Integrand size = 16, antiderivative size = 136 \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{5/2}} \, dx=-\frac {a \sqrt {1-a^2 x^2}}{3 c \left (a^2 c+d\right ) \sqrt {c+d x^2}}+\frac {x \arccos (a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \arccos (a x)}{3 c^2 \sqrt {c+d x^2}}-\frac {2 \arctan \left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{3 c^2 \sqrt {d}} \]
1/3*x*arccos(a*x)/c/(d*x^2+c)^(3/2)-2/3*arctan(d^(1/2)*(-a^2*x^2+1)^(1/2)/ a/(d*x^2+c)^(1/2))/c^2/d^(1/2)+2/3*x*arccos(a*x)/c^2/(d*x^2+c)^(1/2)-1/3*a *(-a^2*x^2+1)^(1/2)/c/(a^2*c+d)/(d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.88 \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {-\frac {a c \sqrt {1-a^2 x^2} \left (c+d x^2\right )}{a^2 c+d}+a x^2 \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,a^2 x^2,-\frac {d x^2}{c}\right )+\left (3 c x+2 d x^3\right ) \arccos (a x)}{3 c^2 \left (c+d x^2\right )^{3/2}} \]
(-((a*c*Sqrt[1 - a^2*x^2]*(c + d*x^2))/(a^2*c + d)) + a*x^2*(c + d*x^2)*Sq rt[1 + (d*x^2)/c]*AppellF1[1, 1/2, 1/2, 2, a^2*x^2, -((d*x^2)/c)] + (3*c*x + 2*d*x^3)*ArcCos[a*x])/(3*c^2*(c + d*x^2)^(3/2))
Time = 0.31 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5171, 27, 435, 87, 66, 218}
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.
\(\displaystyle \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 5171 |
\(\displaystyle a \int \frac {x \left (2 d x^2+3 c\right )}{3 c^2 \sqrt {1-a^2 x^2} \left (d x^2+c\right )^{3/2}}dx+\frac {2 x \arccos (a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \arccos (a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a \int \frac {x \left (2 d x^2+3 c\right )}{\sqrt {1-a^2 x^2} \left (d x^2+c\right )^{3/2}}dx}{3 c^2}+\frac {2 x \arccos (a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \arccos (a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 435 |
\(\displaystyle \frac {a \int \frac {2 d x^2+3 c}{\sqrt {1-a^2 x^2} \left (d x^2+c\right )^{3/2}}dx^2}{6 c^2}+\frac {2 x \arccos (a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \arccos (a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {a \left (2 \int \frac {1}{\sqrt {1-a^2 x^2} \sqrt {d x^2+c}}dx^2-\frac {2 c \sqrt {1-a^2 x^2}}{\left (a^2 c+d\right ) \sqrt {c+d x^2}}\right )}{6 c^2}+\frac {2 x \arccos (a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \arccos (a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {a \left (4 \int \frac {1}{-d x^4-a^2}d\frac {\sqrt {1-a^2 x^2}}{\sqrt {d x^2+c}}-\frac {2 c \sqrt {1-a^2 x^2}}{\left (a^2 c+d\right ) \sqrt {c+d x^2}}\right )}{6 c^2}+\frac {2 x \arccos (a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \arccos (a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {a \left (-\frac {4 \arctan \left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{a \sqrt {d}}-\frac {2 c \sqrt {1-a^2 x^2}}{\left (a^2 c+d\right ) \sqrt {c+d x^2}}\right )}{6 c^2}+\frac {2 x \arccos (a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \arccos (a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
(x*ArcCos[a*x])/(3*c*(c + d*x^2)^(3/2)) + (2*x*ArcCos[a*x])/(3*c^2*Sqrt[c + d*x^2]) + (a*((-2*c*Sqrt[1 - a^2*x^2])/((a^2*c + d)*Sqrt[c + d*x^2]) - ( 4*ArcTan[(Sqrt[d]*Sqrt[1 - a^2*x^2])/(a*Sqrt[c + d*x^2])])/(a*Sqrt[d])))/( 6*c^2)
3.1.32.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2) *(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo l] :> With[{u = IntHide[(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]] /; Fr eeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2 , 0])
\[\int \frac {\arccos \left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {5}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (112) = 224\).
