Integrand size = 12, antiderivative size = 164 \[ \int \frac {1}{(a+b \arcsin (c+d x))^4} \, dx=-\frac {\sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^3}+\frac {c+d x}{6 b^2 d (a+b \arcsin (c+d x))^2}+\frac {\sqrt {1-(c+d x)^2}}{6 b^3 d (a+b \arcsin (c+d x))}-\frac {\operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{6 b^4 d}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{6 b^4 d} \]
1/6*(d*x+c)/b^2/d/(a+b*arcsin(d*x+c))^2+1/6*cos(a/b)*Si((a+b*arcsin(d*x+c) )/b)/b^4/d-1/6*Ci((a+b*arcsin(d*x+c))/b)*sin(a/b)/b^4/d-1/3*(1-(d*x+c)^2)^ (1/2)/b/d/(a+b*arcsin(d*x+c))^3+1/6*(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsin( d*x+c))
Time = 0.45 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(a+b \arcsin (c+d x))^4} \, dx=\frac {-\frac {2 b^3 \sqrt {1-(c+d x)^2}}{(a+b \arcsin (c+d x))^3}+\frac {b^2 (c+d x)}{(a+b \arcsin (c+d x))^2}+\frac {b \sqrt {1-(c+d x)^2}}{a+b \arcsin (c+d x)}-\operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right ) \sin \left (\frac {a}{b}\right )+\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )}{6 b^4 d} \]
((-2*b^3*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^3 + (b^2*(c + d*x) )/(a + b*ArcSin[c + d*x])^2 + (b*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x]) - CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b] + Cos[a/b]*SinIntegral [a/b + ArcSin[c + d*x]])/(6*b^4*d)
Time = 0.80 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5302, 5132, 5222, 5132, 5224, 25, 3042, 3784, 25, 3042, 3780, 3783}
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.
| \(\displaystyle \int \frac {1}{(a+b \arcsin (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 5302 |
| \(\displaystyle \frac {\int \frac {1}{(a+b \arcsin (c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5132 |
| \(\displaystyle \frac {-\frac {\int \frac {c+d x}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}d(c+d x)}{3 b}-\frac {\sqrt {1-(c+d x)^2}}{3 b (a+b \arcsin (c+d x))^3}}{d}\) |
| \(\Big \downarrow \) 5222 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {1}{(a+b \arcsin (c+d x))^2}d(c+d x)}{2 b}-\frac {c+d x}{2 b (a+b \arcsin (c+d x))^2}}{3 b}-\frac {\sqrt {1-(c+d x)^2}}{3 b (a+b \arcsin (c+d x))^3}}{d}\) |
| \(\Big \downarrow \) 5132 |
| \(\displaystyle \frac {-\frac {\frac {-\frac {\int \frac {c+d x}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}d(c+d x)}{b}-\frac {\sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \arcsin (c+d x))^2}}{3 b}-\frac {\sqrt {1-(c+d x)^2}}{3 b (a+b \arcsin (c+d x))^3}}{d}\) |
| \(\Big \downarrow \) 5224 |
| \(\displaystyle \frac {-\frac {\frac {-\frac {\int -\frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {\sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \arcsin (c+d x))^2}}{3 b}-\frac {\sqrt {1-(c+d x)^2}}{3 b (a+b \arcsin (c+d x))^3}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {\sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \arcsin (c+d x))^2}}{3 b}-\frac {\sqrt {1-(c+d x)^2}}{3 b (a+b \arcsin (c+d x))^3}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {\sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \arcsin (c+d x))^2}}{3 b}-\frac {\sqrt {1-(c+d x)^2}}{3 b (a+b \arcsin (c+d x))^3}}{d}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {-\frac {\frac {-\frac {-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))-\cos \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {\sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \arcsin (c+d x))^2}}{3 b}-\frac {\sqrt {1-(c+d x)^2}}{3 b (a+b \arcsin (c+d x))^3}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {-\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {\sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \arcsin (c+d x))^2}}{3 b}-\frac {\sqrt {1-(c+d x)^2}}{3 b (a+b \arcsin (c+d x))^3}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {-\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))-\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}+\frac {\pi }{2}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {\sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \arcsin (c+d x))^2}}{3 b}-\frac {\sqrt {1-(c+d x)^2}}{3 b (a+b \arcsin (c+d x))^3}}{d}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle \frac {-\frac {\frac {-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )-\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}+\frac {\pi }{2}\right )}{a+b \arcsin (c+d x)}d(a+b \arcsin (c+d x))}{b^2}-\frac {\sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \arcsin (c+d x))^2}}{3 b}-\frac {\sqrt {1-(c+d x)^2}}{3 b (a+b \arcsin (c+d x))^3}}{d}\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle \frac {-\frac {\frac {-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )-\sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{b^2}-\frac {\sqrt {1-(c+d x)^2}}{b (a+b \arcsin (c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \arcsin (c+d x))^2}}{3 b}-\frac {\sqrt {1-(c+d x)^2}}{3 b (a+b \arcsin (c+d x))^3}}{d}\) |
(-1/3*Sqrt[1 - (c + d*x)^2]/(b*(a + b*ArcSin[c + d*x])^3) - (-1/2*(c + d*x )/(b*(a + b*ArcSin[c + d*x])^2) + (-(Sqrt[1 - (c + d*x)^2]/(b*(a + b*ArcSi n[c + d*x]))) - (-(CosIntegral[(a + b*ArcSin[c + d*x])/b]*Sin[a/b]) + Cos[ a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/b^2)/(2*b))/(3*b))/d
