\(\int \frac {x^6}{\arccos (a x)^2} \, dx\) [51]

Optimal result

Integrand size = 10, antiderivative size = 82 \[ \int \frac {x^6}{\arccos (a x)^2} \, dx=\frac {x^6 \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {5 \operatorname {CosIntegral}(\arccos (a x))}{64 a^7}-\frac {27 \operatorname {CosIntegral}(3 \arccos (a x))}{64 a^7}-\frac {25 \operatorname {CosIntegral}(5 \arccos (a x))}{64 a^7}-\frac {7 \operatorname {CosIntegral}(7 \arccos (a x))}{64 a^7} \]

[Out]

-5/64*Ci(arccos(a*x))/a^7-27/64*Ci(3*arccos(a*x))/a^7-25/64*Ci(5*arccos(a*x))/a^7-7/64*Ci(7*arccos(a*x))/a^7+x
^6*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4728, 3383} \[ \int \frac {x^6}{\arccos (a x)^2} \, dx=-\frac {5 \operatorname {CosIntegral}(\arccos (a x))}{64 a^7}-\frac {27 \operatorname {CosIntegral}(3 \arccos (a x))}{64 a^7}-\frac {25 \operatorname {CosIntegral}(5 \arccos (a x))}{64 a^7}-\frac {7 \operatorname {CosIntegral}(7 \arccos (a x))}{64 a^7}+\frac {x^6 \sqrt {1-a^2 x^2}}{a \arccos (a x)} \]

[In]

Int[x^6/ArcCos[a*x]^2,x]

[Out]

(x^6*Sqrt[1 - a^2*x^2])/(a*ArcCos[a*x]) - (5*CosIntegral[ArcCos[a*x]])/(64*a^7) - (27*CosIntegral[3*ArcCos[a*x
]])/(64*a^7) - (25*CosIntegral[5*ArcCos[a*x]])/(64*a^7) - (7*CosIntegral[7*ArcCos[a*x]])/(64*a^7)

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 4728

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(-x^m)*Sqrt[1 - c^2*x^2]*((a + b*Arc
Cos[c*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), C
os[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cos[-a/b + x/b]^2), x], x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c},
x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {x^6 \sqrt {1-a^2 x^2}}{a \arccos (a x)}+\frac {\text {Subst}\left (\int \left (-\frac {5 \cos (x)}{64 x}-\frac {27 \cos (3 x)}{64 x}-\frac {25 \cos (5 x)}{64 x}-\frac {7 \cos (7 x)}{64 x}\right ) \, dx,x,\arccos (a x)\right )}{a^7} \\ & = \frac {x^6 \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {5 \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arccos (a x)\right )}{64 a^7}-\frac {7 \text {Subst}\left (\int \frac {\cos (7 x)}{x} \, dx,x,\arccos (a x)\right )}{64 a^7}-\frac {25 \text {Subst}\left (\int \frac {\cos (5 x)}{x} \, dx,x,\arccos (a x)\right )}{64 a^7}-\frac {27 \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arccos (a x)\right )}{64 a^7} \\ & = \frac {x^6 \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {5 \operatorname {CosIntegral}(\arccos (a x))}{64 a^7}-\frac {27 \operatorname {CosIntegral}(3 \arccos (a x))}{64 a^7}-\frac {25 \operatorname {CosIntegral}(5 \arccos (a x))}{64 a^7}-\frac {7 \operatorname {CosIntegral}(7 \arccos (a x))}{64 a^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05 \[ \int \frac {x^6}{\arccos (a x)^2} \, dx=-\frac {-64 a^6 x^6 \sqrt {1-a^2 x^2}+5 \arccos (a x) \operatorname {CosIntegral}(\arccos (a x))+27 \arccos (a x) \operatorname {CosIntegral}(3 \arccos (a x))+25 \arccos (a x) \operatorname {CosIntegral}(5 \arccos (a x))+7 \arccos (a x) \operatorname {CosIntegral}(7 \arccos (a x))}{64 a^7 \arccos (a x)} \]

[In]

Integrate[x^6/ArcCos[a*x]^2,x]

[Out]

