Integrand size = 24, antiderivative size = 381 \[ \int \left (d-c^2 d x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=\frac {2}{3} d x \sqrt {a+b \arcsin (c x)}+\frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}-\frac {\sqrt {b} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{12 c}-\frac {\sqrt {b} d \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{3 c}-\frac {\sqrt {b} d \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{12 c}+\frac {\sqrt {b} d \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{12 c}+\frac {\sqrt {b} d \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 c}+\frac {\sqrt {b} d \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{12 c} \] Output:
2/3*d*x*(a+b*arcsin(c*x))^(1/2)+1/3*d*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^(1/ 2)-3/8*b^(1/2)*d*2^(1/2)*Pi^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b* arcsin(c*x))^(1/2)/b^(1/2))/c-1/72*b^(1/2)*d*6^(1/2)*Pi^(1/2)*cos(3*a/b)*F resnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))/c+3/8*b^(1/2)*d* 2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2) )*sin(a/b)/c+1/72*b^(1/2)*d*6^(1/2)*Pi^(1/2)*FresnelC(6^(1/2)/Pi^(1/2)*(a+ b*arcsin(c*x))^(1/2)/b^(1/2))*sin(3*a/b)/c
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.60 \[ \int \left (d-c^2 d x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=\frac {b d e^{-\frac {3 i a}{b}} \left (27 e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+27 e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c x))}{b}\right )+\sqrt {3} \left (\sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )\right )\right )}{72 c \sqrt {a+b \arcsin (c x)}} \] Input:
Integrate[(d - c^2*d*x^2)*Sqrt[a + b*ArcSin[c*x]],x]
Output:
(b*d*(27*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((- I)*(a + b*ArcSin[c*x]))/b] + 27*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x] ))/b]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b] + Sqrt[3]*(Sqrt[((-I)*(a + b*A rcSin[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + E^(((6*I)*a)/ b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcSin[c*x]))/ b])))/(72*c*E^(((3*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])
Time = 1.68 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5158, 5130, 5224, 25, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833, 4906, 2009}
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.
\(\displaystyle \int \left (d-c^2 d x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx\) |
\(\Big \downarrow \) 5158 |
\(\displaystyle -\frac {1}{6} b c d \int \frac {x \sqrt {1-c^2 x^2}}{\sqrt {a+b \arcsin (c x)}}dx+\frac {2}{3} d \int \sqrt {a+b \arcsin (c x)}dx+\frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}\) |
\(\Big \downarrow \) 5130 |
\(\displaystyle \frac {2}{3} d \left (x \sqrt {a+b \arcsin (c x)}-\frac {1}{2} b c \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}dx\right )-\frac {1}{6} b c d \int \frac {x \sqrt {1-c^2 x^2}}{\sqrt {a+b \arcsin (c x)}}dx+\frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}\) |
\(\Big \downarrow \) 5224 |
\(\displaystyle \frac {2}{3} d \left (x \sqrt {a+b \arcsin (c x)}-\frac {\int -\frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{2 c}\right )-\frac {d \int -\frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{6 c}+\frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2}{3} d \left (\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{2 c}+x \sqrt {a+b \arcsin (c x)}\right )+\frac {d \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{6 c}+\frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} d \left (\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{2 c}+x \sqrt {a+b \arcsin (c x)}\right )+\frac {d \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{6 c}+\frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}\) |
\(\Big \downarrow \) 3787 |
\(\displaystyle \frac {d \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{6 c}+\frac {2}{3} d \left (x \sqrt {a+b \arcsin (c x)}-\frac {-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))-\cos \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{2 c}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {d \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{6 c}+\frac {2}{3} d \left (x \sqrt {a+b \arcsin (c x)}-\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{2 c}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {d \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{6 c}+\frac {2}{3} d \left (x \sqrt {a+b \arcsin (c x)}-\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))-\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{2 c}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}\) |
\(\Big \downarrow \) 3785 |
\(\displaystyle \frac {d \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{6 c}+\frac {2}{3} d \left (x \sqrt {a+b \arcsin (c x)}-\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}}{2 c}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle \frac {d \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{6 c}+\frac {2}{3} d \left (x \sqrt {a+b \arcsin (c x)}-\frac {2 \cos \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}}{2 c}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {2}{3} d \left (x \sqrt {a+b \arcsin (c x)}-\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}}{2 c}\right )+\frac {d \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{6 c}+\frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {d \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{6 c}+\frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}+\frac {2}{3} d \left (x \sqrt {a+b \arcsin (c x)}-\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 c}\right )\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {d \int \left (\frac {\sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 \sqrt {a+b \arcsin (c x)}}+\frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{4 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{6 c}+\frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}+\frac {2}{3} d \left (x \sqrt {a+b \arcsin (c x)}-\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 c}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}+\frac {2}{3} d \left (x \sqrt {a+b \arcsin (c x)}-\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 c}\right )-\frac {d \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )\right )}{6 c}\) |