3.3.37.3.1 Defintions of rubi rules used
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[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[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]
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] + Simp[c/(b*(n + 1)) Int[x*((a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a , b, c}, x] && LtQ[n, -1]
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] - 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* 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]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x ^2)^p] Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]
Time = 0.56 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.65
| method | result | size |
| derivativedivides | \(\frac {-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{3 \left (a +b \arcsin \left (d x +c \right )\right )^{3} b}+\frac {\arcsin \left (d x +c \right )^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b^{2}-\arcsin \left (d x +c \right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b^{2}+2 \arcsin \left (d x +c \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a b -2 \arcsin \left (d x +c \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a b +\sqrt {1-\left (d x +c \right )^{2}}\, \arcsin \left (d x +c \right ) b^{2}+\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{2}-\sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{2}+\sqrt {1-\left (d x +c \right )^{2}}\, a b +\left (d x +c \right ) b^{2}}{6 \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{4}}}{d}\) | \(270\) |
| default | \(\frac {-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{3 \left (a +b \arcsin \left (d x +c \right )\right )^{3} b}+\frac {\arcsin \left (d x +c \right )^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b^{2}-\arcsin \left (d x +c \right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b^{2}+2 \arcsin \left (d x +c \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a b -2 \arcsin \left (d x +c \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a b +\sqrt {1-\left (d x +c \right )^{2}}\, \arcsin \left (d x +c \right ) b^{2}+\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{2}-\sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{2}+\sqrt {1-\left (d x +c \right )^{2}}\, a b +\left (d x +c \right ) b^{2}}{6 \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{4}}}{d}\) | \(270\) |
1/d*(-1/3*(1-(d*x+c)^2)^(1/2)/(a+b*arcsin(d*x+c))^3/b+1/6*(arcsin(d*x+c)^2 *cos(a/b)*Si(arcsin(d*x+c)+a/b)*b^2-arcsin(d*x+c)^2*sin(a/b)*Ci(arcsin(d*x +c)+a/b)*b^2+2*arcsin(d*x+c)*cos(a/b)*Si(arcsin(d*x+c)+a/b)*a*b-2*arcsin(d *x+c)*sin(a/b)*Ci(arcsin(d*x+c)+a/b)*a*b+(1-(d*x+c)^2)^(1/2)*arcsin(d*x+c) *b^2+cos(a/b)*Si(arcsin(d*x+c)+a/b)*a^2-sin(a/b)*Ci(arcsin(d*x+c)+a/b)*a^2 +(1-(d*x+c)^2)^(1/2)*a*b+(d*x+c)*b^2)/(a+b*arcsin(d*x+c))^2/b^4)
\[ \int \frac {1}{(a+b \arcsin (c+d x))^4} \, dx=\int { \frac {1}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]
integral(1/(b^4*arcsin(d*x + c)^4 + 4*a*b^3*arcsin(d*x + c)^3 + 6*a^2*b^2* arcsin(d*x + c)^2 + 4*a^3*b*arcsin(d*x + c) + a^4), x)
\[ \int \frac {1}{(a+b \arcsin (c+d x))^4} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{4}}\, dx \]
Timed out. \[ \int \frac {1}{(a+b \arcsin (c+d x))^4} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1112 vs. \(2 (150) = 300\).
Time = 0.31 (sec) , antiderivative size = 1112, normalized size of antiderivative = 6.78 \[ \int \frac {1}{(a+b \arcsin (c+d x))^4} \, dx=\text {Too large to display} \]
-1/6*b^3*arcsin(d*x + c)^3*cos_integral(a/b + arcsin(d*x + c))*sin(a/b)/(b ^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin( d*x + c) + a^3*b^4*d) + 1/6*b^3*arcsin(d*x + c)^3*cos(a/b)*sin_integral(a/ b + arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^ 2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 1/2*a*b^2*arcsin(d*x + c)^2 *cos_integral(a/b + arcsin(d*x + c))*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3 *a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 1/ 2*a*b^2*arcsin(d*x + c)^2*cos(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^ 7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d *x + c) + a^3*b^4*d) - 1/2*a^2*b*arcsin(d*x + c)*cos_integral(a/b + arcsin (d*x + c))*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 1/2*a^2*b*arcsin(d*x + c)*co s(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^7*d*arcsin(d*x + c)^3 + 3*a* b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 1/6*s qrt(-(d*x + c)^2 + 1)*b^3*arcsin(d*x + c)^2/(b^7*d*arcsin(d*x + c)^3 + 3*a *b^6*d*arcsin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 1/6* (d*x + c)*b^3*arcsin(d*x + c)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*arcsin( d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) - 1/6*a^3*cos_integr al(a/b + arcsin(d*x + c))*sin(a/b)/(b^7*d*arcsin(d*x + c)^3 + 3*a*b^6*d*ar csin(d*x + c)^2 + 3*a^2*b^5*d*arcsin(d*x + c) + a^3*b^4*d) + 1/6*a^3*co...
Timed out. \[ \int \frac {1}{(a+b \arcsin (c+d x))^4} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4} \,d x \]