-1/64*(-64*a^6*x^6*Sqrt[1 - a^2*x^2] + 5*ArcCos[a*x]*CosIntegral[ArcCos[a*x]] + 27*ArcCos[a*x]*CosIntegral[3*A
rcCos[a*x]] + 25*ArcCos[a*x]*CosIntegral[5*ArcCos[a*x]] + 7*ArcCos[a*x]*CosIntegral[7*ArcCos[a*x]])/(a^7*ArcCo
s[a*x])

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.28

method	result	size
derivativedivides	\(\frac {\frac {9 \sin \left (3 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {27 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{64}+\frac {5 \sin \left (5 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {25 \,\operatorname {Ci}\left (5 \arccos \left (a x \right )\right )}{64}+\frac {\sin \left (7 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {7 \,\operatorname {Ci}\left (7 \arccos \left (a x \right )\right )}{64}+\frac {5 \sqrt {-a^{2} x^{2}+1}}{64 \arccos \left (a x \right )}-\frac {5 \,\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{64}}{a^{7}}\)	\(105\)
default	\(\frac {\frac {9 \sin \left (3 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {27 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{64}+\frac {5 \sin \left (5 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {25 \,\operatorname {Ci}\left (5 \arccos \left (a x \right )\right )}{64}+\frac {\sin \left (7 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {7 \,\operatorname {Ci}\left (7 \arccos \left (a x \right )\right )}{64}+\frac {5 \sqrt {-a^{2} x^{2}+1}}{64 \arccos \left (a x \right )}-\frac {5 \,\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{64}}{a^{7}}\)	\(105\)

[In]

int(x^6/arccos(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/a^7*(9/64/arccos(a*x)*sin(3*arccos(a*x))-27/64*Ci(3*arccos(a*x))+5/64/arccos(a*x)*sin(5*arccos(a*x))-25/64*C
i(5*arccos(a*x))+1/64*sin(7*arccos(a*x))/arccos(a*x)-7/64*Ci(7*arccos(a*x))+5/64*(-a^2*x^2+1)^(1/2)/arccos(a*x
)-5/64*Ci(arccos(a*x)))

Fricas [F]

\[ \int \frac {x^6}{\arccos (a x)^2} \, dx=\int { \frac {x^{6}}{\arccos \left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^6/arccos(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^6/arccos(a*x)^2, x)

Sympy [F]

\[ \int \frac {x^6}{\arccos (a x)^2} \, dx=\int \frac {x^{6}}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx \]

[In]

integrate(x**6/acos(a*x)**2,x)

[Out]

Integral(x**6/acos(a*x)**2, x)

Maxima [F]

\[ \int \frac {x^6}{\arccos (a x)^2} \, dx=\int { \frac {x^{6}}{\arccos \left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^6/arccos(a*x)^2,x, algorithm="maxima")

[Out]

(sqrt(a*x + 1)*sqrt(-a*x + 1)*x^6 - a*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)*integrate((7*a^2*x^7 - 6*x^5)
*sqrt(a*x + 1)*sqrt(-a*x + 1)/((a^3*x^2 - a)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)), x))/(a*arctan2(sqrt(
a*x + 1)*sqrt(-a*x + 1), a*x))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.88 \[ \int \frac {x^6}{\arccos (a x)^2} \, dx=\frac {\sqrt {-a^{2} x^{2} + 1} x^{6}}{a \arccos \left (a x\right )} - \frac {7 \, \operatorname {Ci}\left (7 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {25 \, \operatorname {Ci}\left (5 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {27 \, \operatorname {Ci}\left (3 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {5 \, \operatorname {Ci}\left (\arccos \left (a x\right )\right )}{64 \, a^{7}} \]

[In]

integrate(x^6/arccos(a*x)^2,x, algorithm="giac")

[Out]

sqrt(-a^2*x^2 + 1)*x^6/(a*arccos(a*x)) - 7/64*cos_integral(7*arccos(a*x))/a^7 - 25/64*cos_integral(5*arccos(a*
x))/a^7 - 27/64*cos_integral(3*arccos(a*x))/a^7 - 5/64*cos_integral(arccos(a*x))/a^7

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\arccos (a x)^2} \, dx=\int \frac {x^6}{{\mathrm {acos}\left (a\,x\right )}^2} \,d x \]

[In]

int(x^6/acos(a*x)^2,x)

[Out]

int(x^6/acos(a*x)^2, x)