Integrand size = 14, antiderivative size = 218 \[ \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx=-\frac {2 \sqrt {1-(c+d x)^2}}{5 b d (a+b \arcsin (c+d x))^{5/2}}+\frac {4 (c+d x)}{15 b^2 d (a+b \arcsin (c+d x))^{3/2}}+\frac {8 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \arcsin (c+d x)}}+\frac {8 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {8 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{15 b^{7/2} d} \]
4/15*(d*x+c)/b^2/d/(a+b*arcsin(d*x+c))^(3/2)+8/15*cos(a/b)*FresnelS(2^(1/2 )/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(7/2)/d-8 /15*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(a/b)* 2^(1/2)*Pi^(1/2)/b^(7/2)/d-2/5*(1-(d*x+c)^2)^(1/2)/b/d/(a+b*arcsin(d*x+c)) ^(5/2)+8/15*(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsin(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\frac {-6 b^2 e^{i \arcsin (c+d x)}+4 e^{-\frac {i a}{b}} (a+b \arcsin (c+d x)) \left (e^{\frac {i (a+b \arcsin (c+d x))}{b}} (2 a+b (-i+2 \arcsin (c+d x)))-2 i b \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )+e^{-i \arcsin (c+d x)} \left (8 a^2+4 a b (i+4 \arcsin (c+d x))+2 b^2 \left (-3+2 i \arcsin (c+d x)+4 \arcsin (c+d x)^2\right )-8 e^{\frac {i (a+b \arcsin (c+d x))}{b}} (a+b \arcsin (c+d x))^2 \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )}{30 b^3 d (a+b \arcsin (c+d x))^{5/2}} \]
(-6*b^2*E^(I*ArcSin[c + d*x]) + (4*(a + b*ArcSin[c + d*x])*(E^((I*(a + b*A rcSin[c + d*x]))/b)*(2*a + b*(-I + 2*ArcSin[c + d*x])) - (2*I)*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/ b]))/E^((I*a)/b) + (8*a^2 + 4*a*b*(I + 4*ArcSin[c + d*x]) + 2*b^2*(-3 + (2 *I)*ArcSin[c + d*x] + 4*ArcSin[c + d*x]^2) - 8*E^((I*(a + b*ArcSin[c + d*x ]))/b)*(a + b*ArcSin[c + d*x])^2*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma [1/2, (I*(a + b*ArcSin[c + d*x]))/b])/E^(I*ArcSin[c + d*x]))/(30*b^3*d*(a + b*ArcSin[c + d*x])^(5/2))
Time = 1.04 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.02, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5302, 5132, 5222, 5132, 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.
| \(\displaystyle \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 5302 |
| \(\displaystyle \frac {\int \frac {1}{(a+b \arcsin (c+d x))^{7/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5132 |
| \(\displaystyle \frac {-\frac {2 \int \frac {c+d x}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}d(c+d x)}{5 b}-\frac {2 \sqrt {1-(c+d x)^2}}{5 b (a+b \arcsin (c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 5222 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \int \frac {1}{(a+b \arcsin (c+d x))^{3/2}}d(c+d x)}{3 b}-\frac {2 (c+d x)}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2}}{5 b (a+b \arcsin (c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 5132 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \left (-\frac {2 \int \frac {c+d x}{\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}d(c+d x)}{b}-\frac {2 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2}}{5 b (a+b \arcsin (c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 5224 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \left (-\frac {2 \int -\frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2}}{5 b (a+b \arcsin (c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \left (\frac {2 \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2}}{5 b (a+b \arcsin (c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \left (\frac {2 \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2}}{5 b (a+b \arcsin (c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \left (-\frac {2 \left (-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\cos \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{b^2}-\frac {2 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2}}{5 b (a+b \arcsin (c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \left (-\frac {2 \left (\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{b^2}-\frac {2 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2}}{5 b (a+b \arcsin (c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \left (-\frac {2 \left (\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{b^2}-\frac {2 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2}}{5 b (a+b \arcsin (c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \left (-\frac {2 \left (\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )}{b^2}-\frac {2 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2}}{5 b (a+b \arcsin (c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \left (-\frac {2 \left (2 \cos \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )d\sqrt {a+b \arcsin (c+d x)}-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )}{b^2}-\frac {2 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2}}{5 b (a+b \arcsin (c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \left (-\frac {2 \left (\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )}{b^2}-\frac {2 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2}}{5 b (a+b \arcsin (c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \left (-\frac {2 \left (\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2}}{5 b (a+b \arcsin (c+d x))^{5/2}}}{d}\) |
((-2*Sqrt[1 - (c + d*x)^2])/(5*b*(a + b*ArcSin[c + d*x])^(5/2)) - (2*((-2* (c + d*x))/(3*b*(a + b*ArcSin[c + d*x])^(3/2)) + (2*((-2*Sqrt[1 - (c + d*x )^2])/(b*Sqrt[a + b*ArcSin[c + d*x]]) - (2*(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]))/ b^2))/(3*b)))/(5*b))/d
3.2.71.3.1 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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(623\) vs. \(2(178)=356\).
Time = 0.78 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.86
| method | result | size |
| default | \(\frac {-\frac {8 \arcsin \left (d x +c \right )^{2} \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 ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b^{2}}{15}-\frac {8 \arcsin \left (d x +c \right )^{2} \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 ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b^{2}}{15}-\frac {16 \arcsin \left (d x +c \right ) \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 ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, a b}{15}-\frac {16 \arcsin \left (d x +c \right ) \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 ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, a b}{15}-\frac {8 \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 ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, a^{2}}{15}-\frac {8 \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 ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, a^{2}}{15}+\frac {8 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2}}{15}+\frac {16 \arcsin \left (d x +c \right ) \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a b}{15}-\frac {4 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2}}{15}+\frac {8 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2}}{15}-\frac {2 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2}}{5}-\frac {4 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a b}{15}}{d \,b^{3} \left (a +b \arcsin \left (d x +c \right )\right )^{\frac {5}{2}}}\) | \(624\) |
2/15/d/b^3*(-4*arcsin(d*x+c)^2*(a+b*arcsin(d*x+c))^(1/2)*cos(a/b)*FresnelS (2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^(1/ 2)*(-1/b)^(1/2)*b^2-4*arcsin(d*x+c)^2*(a+b*arcsin(d*x+c))^(1/2)*sin(a/b)*F resnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2) *Pi^(1/2)*(-1/b)^(1/2)*b^2-8*arcsin(d*x+c)*(a+b*arcsin(d*x+c))^(1/2)*cos(a /b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^ (1/2)*Pi^(1/2)*(-1/b)^(1/2)*a*b-8*arcsin(d*x+c)*(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)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*a*b-4*(a+b*arcsin(d*x+c))^(1/2)*cos(a/b)* FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2 )*Pi^(1/2)*(-1/b)^(1/2)*a^2-4*(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)*2^(1/2)*Pi^(1/2 )*(-1/b)^(1/2)*a^2+4*arcsin(d*x+c)^2*cos(-(a+b*arcsin(d*x+c))/b+a/b)*b^2+8 *arcsin(d*x+c)*cos(-(a+b*arcsin(d*x+c))/b+a/b)*a*b-2*arcsin(d*x+c)*sin(-(a +b*arcsin(d*x+c))/b+a/b)*b^2+4*cos(-(a+b*arcsin(d*x+c))/b+a/b)*a^2-3*cos(- (a+b*arcsin(d*x+c))/b+a/b)*b^2-2*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a*b)/(a+b *arcsin(d*x+c))^(5/2)
Exception generated. \[ \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]