Integrand size = 14, antiderivative size = 133 \[ \int \sqrt {a+b \arcsin (c+d x)} \, dx=\frac {(c+d x) \sqrt {a+b \arcsin (c+d x)}}{d}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{d}+\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{d} \]
-1/2*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)) *b^(1/2)*2^(1/2)*Pi^(1/2)/d+1/2*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+ c))^(1/2)/b^(1/2))*sin(a/b)*b^(1/2)*2^(1/2)*Pi^(1/2)/d+(d*x+c)*(a+b*arcsin (d*x+c))^(1/2)/d
Result contains complex when optimal does not.
Time = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.97 \[ \int \sqrt {a+b \arcsin (c+d x)} \, dx=\frac {b e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )}{2 d \sqrt {a+b \arcsin (c+d x)}} \]
(b*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[ c + d*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[ 3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(2*d*E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])
Time = 0.68 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {5302, 5130, 5224, 25, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833}
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.
| \(\displaystyle \int \sqrt {a+b \arcsin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 5302 |
| \(\displaystyle \frac {\int \sqrt {a+b \arcsin (c+d x)}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle \frac {(c+d x) \sqrt {a+b \arcsin (c+d x)}-\frac {1}{2} b \int \frac {c+d x}{\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5224 |
| \(\displaystyle \frac {(c+d x) \sqrt {a+b \arcsin (c+d x)}-\frac {1}{2} \int -\frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+(c+d x) \sqrt {a+b \arcsin (c+d x)}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+(c+d x) \sqrt {a+b \arcsin (c+d x)}}{d}\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+\cos \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )+(c+d x) \sqrt {a+b \arcsin (c+d x)}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )+(c+d x) \sqrt {a+b \arcsin (c+d x)}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )+(c+d x) \sqrt {a+b \arcsin (c+d x)}}{d}\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle \frac {\frac {1}{2} \left (2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )d\sqrt {a+b \arcsin (c+d x)}-\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )+(c+d x) \sqrt {a+b \arcsin (c+d x)}}{d}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle \frac {\frac {1}{2} \left (2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )d\sqrt {a+b \arcsin (c+d x)}-2 \cos \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )+(c+d x) \sqrt {a+b \arcsin (c+d x)}}{d}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {\frac {1}{2} \left (2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )d\sqrt {a+b \arcsin (c+d x)}-\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )+(c+d x) \sqrt {a+b \arcsin (c+d x)}}{d}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )+(c+d x) \sqrt {a+b \arcsin (c+d x)}}{d}\) |
((c + d*x)*Sqrt[a + b*ArcSin[c + d*x]] + (-(Sqrt[b]*Sqrt[2*Pi]*Cos[a/b]*Fr esnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]) + Sqrt[b]*Sqrt[2 *Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/ 2)/d
3.3.43.3.1 Defintions of rubi rules used
Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[2/d Subst[Int[Cos[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f }, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos [(d*e - c*f)/d] Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( d*e - c*f)/d] Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d , e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cSin[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
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 = 0.32 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.53
| method | result | size |
| default | \(-\frac {-\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b -\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b +2 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b +2 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a}{2 d \sqrt {a +b \arcsin \left (d x +c \right )}}\) | \(203\) |
-1/2/d/(a+b*arcsin(d*x+c))^(1/2)*(-2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcs in(d*x+c))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcs in(d*x+c))^(1/2)/b)*b-2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1 /2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1 /2)/b)*b+2*arcsin(d*x+c)*sin(-(a+b*arcsin(d*x+c))/b+a/b)*b+2*sin(-(a+b*arc sin(d*x+c))/b+a/b)*a)
Exception generated. \[ \int \sqrt {a+b \arcsin (c+d x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \sqrt {a+b \arcsin (c+d x)} \, dx=\int \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx \]
\[ \int \sqrt {a+b \arcsin (c+d x)} \, dx=\int { \sqrt {b \arcsin \left (d x + c\right ) + a} \,d x } \]
Result contains complex when optimal does not.
Time = 0.64 (sec) , antiderivative size = 563, normalized size of antiderivative = 4.23 \[ \int \sqrt {a+b \arcsin (c+d x)} \, dx=\frac {\sqrt {2} \sqrt {\pi } a b \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{2 \, {\left (\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} d} + \frac {i \, \sqrt {2} \sqrt {\pi } b^{2} \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{4 \, {\left (\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} d} + \frac {\sqrt {2} \sqrt {\pi } a b \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{2 \, {\left (-\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} d} - \frac {i \, \sqrt {2} \sqrt {\pi } b^{2} \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{4 \, {\left (-\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} d} - \frac {\sqrt {\pi } a \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{d {\left (\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} - \frac {\sqrt {\pi } a \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{d {\left (-\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} - \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} e^{\left (i \, \arcsin \left (d x + c\right )\right )}}{2 \, d} + \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} e^{\left (-i \, \arcsin \left (d x + c\right )\right )}}{2 \, d} \]
1/2*sqrt(2)*sqrt(pi)*a*b*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sq rt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I* a/b)/((I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*d) + 1/4*I*sqrt(2)*sqrt(pi)*b^ 2*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2 )*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^2/sqrt(abs(b )) + b*sqrt(abs(b)))*d) + 1/2*sqrt(2)*sqrt(pi)*a*b*erf(1/2*I*sqrt(2)*sqrt( b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*d) - 1/4*I*sqrt(2)*sqrt(pi)*b^2*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a) /sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^ (-I*a/b)/((-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*d) - sqrt(pi)*a*erf(-1/2* I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*ar csin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(d*(I*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b)))) - sqrt(pi)*a*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs( b))/b)*e^(-I*a/b)/(d*(-I*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b)))) - 1/2*I*sqrt(b*arcsin(d*x + c) + a)*e^(I*arcsin(d*x + c))/d + 1/2*I*sqrt(b* arcsin(d*x + c) + a)*e^(-I*arcsin(d*x + c))/d
Timed out. \[ \int \sqrt {a+b \arcsin (c+d x)} \, dx=\int \sqrt {a+b\,\mathrm {asin}\left (c+d\,x\right )} \,d x \]