\(\int \frac {\arccos (a x)}{(c+d x^2)^{7/2}} \, dx\) [33]

Optimal result

Integrand size = 16, antiderivative size = 211 \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{7/2}} \, dx=-\frac {a \sqrt {1-a^2 x^2}}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}-\frac {2 a \left (3 a^2 c+2 d\right ) \sqrt {1-a^2 x^2}}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \arccos (a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \arccos (a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \arccos (a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {8 \arctan \left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{15 c^3 \sqrt {d}} \]

[Out]

1/5*x*arccos(a*x)/c/(d*x^2+c)^(5/2)+4/15*x*arccos(a*x)/c^2/(d*x^2+c)^(3/2)-8/15*arctan(d^(1/2)*(-a^2*x^2+1)^(1
/2)/a/(d*x^2+c)^(1/2))/c^3/d^(1/2)-1/15*a*(-a^2*x^2+1)^(1/2)/c/(a^2*c+d)/(d*x^2+c)^(3/2)+8/15*x*arccos(a*x)/c^
3/(d*x^2+c)^(1/2)-2/15*a*(3*a^2*c+2*d)*(-a^2*x^2+1)^(1/2)/c^2/(a^2*c+d)^2/(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {198, 197, 4756, 12, 6847, 963, 79, 65, 223, 209} \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{7/2}} \, dx=-\frac {8 \arctan \left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{15 c^3 \sqrt {d}}-\frac {2 a \sqrt {1-a^2 x^2} \left (3 a^2 c+2 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}-\frac {a \sqrt {1-a^2 x^2}}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {8 x \arccos (a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \arccos (a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \arccos (a x)}{5 c \left (c+d x^2\right )^{5/2}} \]

[In]

Int[ArcCos[a*x]/(c + d*x^2)^(7/2),x]

[Out]

-1/15*(a*Sqrt[1 - a^2*x^2])/(c*(a^2*c + d)*(c + d*x^2)^(3/2)) - (2*a*(3*a^2*c + 2*d)*Sqrt[1 - a^2*x^2])/(15*c^
2*(a^2*c + d)^2*Sqrt[c + d*x^2]) + (x*ArcCos[a*x])/(5*c*(c + d*x^2)^(5/2)) + (4*x*ArcCos[a*x])/(15*c^2*(c + d*
x^2)^(3/2)) + (8*x*ArcCos[a*x])/(15*c^3*Sqrt[c + d*x^2]) - (8*ArcTan[(Sqrt[d]*Sqrt[1 - a^2*x^2])/(a*Sqrt[c + d
*x^2])])/(15*c^3*Sqrt[d])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(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] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 963

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p,
d + e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g))), x] + Dist[1/((m + 1)*(e*f -
d*g)), Int[(d + e*x)^(m + 1)*(f + g*x)^n*ExpandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /;
FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&& IGtQ[p, 0] && LtQ[m, -1]

