\(\int \frac {1}{(a+b \arcsin (c+d x))^3} \, dx\) [231]

Optimal result

Integrand size = 12, antiderivative size = 127 \[ \int \frac {1}{(a+b \arcsin (c+d x))^3} \, dx=-\frac {\sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}+\frac {c+d x}{2 b^2 d (a+b \arcsin (c+d x))}-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{2 b^3 d}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{2 b^3 d} \]

[Out]

1/2*(d*x+c)/b^2/d/(a+b*arcsin(d*x+c))-1/2*Ci((a+b*arcsin(d*x+c))/b)*cos(a/b)/b^3/d-1/2*Si((a+b*arcsin(d*x+c))/
b)*sin(a/b)/b^3/d-1/2*(1-(d*x+c)^2)^(1/2)/b/d/(a+b*arcsin(d*x+c))^2

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4887, 4717, 4807, 4719, 3384, 3380, 3383} \[ \int \frac {1}{(a+b \arcsin (c+d x))^3} \, dx=-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{2 b^3 d}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{2 b^3 d}+\frac {c+d x}{2 b^2 d (a+b \arcsin (c+d x))}-\frac {\sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2} \]

[In]

Int[(a + b*ArcSin[c + d*x])^(-3),x]

[Out]

-1/2*Sqrt[1 - (c + d*x)^2]/(b*d*(a + b*ArcSin[c + d*x])^2) + (c + d*x)/(2*b^2*d*(a + b*ArcSin[c + d*x])) - (Co
s[a/b]*CosIntegral[(a + b*ArcSin[c + d*x])/b])/(2*b^3*d) - (Sin[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(
2*b^3*d)

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 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 3384

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

Rule 4717

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 - c^2*x^2]*((a + b*ArcSin[c*x])^(n + 1)/
(b*c*(n + 1))), x] + Dist[c/(b*(n + 1)), Int[x*((a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; Free
Q[{a, b, c}, x] && LtQ[n, -1]

Rule 4719

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cos[-a/b + x/b], x], x,
a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 4807

Int[(((a_.) + ArcSin[(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*ArcSin[c*x])^(n + 1), x] - Dist[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*ArcSin[c*x])^(n + 1), x], x] /;
FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]

Rule 4887

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcSin[x])^n, x],
x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{(a+b \arcsin (x))^3} \, dx,x,c+d x\right )}{d} \\ & = -\frac {\sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}-\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} (a+b \arcsin (x))^2} \, dx,x,c+d x\right )}{2 b d} \\ & = -\frac {\sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}+\frac {c+d x}{2 b^2 d (a+b \arcsin (c+d x))}-\frac {\text {Subst}\left (\int \frac {1}{a+b \arcsin (x)} \, dx,x,c+d x\right )}{2 b^2 d} \\ & = -\frac {\sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}+\frac {c+d x}{2 b^2 d (a+b \arcsin (c+d x))}-\frac {\text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{2 b^3 d} \\ & = -\frac {\sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}+\frac {c+d x}{2 b^2 d (a+b \arcsin (c+d x))}-\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{2 b^3 d}-\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{2 b^3 d} \\ & = -\frac {\sqrt {1-(c+d x)^2}}{2 b d (a+b \arcsin (c+d x))^2}+\frac {c+d x}{2 b^2 d (a+b \arcsin (c+d x))}-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{2 b^3 d}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{2 b^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(a+b \arcsin (c+d x))^3} \, dx=-\frac {\frac {b^2 \sqrt {1-(c+d x)^2}}{(a+b \arcsin (c+d x))^2}-\frac {b (c+d x)}{a+b \arcsin (c+d x)}+\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )}{2 b^3 d} \]

[In]

Integrate[(a + b*ArcSin[c + d*x])^(-3),x]

[Out]

-1/2*((b^2*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^2 - (b*(c + d*x))/(a + b*ArcSin[c + d*x]) + Cos[a/b]
*CosIntegral[a/b + ArcSin[c + d*x]] + Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x]])/(b^3*d)

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.24

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{2 \left (a +b \arcsin \left (d x +c \right )\right )^{2} b}-\frac {\arcsin \left (d x +c \right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\arcsin \left (d x +c \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b +\operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a +\sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a -\left (d x +c \right ) b}{2 \left (a +b \arcsin \left (d x +c \right )\right ) b^{3}}}{d}\)	\(158\)
default	\(\frac {-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{2 \left (a +b \arcsin \left (d x +c \right )\right )^{2} b}-\frac {\arcsin \left (d x +c \right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\arcsin \left (d x +c \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b +\operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a +\sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a -\left (d x +c \right ) b}{2 \left (a +b \arcsin \left (d x +c \right )\right ) b^{3}}}{d}\)	\(158\)

