right )}}{b^{2}} - \frac {2 \, {\left (b x + a\right )} c \arccos \left (-\frac {c}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{b^{2}}\right )}}{2 \, c} \]
-1/2*b*(c^2*(log(sqrt(-c^2/(b*x + a)^2 + 1) + 1) - log(-sqrt(-c^2/(b*x + a )^2 + 1) + 1))/b^2 - 2*(b*x + a)*c*arccos(-c/((b*x + a)*(a/(b*x + a) - 1) - a))/b^2)/c
Time = 0.69 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.90 \[ \int \arccos \left (\frac {c}{a+b x}\right ) \, dx=\frac {\mathrm {acos}\left (\frac {c}{a+b\,x}\right )\,\left (a+b\,x\right )}{b}-\frac {c\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {c^2}{{\left (a+b\,x\right )}^2}}}\right )}{b} \]