Integrand size = 8, antiderivative size = 65 \[ \int \frac {1}{\arccos (a+b x)^3} \, dx=\frac {\sqrt {1-(a+b x)^2}}{2 b \arccos (a+b x)^2}+\frac {a+b x}{2 b \arccos (a+b x)}+\frac {\text {Si}(\arccos (a+b x))}{2 b} \] Output:
1/2*(1-(b*x+a)^2)^(1/2)/b/arccos(b*x+a)^2+1/2*(b*x+a)/b/arccos(b*x+a)+1/2* Si(arccos(b*x+a))/b
Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\arccos (a+b x)^3} \, dx=\frac {\sqrt {1-(a+b x)^2}}{2 b \arccos (a+b x)^2}+\frac {a+b x}{2 b \arccos (a+b x)}+\frac {\text {Si}(\arccos (a+b x))}{2 b} \] Input:
Integrate[ArcCos[a + b*x]^(-3),x]
Output:
Sqrt[1 - (a + b*x)^2]/(2*b*ArcCos[a + b*x]^2) + (a + b*x)/(2*b*ArcCos[a + b*x]) + SinIntegral[ArcCos[a + b*x]]/(2*b)
Time = 0.42 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5303, 5133, 5223, 5135, 3042, 3780}
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.
\(\displaystyle \int \frac {1}{\arccos (a+b x)^3} \, dx\) |
\(\Big \downarrow \) 5303 |
\(\displaystyle \frac {\int \frac {1}{\arccos (a+b x)^3}d(a+b x)}{b}\) |
\(\Big \downarrow \) 5133 |
\(\displaystyle \frac {\frac {1}{2} \int \frac {a+b x}{\sqrt {1-(a+b x)^2} \arccos (a+b x)^2}d(a+b x)+\frac {\sqrt {1-(a+b x)^2}}{2 \arccos (a+b x)^2}}{b}\) |
\(\Big \downarrow \) 5223 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {a+b x}{\arccos (a+b x)}-\int \frac {1}{\arccos (a+b x)}d(a+b x)\right )+\frac {\sqrt {1-(a+b x)^2}}{2 \arccos (a+b x)^2}}{b}\) |
\(\Big \downarrow \) 5135 |
\(\displaystyle \frac {\frac {1}{2} \left (\int \frac {\sqrt {1-(a+b x)^2}}{\arccos (a+b x)}d\arccos (a+b x)+\frac {a+b x}{\arccos (a+b x)}\right )+\frac {\sqrt {1-(a+b x)^2}}{2 \arccos (a+b x)^2}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{2} \left (\int \frac {\sin (\arccos (a+b x))}{\arccos (a+b x)}d\arccos (a+b x)+\frac {a+b x}{\arccos (a+b x)}\right )+\frac {\sqrt {1-(a+b x)^2}}{2 \arccos (a+b x)^2}}{b}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \frac {\frac {1}{2} \left (\text {Si}(\arccos (a+b x))+\frac {a+b x}{\arccos (a+b x)}\right )+\frac {\sqrt {1-(a+b x)^2}}{2 \arccos (a+b x)^2}}{b}\) |
Input:
Int[ArcCos[a + b*x]^(-3),x]
Output:
(Sqrt[1 - (a + b*x)^2]/(2*ArcCos[a + b*x]^2) + ((a + b*x)/ArcCos[a + b*x] + SinIntegral[ArcCos[a + b*x]])/2)/b
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-Sqrt[1 - c ^2*x^2])*((a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c/(b*(n + 1 )) Int[x*((a + b*ArcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ [{a, b, c}, x] && LtQ[n, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[-(b*c)^(-1) Subst[Int[x^n*Sin[-a/b + x/b], x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-(f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c ^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcCos[c*x])^(n + 1), x] + Simp[f*(m/(b*c*( n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[(f*x)^(m - 1)*(a + b *ArcCos[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2 *d + e, 0] && LtQ[n, -1]
Int[((a_.) + ArcCos[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcCos[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]
Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {1-\left (b x +a \right )^{2}}}{2 \arccos \left (b x +a \right )^{2}}+\frac {b x +a}{2 \arccos \left (b x +a \right )}+\frac {\operatorname {Si}\left (\arccos \left (b x +a \right )\right )}{2}}{b}\) | \(53\) |
default | \(\frac {\frac {\sqrt {1-\left (b x +a \right )^{2}}}{2 \arccos \left (b x +a \right )^{2}}+\frac {b x +a}{2 \arccos \left (b x +a \right )}+\frac {\operatorname {Si}\left (\arccos \left (b x +a \right )\right )}{2}}{b}\) | \(53\) |
Input:
int(1/arccos(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/b*(1/2/arccos(b*x+a)^2*(1-(b*x+a)^2)^(1/2)+1/2*(b*x+a)/arccos(b*x+a)+1/2 *Si(arccos(b*x+a)))
\[ \int \frac {1}{\arccos (a+b x)^3} \, dx=\int { \frac {1}{\arccos \left (b x + a\right )^{3}} \,d x } \] Input:
integrate(1/arccos(b*x+a)^3,x, algorithm="fricas")
Output:
integral(arccos(b*x + a)^(-3), x)
\[ \int \frac {1}{\arccos (a+b x)^3} \, dx=\int \frac {1}{\operatorname {acos}^{3}{\left (a + b x \right )}}\, dx \] Input:
integrate(1/acos(b*x+a)**3,x)
Output:
Integral(acos(a + b*x)**(-3), x)
\[ \int \frac {1}{\arccos (a+b x)^3} \, dx=\int { \frac {1}{\arccos \left (b x + a\right )^{3}} \,d x } \] Input:
integrate(1/arccos(b*x+a)^3,x, algorithm="maxima")
Output:
-1/2*(b*arctan2(sqrt(b*x + a + 1)*sqrt(-b*x - a + 1), b*x + a)^2*integrate (1/arctan2(sqrt(b*x + a + 1)*sqrt(-b*x - a + 1), b*x + a), x) - (b*x + a)* arctan2(sqrt(b*x + a + 1)*sqrt(-b*x - a + 1), b*x + a) - sqrt(b*x + a + 1) *sqrt(-b*x - a + 1))/(b*arctan2(sqrt(b*x + a + 1)*sqrt(-b*x - a + 1), b*x + a)^2)
Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\arccos (a+b x)^3} \, dx=\frac {\operatorname {Si}\left (\arccos \left (b x + a\right )\right )}{2 \, b} + \frac {b x + a}{2 \, b \arccos \left (b x + a\right )} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{2 \, b \arccos \left (b x + a\right )^{2}} \] Input:
integrate(1/arccos(b*x+a)^3,x, algorithm="giac")
Output:
1/2*sin_integral(arccos(b*x + a))/b + 1/2*(b*x + a)/(b*arccos(b*x + a)) + 1/2*sqrt(-(b*x + a)^2 + 1)/(b*arccos(b*x + a)^2)
Timed out. \[ \int \frac {1}{\arccos (a+b x)^3} \, dx=\int \frac {1}{{\mathrm {acos}\left (a+b\,x\right )}^3} \,d x \] Input:
int(1/acos(a + b*x)^3,x)
Output:
int(1/acos(a + b*x)^3, x)
\[ \int \frac {1}{\arccos (a+b x)^3} \, dx=\int \frac {1}{\mathit {acos} \left (b x +a \right )^{3}}d x \] Input:
int(1/acos(b*x+a)^3,x)
Output:
int(1/acos(a + b*x)**3,x)