Input:
Int[(d - c^2*d*x^2)*Sqrt[a + b*ArcSin[c*x]],x]
Output:
(d*x*(1 - c^2*x^2)*Sqrt[a + b*ArcSin[c*x]])/3 + (2*d*(x*Sqrt[a + b*ArcSin[ c*x]] - (Sqrt[b]*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSi n[c*x]])/Sqrt[b]] - Sqrt[b]*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*Arc Sin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*c)))/3 - (d*((Sqrt[b]*Sqrt[Pi/2]*Cos[a/b] *FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/2 + (Sqrt[b]*Sqrt [Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]] )/2 - (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sq rt[b]]*Sin[a/b])/2 - (Sqrt[b]*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*A rcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/2))/(6*c)
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[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG tQ[p, 0]
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_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (S imp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c , d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 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]
Time = 0.10 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.96
method | result | size |
default | \(-\frac {d \left (-27 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b -27 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b -\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b -\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b +54 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b +54 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a +6 \arcsin \left (c x \right ) \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b +6 \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a \right )}{72 c \sqrt {a +b \arcsin \left (c x \right )}}\) | \(364\) |
Input:
int((-c^2*d*x^2+d)*(a+b*arcsin(c*x))^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/72/c*d/(a+b*arcsin(c*x))^(1/2)*(-27*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b* arcsin(c*x))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*ar csin(c*x))^(1/2)/b)*b-27*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^( 1/2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/ 2)/b)*b-2^(1/2)*Pi^(1/2)*(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(3*a/b)*F resnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b-2^(1/ 2)*Pi^(1/2)*(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(3*a/b)*FresnelC(3*2^( 1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b+54*arcsin(c*x)*sin (-(a+b*arcsin(c*x))/b+a/b)*b+54*sin(-(a+b*arcsin(c*x))/b+a/b)*a+6*arcsin(c *x)*sin(-3*(a+b*arcsin(c*x))/b+3*a/b)*b+6*sin(-3*(a+b*arcsin(c*x))/b+3*a/b )*a)
Exception generated. \[ \int \left (d-c^2 d x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*d*x^2+d)*(a+b*arcsin(c*x))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \left (d-c^2 d x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=- d \left (\int c^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c x \right )}}\, dx + \int \left (- \sqrt {a + b \operatorname {asin}{\left (c x \right )}}\right )\, dx\right ) \] Input:
integrate((-c**2*d*x**2+d)*(a+b*asin(c*x))**(1/2),x)
Output:
-d*(Integral(c**2*x**2*sqrt(a + b*asin(c*x)), x) + Integral(-sqrt(a + b*as in(c*x)), x))
\[ \int \left (d-c^2 d x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=\int { -{\left (c^{2} d x^{2} - d\right )} \sqrt {b \arcsin \left (c x\right ) + a} \,d x } \] Input:
integrate((-c^2*d*x^2+d)*(a+b*arcsin(c*x))^(1/2),x, algorithm="maxima")
Output:
-integrate((c^2*d*x^2 - d)*sqrt(b*arcsin(c*x) + a), x)
Result contains complex when optimal does not.
Time = 1.10 (sec) , antiderivative size = 1103, normalized size of antiderivative = 2.90 \[ \int \left (d-c^2 d x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=\text {Too large to display} \] Input:
integrate((-c^2*d*x^2+d)*(a+b*arcsin(c*x))^(1/2),x, algorithm="giac")
Output:
3/8*sqrt(2)*sqrt(pi)*a*b^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sq rt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b) /((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 3/16*I*sqrt(2)*sqrt(pi)*b^3 *d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*s qrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^ 2*sqrt(abs(b)))*c) + 3/8*sqrt(2)*sqrt(pi)*a*b^2*d*erf(1/2*I*sqrt(2)*sqrt(b *arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt( abs(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) - 3/16* I*sqrt(2)*sqrt(pi)*b^3*d*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(ab s(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((- I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 1/4*sqrt(pi)*a*b^(3/2)*d*erf(- 1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin( c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b) )*c) + 1/24*I*sqrt(pi)*b^(5/2)*d*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/ sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b )/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*c) + 1/4*sqrt(pi)*a*b^(3/2)*d*erf( -1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin (c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/((sqrt(6)*b^2 - I*sqrt(6)*b^3/abs( b))*c) - 1/24*I*sqrt(pi)*b^(5/2)*d*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a )/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3...
Timed out. \[ \int \left (d-c^2 d x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=\int \sqrt {a+b\,\mathrm {asin}\left (c\,x\right )}\,\left (d-c^2\,d\,x^2\right ) \,d x \] Input:
int((a + b*asin(c*x))^(1/2)*(d - c^2*d*x^2),x)
Output:
int((a + b*asin(c*x))^(1/2)*(d - c^2*d*x^2), x)
\[ \int \left (d-c^2 d x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=d \left (\int \sqrt {\mathit {asin} \left (c x \right ) b +a}d x -\left (\int \sqrt {\mathit {asin} \left (c x \right ) b +a}\, x^{2}d x \right ) c^{2}\right ) \] Input:
int((-c^2*d*x^2+d)*(a+b*asin(c*x))^(1/2),x)
Output:
d*(int(sqrt(asin(c*x)*b + a),x) - int(sqrt(asin(c*x)*b + a)*x**2,x)*c**2)