Time = 0.31 (sec) , antiderivative size = 580, normalized size of antiderivative = 4.26 \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\left [-\frac {{\left (a^{2} c^{3} + {\left (a^{2} c d^{2} + d^{3}\right )} x^{4} + c^{2} d + 2 \, {\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\right )} \sqrt {-d} \log \left (8 \, a^{4} d^{2} x^{4} + a^{4} c^{2} - 6 \, a^{2} c d + 8 \, {\left (a^{4} c d - a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{3} d x^{2} + a^{3} c - a d\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {d x^{2} + c} \sqrt {-d} + d^{2}\right ) - 2 \, \sqrt {d x^{2} + c} {\left ({\left (2 \, {\left (a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \, {\left (a^{2} c^{2} d + c d^{2}\right )} x\right )} \arccos \left (a x\right ) - {\left (a c d^{2} x^{2} + a c^{2} d\right )} \sqrt {-a^{2} x^{2} + 1}\right )}}{6 \, {\left (a^{2} c^{5} d + c^{4} d^{2} + {\left (a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}, -\frac {{\left (a^{2} c^{3} + {\left (a^{2} c d^{2} + d^{3}\right )} x^{4} + c^{2} d + 2 \, {\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\right )} \sqrt {d} \arctan \left (\frac {{\left (2 \, a^{2} d x^{2} + a^{2} c - d\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {d x^{2} + c} \sqrt {d}}{2 \, {\left (a^{3} d^{2} x^{4} - a c d + {\left (a^{3} c d - a d^{2}\right )} x^{2}\right )}}\right ) - \sqrt {d x^{2} + c} {\left ({\left (2 \, {\left (a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \, {\left (a^{2} c^{2} d + c d^{2}\right )} x\right )} \arccos \left (a x\right ) - {\left (a c d^{2} x^{2} + a c^{2} d\right )} \sqrt {-a^{2} x^{2} + 1}\right )}}{3 \, {\left (a^{2} c^{5} d + c^{4} d^{2} + {\left (a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}\right ] \]
[-1/6*((a^2*c^3 + (a^2*c*d^2 + d^3)*x^4 + c^2*d + 2*(a^2*c^2*d + c*d^2)*x^ 2)*sqrt(-d)*log(8*a^4*d^2*x^4 + a^4*c^2 - 6*a^2*c*d + 8*(a^4*c*d - a^2*d^2 )*x^2 - 4*(2*a^3*d*x^2 + a^3*c - a*d)*sqrt(-a^2*x^2 + 1)*sqrt(d*x^2 + c)*s qrt(-d) + d^2) - 2*sqrt(d*x^2 + c)*((2*(a^2*c*d^2 + d^3)*x^3 + 3*(a^2*c^2* d + c*d^2)*x)*arccos(a*x) - (a*c*d^2*x^2 + a*c^2*d)*sqrt(-a^2*x^2 + 1)))/( a^2*c^5*d + c^4*d^2 + (a^2*c^3*d^3 + c^2*d^4)*x^4 + 2*(a^2*c^4*d^2 + c^3*d ^3)*x^2), -1/3*((a^2*c^3 + (a^2*c*d^2 + d^3)*x^4 + c^2*d + 2*(a^2*c^2*d + c*d^2)*x^2)*sqrt(d)*arctan(1/2*(2*a^2*d*x^2 + a^2*c - d)*sqrt(-a^2*x^2 + 1 )*sqrt(d*x^2 + c)*sqrt(d)/(a^3*d^2*x^4 - a*c*d + (a^3*c*d - a*d^2)*x^2)) - sqrt(d*x^2 + c)*((2*(a^2*c*d^2 + d^3)*x^3 + 3*(a^2*c^2*d + c*d^2)*x)*arcc os(a*x) - (a*c*d^2*x^2 + a*c^2*d)*sqrt(-a^2*x^2 + 1)))/(a^2*c^5*d + c^4*d^ 2 + (a^2*c^3*d^3 + c^2*d^4)*x^4 + 2*(a^2*c^4*d^2 + c^3*d^3)*x^2)]
\[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {acos}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(d-a^2*c>0)', see `assume?` for m ore detail
Time = 0.37 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.12 \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{5/2}} \, dx=-\frac {1}{3} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} {\left | a \right |}}{{\left (a^{4} c^{5} + a^{2} c^{4} d\right )} \sqrt {a^{2} c + {\left (a^{2} x^{2} - 1\right )} d + d}} - \frac {2 \, {\left | a \right |} \log \left ({\left | -\sqrt {-a^{2} x^{2} + 1} \sqrt {-d} + \sqrt {a^{2} c + {\left (a^{2} x^{2} - 1\right )} d + d} \right |}\right )}{a^{2} c^{2} \sqrt {-d}}\right )} a + \frac {x {\left (\frac {2 \, d x^{2}}{c^{2}} + \frac {3}{c}\right )} \arccos \left (a x\right )}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \]
-1/3*(sqrt(-a^2*x^2 + 1)*a^2*c^3*abs(a)/((a^4*c^5 + a^2*c^4*d)*sqrt(a^2*c + (a^2*x^2 - 1)*d + d)) - 2*abs(a)*log(abs(-sqrt(-a^2*x^2 + 1)*sqrt(-d) + sqrt(a^2*c + (a^2*x^2 - 1)*d + d)))/(a^2*c^2*sqrt(-d)))*a + 1/3*x*(2*d*x^2 /c^2 + 3/c)*arccos(a*x)/(d*x^2 + c)^(3/2)
Timed out. \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\mathrm {acos}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]