Rule 4756

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(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]] /; F
reeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps \begin{align*} \text {integral}& = \frac {x \arccos (a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \arccos (a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \arccos (a x)}{15 c^3 \sqrt {c+d x^2}}+a \int \frac {x \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{15 c^3 \sqrt {1-a^2 x^2} \left (c+d x^2\right )^{5/2}} \, dx \\ & = \frac {x \arccos (a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \arccos (a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \arccos (a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {a \int \frac {x \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{\sqrt {1-a^2 x^2} \left (c+d x^2\right )^{5/2}} \, dx}{15 c^3} \\ & = \frac {x \arccos (a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \arccos (a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \arccos (a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {a \text {Subst}\left (\int \frac {15 c^2+20 c d x+8 d^2 x^2}{\sqrt {1-a^2 x} (c+d x)^{5/2}} \, dx,x,x^2\right )}{30 c^3} \\ & = -\frac {a \sqrt {1-a^2 x^2}}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {x \arccos (a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \arccos (a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \arccos (a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {a \text {Subst}\left (\int \frac {-3 c \left (7 a^2 c+6 d\right )-12 d \left (a^2 c+d\right ) x}{\sqrt {1-a^2 x} (c+d x)^{3/2}} \, dx,x,x^2\right )}{45 c^3 \left (a^2 c+d\right )} \\ & = -\frac {a \sqrt {1-a^2 x^2}}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}-\frac {2 a \left (3 a^2 c+2 d\right ) \sqrt {1-a^2 x^2}}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \arccos (a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \arccos (a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \arccos (a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {(4 a) \text {Subst}\left (\int \frac {1}{\sqrt {1-a^2 x} \sqrt {c+d x}} \, dx,x,x^2\right )}{15 c^3} \\ & = -\frac {a \sqrt {1-a^2 x^2}}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}-\frac {2 a \left (3 a^2 c+2 d\right ) \sqrt {1-a^2 x^2}}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \arccos (a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \arccos (a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \arccos (a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {8 \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d}{a^2}-\frac {d x^2}{a^2}}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{15 a c^3} \\ & = -\frac {a \sqrt {1-a^2 x^2}}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}-\frac {2 a \left (3 a^2 c+2 d\right ) \sqrt {1-a^2 x^2}}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \arccos (a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \arccos (a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \arccos (a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {8 \text {Subst}\left (\int \frac {1}{1+\frac {d x^2}{a^2}} \, dx,x,\frac {\sqrt {1-a^2 x^2}}{\sqrt {c+d x^2}}\right )}{15 a c^3} \\ & = -\frac {a \sqrt {1-a^2 x^2}}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}-\frac {2 a \left (3 a^2 c+2 d\right ) \sqrt {1-a^2 x^2}}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \arccos (a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \arccos (a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \arccos (a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {8 \arctan \left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{15 c^3 \sqrt {d}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.31 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.77 \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {-\frac {a c \sqrt {1-a^2 x^2} \left (c+d x^2\right ) \left (d \left (5 c+4 d x^2\right )+a^2 c \left (7 c+6 d x^2\right )\right )}{\left (a^2 c+d\right )^2}+4 a x^2 \left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,a^2 x^2,-\frac {d x^2}{c}\right )+x \left (15 c^2+20 c d x^2+8 d^2 x^4\right ) \arccos (a x)}{15 c^3 \left (c+d x^2\right )^{5/2}} \]

[In]

Integrate[ArcCos[a*x]/(c + d*x^2)^(7/2),x]

[Out]

(-((a*c*Sqrt[1 - a^2*x^2]*(c + d*x^2)*(d*(5*c + 4*d*x^2) + a^2*c*(7*c + 6*d*x^2)))/(a^2*c + d)^2) + 4*a*x^2*(c
+ d*x^2)^2*Sqrt[1 + (d*x^2)/c]*AppellF1[1, 1/2, 1/2, 2, a^2*x^2, -((d*x^2)/c)] + x*(15*c^2 + 20*c*d*x^2 + 8*d
^2*x^4)*ArcCos[a*x])/(15*c^3*(c + d*x^2)^(5/2))

Maple [F]

\[\int \frac {\arccos \left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {7}{2}}}d x\]

[In]

int(arccos(a*x)/(d*x^2+c)^(7/2),x)

[Out]

int(arccos(a*x)/(d*x^2+c)^(7/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 523 vs. \(2 (177) = 354\).

