\(\int \frac {x^6}{\arccos (a x)} \, dx\) [42]

Optimal result

Integrand size = 10, antiderivative size = 55 \[ \int \frac {x^6}{\arccos (a x)} \, dx=-\frac {5 \text {Si}(\arccos (a x))}{64 a^7}-\frac {9 \text {Si}(3 \arccos (a x))}{64 a^7}-\frac {5 \text {Si}(5 \arccos (a x))}{64 a^7}-\frac {\text {Si}(7 \arccos (a x))}{64 a^7} \]

[Out]

-5/64*Si(arccos(a*x))/a^7-9/64*Si(3*arccos(a*x))/a^7-5/64*Si(5*arccos(a*x))/a^7-1/64*Si(7*arccos(a*x))/a^7

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4732, 4491, 3380} \[ \int \frac {x^6}{\arccos (a x)} \, dx=-\frac {5 \text {Si}(\arccos (a x))}{64 a^7}-\frac {9 \text {Si}(3 \arccos (a x))}{64 a^7}-\frac {5 \text {Si}(5 \arccos (a x))}{64 a^7}-\frac {\text {Si}(7 \arccos (a x))}{64 a^7} \]

[In]

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

[Out]

(-5*SinIntegral[ArcCos[a*x]])/(64*a^7) - (9*SinIntegral[3*ArcCos[a*x]])/(64*a^7) - (5*SinIntegral[5*ArcCos[a*x
]])/(64*a^7) - SinIntegral[7*ArcCos[a*x]]/(64*a^7)

Rule 3380

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

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4732

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\cos ^6(x) \sin (x)}{x} \, dx,x,\arccos (a x)\right )}{a^7} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {5 \sin (x)}{64 x}+\frac {9 \sin (3 x)}{64 x}+\frac {5 \sin (5 x)}{64 x}+\frac {\sin (7 x)}{64 x}\right ) \, dx,x,\arccos (a x)\right )}{a^7} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin (7 x)}{x} \, dx,x,\arccos (a x)\right )}{64 a^7}-\frac {5 \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arccos (a x)\right )}{64 a^7}-\frac {5 \text {Subst}\left (\int \frac {\sin (5 x)}{x} \, dx,x,\arccos (a x)\right )}{64 a^7}-\frac {9 \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\arccos (a x)\right )}{64 a^7} \\ & = -\frac {5 \text {Si}(\arccos (a x))}{64 a^7}-\frac {9 \text {Si}(3 \arccos (a x))}{64 a^7}-\frac {5 \text {Si}(5 \arccos (a x))}{64 a^7}-\frac {\text {Si}(7 \arccos (a x))}{64 a^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73 \[ \int \frac {x^6}{\arccos (a x)} \, dx=-\frac {5 \text {Si}(\arccos (a x))+9 \text {Si}(3 \arccos (a x))+5 \text {Si}(5 \arccos (a x))+\text {Si}(7 \arccos (a x))}{64 a^7} \]

[In]

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

[Out]

-1/64*(5*SinIntegral[ArcCos[a*x]] + 9*SinIntegral[3*ArcCos[a*x]] + 5*SinIntegral[5*ArcCos[a*x]] + SinIntegral[
7*ArcCos[a*x]])/a^7

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {-\frac {9 \,\operatorname {Si}\left (3 \arccos \left (a x \right )\right )}{64}-\frac {5 \,\operatorname {Si}\left (5 \arccos \left (a x \right )\right )}{64}-\frac {\operatorname {Si}\left (7 \arccos \left (a x \right )\right )}{64}-\frac {5 \,\operatorname {Si}\left (\arccos \left (a x \right )\right )}{64}}{a^{7}}\)	\(40\)
default	\(\frac {-\frac {9 \,\operatorname {Si}\left (3 \arccos \left (a x \right )\right )}{64}-\frac {5 \,\operatorname {Si}\left (5 \arccos \left (a x \right )\right )}{64}-\frac {\operatorname {Si}\left (7 \arccos \left (a x \right )\right )}{64}-\frac {5 \,\operatorname {Si}\left (\arccos \left (a x \right )\right )}{64}}{a^{7}}\)	\(40\)

[In]

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

[Out]

1/a^7*(-9/64*Si(3*arccos(a*x))-5/64*Si(5*arccos(a*x))-1/64*Si(7*arccos(a*x))-5/64*Si(arccos(a*x)))

Fricas [F]

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

[In]

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

[Out]

integral(x^6/arccos(a*x), x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**6/acos(a*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^6/arccos(a*x), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {x^6}{\arccos (a x)} \, dx=-\frac {\operatorname {Si}\left (7 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {5 \, \operatorname {Si}\left (5 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {9 \, \operatorname {Si}\left (3 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {5 \, \operatorname {Si}\left (\arccos \left (a x\right )\right )}{64 \, a^{7}} \]

[In]

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

[Out]

-1/64*sin_integral(7*arccos(a*x))/a^7 - 5/64*sin_integral(5*arccos(a*x))/a^7 - 9/64*sin_integral(3*arccos(a*x)
)/a^7 - 5/64*sin_integral(arccos(a*x))/a^7

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^6/acos(a*x), x)