[In]

int(1/(a+b*arcsin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2*(1-(d*x+c)^2)^(1/2)/(a+b*arcsin(d*x+c))^2/b-1/2*(arcsin(d*x+c)*Ci(arcsin(d*x+c)+a/b)*cos(a/b)*b+arcs
in(d*x+c)*sin(a/b)*Si(arcsin(d*x+c)+a/b)*b+Ci(arcsin(d*x+c)+a/b)*cos(a/b)*a+sin(a/b)*Si(arcsin(d*x+c)+a/b)*a-(
d*x+c)*b)/(a+b*arcsin(d*x+c))/b^3)

Fricas [F]

\[ \int \frac {1}{(a+b \arcsin (c+d x))^3} \, dx=\int { \frac {1}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(1/(a+b*arcsin(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(1/(b^3*arcsin(d*x + c)^3 + 3*a*b^2*arcsin(d*x + c)^2 + 3*a^2*b*arcsin(d*x + c) + a^3), x)

Sympy [F]

\[ \int \frac {1}{(a+b \arcsin (c+d x))^3} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{3}}\, dx \]

[In]

integrate(1/(a+b*asin(d*x+c))**3,x)

[Out]

Integral((a + b*asin(c + d*x))**(-3), x)

Maxima [F]

\[ \int \frac {1}{(a+b \arcsin (c+d x))^3} \, dx=\int { \frac {1}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(1/(a+b*arcsin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/2*(a*d*x - sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)*b + a*c + (b*d*x + b*c)*arctan2(d*x + c, sqrt(d*x + c + 1)*s
qrt(-d*x - c + 1)) - 2*(b^4*d*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + 2*a*b^3*d*arctan2(d*x
+ c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + a^2*b^2*d)*integrate(1/2/(b^3*arctan2(d*x + c, sqrt(d*x + c + 1)
*sqrt(-d*x - c + 1)) + a*b^2), x))/(b^4*d*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 + 2*a*b^3*d
*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) + a^2*b^2*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (117) = 234\).

Time = 0.31 (sec) , antiderivative size = 547, normalized size of antiderivative = 4.31 \[ \int \frac {1}{(a+b \arcsin (c+d x))^3} \, dx=-\frac {b^{2} \arcsin \left (d x + c\right )^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right )}{2 \, {\left (b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d\right )}} - \frac {b^{2} \arcsin \left (d x + c\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right )}{2 \, {\left (b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d\right )}} - \frac {a b \arcsin \left (d x + c\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right )}{b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d} - \frac {a b \arcsin \left (d x + c\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right )}{b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d} + \frac {{\left (d x + c\right )} b^{2} \arcsin \left (d x + c\right )}{2 \, {\left (b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d\right )}} - \frac {a^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right )}{2 \, {\left (b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d\right )}} - \frac {a^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right )}{2 \, {\left (b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d\right )}} + \frac {{\left (d x + c\right )} a b}{2 \, {\left (b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d\right )}} - \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} b^{2}}{2 \, {\left (b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d\right )}} \]

[In]

integrate(1/(a+b*arcsin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/2*b^2*arcsin(d*x + c)^2*cos(a/b)*cos_integral(a/b + arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*a
rcsin(d*x + c) + a^2*b^3*d) - 1/2*b^2*arcsin(d*x + c)^2*sin(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^5*d*ar
csin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - a*b*arcsin(d*x + c)*cos(a/b)*cos_integral(a/b + arc
sin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - a*b*arcsin(d*x + c)*sin(a/b)
*sin_integral(a/b + arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) + 1/2*(
d*x + c)*b^2*arcsin(d*x + c)/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 1/2*a^2*cos(a
/b)*cos_integral(a/b + arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 1/
2*a^2*sin(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*
b^3*d) + 1/2*(d*x + c)*a*b/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d) - 1/2*sqrt(-(d*x
+ c)^2 + 1)*b^2/(b^5*d*arcsin(d*x + c)^2 + 2*a*b^4*d*arcsin(d*x + c) + a^2*b^3*d)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \arcsin (c+d x))^3} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3} \,d x \]

[In]

int(1/(a + b*asin(c + d*x))^3,x)

[Out]

int(1/(a + b*asin(c + d*x))^3, x)