Time = 0.35 (sec) , antiderivative size = 1066, normalized size of antiderivative = 5.05 \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\left [-\frac {2 \, {\left (a^{4} c^{5} + 2 \, a^{2} c^{4} d + {\left (a^{4} c^{2} d^{3} + 2 \, a^{2} c d^{4} + d^{5}\right )} x^{6} + c^{3} d^{2} + 3 \, {\left (a^{4} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + c d^{4}\right )} x^{4} + 3 \, {\left (a^{4} c^{4} d + 2 \, a^{2} c^{3} d^{2} + c^{2} d^{3}\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 ) - \sqrt {d x^{2} + c} {\left ({\left (8 \, {\left (a^{4} c^{2} d^{3} + 2 \, a^{2} c d^{4} + d^{5}\right )} x^{5} + 20 \, {\left (a^{4} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + c d^{4}\right )} x^{3} + 15 \, {\left (a^{4} c^{4} d + 2 \, a^{2} c^{3} d^{2} + c^{2} d^{3}\right )} x\right )} \arccos \left (a x\right ) - {\left (7 \, a^{3} c^{4} d + 5 \, a c^{3} d^{2} + 2 \, {\left (3 \, a^{3} c^{2} d^{3} + 2 \, a c d^{4}\right )} x^{4} + {\left (13 \, a^{3} c^{3} d^{2} + 9 \, a c^{2} d^{3}\right )} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1}\right )}}{15 \, {\left (a^{4} c^{8} d + 2 \, a^{2} c^{7} d^{2} + c^{6} d^{3} + {\left (a^{4} c^{5} d^{4} + 2 \, a^{2} c^{4} d^{5} + c^{3} d^{6}\right )} x^{6} + 3 \, {\left (a^{4} c^{6} d^{3} + 2 \, a^{2} c^{5} d^{4} + c^{4} d^{5}\right )} x^{4} + 3 \, {\left (a^{4} c^{7} d^{2} + 2 \, a^{2} c^{6} d^{3} + c^{5} d^{4}\right )} x^{2}\right )}}, -\frac {4 \, {\left (a^{4} c^{5} + 2 \, a^{2} c^{4} d + {\left (a^{4} c^{2} d^{3} + 2 \, a^{2} c d^{4} + d^{5}\right )} x^{6} + c^{3} d^{2} + 3 \, {\left (a^{4} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + c d^{4}\right )} x^{4} + 3 \, {\left (a^{4} c^{4} d + 2 \, a^{2} c^{3} d^{2} + c^{2} d^{3}\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 (8 \, {\left (a^{4} c^{2} d^{3} + 2 \, a^{2} c d^{4} + d^{5}\right )} x^{5} + 20 \, {\left (a^{4} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + c d^{4}\right )} x^{3} + 15 \, {\left (a^{4} c^{4} d + 2 \, a^{2} c^{3} d^{2} + c^{2} d^{3}\right )} x\right )} \arccos \left (a x\right ) - {\left (7 \, a^{3} c^{4} d + 5 \, a c^{3} d^{2} + 2 \, {\left (3 \, a^{3} c^{2} d^{3} + 2 \, a c d^{4}\right )} x^{4} + {\left (13 \, a^{3} c^{3} d^{2} + 9 \, a c^{2} d^{3}\right )} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1}\right )}}{15 \, {\left (a^{4} c^{8} d + 2 \, a^{2} c^{7} d^{2} + c^{6} d^{3} + {\left (a^{4} c^{5} d^{4} + 2 \, a^{2} c^{4} d^{5} + c^{3} d^{6}\right )} x^{6} + 3 \, {\left (a^{4} c^{6} d^{3} + 2 \, a^{2} c^{5} d^{4} + c^{4} d^{5}\right )} x^{4} + 3 \, {\left (a^{4} c^{7} d^{2} + 2 \, a^{2} c^{6} d^{3} + c^{5} d^{4}\right )} x^{2}\right )}}\right ] \]

[In]

integrate(arccos(a*x)/(d*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

[-1/15*(2*(a^4*c^5 + 2*a^2*c^4*d + (a^4*c^2*d^3 + 2*a^2*c*d^4 + d^5)*x^6 + c^3*d^2 + 3*(a^4*c^3*d^2 + 2*a^2*c^
2*d^3 + c*d^4)*x^4 + 3*(a^4*c^4*d + 2*a^2*c^3*d^2 + c^2*d^3)*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)*sqrt(-d) +
d^2) - sqrt(d*x^2 + c)*((8*(a^4*c^2*d^3 + 2*a^2*c*d^4 + d^5)*x^5 + 20*(a^4*c^3*d^2 + 2*a^2*c^2*d^3 + c*d^4)*x
^3 + 15*(a^4*c^4*d + 2*a^2*c^3*d^2 + c^2*d^3)*x)*arccos(a*x) - (7*a^3*c^4*d + 5*a*c^3*d^2 + 2*(3*a^3*c^2*d^3 +
2*a*c*d^4)*x^4 + (13*a^3*c^3*d^2 + 9*a*c^2*d^3)*x^2)*sqrt(-a^2*x^2 + 1)))/(a^4*c^8*d + 2*a^2*c^7*d^2 + c^6*d^
3 + (a^4*c^5*d^4 + 2*a^2*c^4*d^5 + c^3*d^6)*x^6 + 3*(a^4*c^6*d^3 + 2*a^2*c^5*d^4 + c^4*d^5)*x^4 + 3*(a^4*c^7*d
^2 + 2*a^2*c^6*d^3 + c^5*d^4)*x^2), -1/15*(4*(a^4*c^5 + 2*a^2*c^4*d + (a^4*c^2*d^3 + 2*a^2*c*d^4 + d^5)*x^6 +
c^3*d^2 + 3*(a^4*c^3*d^2 + 2*a^2*c^2*d^3 + c*d^4)*x^4 + 3*(a^4*c^4*d + 2*a^2*c^3*d^2 + c^2*d^3)*x^2)*sqrt(d)*a
rctan(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)*((8*(a^4*c^2*d^3 + 2*a^2*c*d^4 + d^5)*x^5 + 20*(a^4*c^3*d^2 + 2*a^2*c^2*d^3
+ c*d^4)*x^3 + 15*(a^4*c^4*d + 2*a^2*c^3*d^2 + c^2*d^3)*x)*arccos(a*x) - (7*a^3*c^4*d + 5*a*c^3*d^2 + 2*(3*a^3
*c^2*d^3 + 2*a*c*d^4)*x^4 + (13*a^3*c^3*d^2 + 9*a*c^2*d^3)*x^2)*sqrt(-a^2*x^2 + 1)))/(a^4*c^8*d + 2*a^2*c^7*d^
2 + c^6*d^3 + (a^4*c^5*d^4 + 2*a^2*c^4*d^5 + c^3*d^6)*x^6 + 3*(a^4*c^6*d^3 + 2*a^2*c^5*d^4 + c^4*d^5)*x^4 + 3*
(a^4*c^7*d^2 + 2*a^2*c^6*d^3 + c^5*d^4)*x^2)]

Sympy [F]

\[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\operatorname {acos}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate(acos(a*x)/(d*x**2+c)**(7/2),x)

[Out]

Integral(acos(a*x)/(c + d*x**2)**(7/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(arccos(a*x)/(d*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(d-a^2*c>0)', see `assume?` for
more detail

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.34 \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{7/2}} \, dx=-\frac {1}{15} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (\frac {2 \, {\left (3 \, a^{6} c^{8} d^{2} + 2 \, a^{4} c^{7} d^{3}\right )} {\left (a^{2} x^{2} - 1\right )}}{a^{6} c^{11} d {\left | a \right |} + 2 \, a^{4} c^{10} d^{2} {\left | a \right |} + a^{2} c^{9} d^{3} {\left | a \right |}} + \frac {7 \, a^{8} c^{9} d + 11 \, a^{6} c^{8} d^{2} + 4 \, a^{4} c^{7} d^{3}}{a^{6} c^{11} d {\left | a \right |} + 2 \, a^{4} c^{10} d^{2} {\left | a \right |} + a^{2} c^{9} d^{3} {\left | a \right |}}\right )}}{{\left (a^{2} c + {\left (a^{2} x^{2} - 1\right )} d + d\right )}^{\frac {3}{2}}} - \frac {8 \, \log \left ({\left | -\sqrt {-a^{2} x^{2} + 1} \sqrt {-d} + \sqrt {a^{2} c + {\left (a^{2} x^{2} - 1\right )} d + d} \right |}\right )}{c^{3} \sqrt {-d} {\left | a \right |}}\right )} + \frac {{\left (4 \, x^{2} {\left (\frac {2 \, d^{2} x^{2}}{c^{3}} + \frac {5 \, d}{c^{2}}\right )} + \frac {15}{c}\right )} x \arccos \left (a x\right )}{15 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \]

[In]

integrate(arccos(a*x)/(d*x^2+c)^(7/2),x, algorithm="giac")

[Out]

-1/15*a*(sqrt(-a^2*x^2 + 1)*(2*(3*a^6*c^8*d^2 + 2*a^4*c^7*d^3)*(a^2*x^2 - 1)/(a^6*c^11*d*abs(a) + 2*a^4*c^10*d
^2*abs(a) + a^2*c^9*d^3*abs(a)) + (7*a^8*c^9*d + 11*a^6*c^8*d^2 + 4*a^4*c^7*d^3)/(a^6*c^11*d*abs(a) + 2*a^4*c^
10*d^2*abs(a) + a^2*c^9*d^3*abs(a)))/(a^2*c + (a^2*x^2 - 1)*d + d)^(3/2) - 8*log(abs(-sqrt(-a^2*x^2 + 1)*sqrt(
-d) + sqrt(a^2*c + (a^2*x^2 - 1)*d + d)))/(c^3*sqrt(-d)*abs(a))) + 1/15*(4*x^2*(2*d^2*x^2/c^3 + 5*d/c^2) + 15/
c)*x*arccos(a*x)/(d*x^2 + c)^(5/2)

Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\mathrm {acos}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{7/2}} \,d x \]

[In]

int(acos(a*x)/(c + d*x^2)^(7/2),x)

[Out]

int(acos(a*x)/(c + d*x^2)^(7/2), x)