\(\int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx\) [99]

Optimal result

Integrand size = 12, antiderivative size = 171 \[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=-\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}-\frac {5 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7}+\frac {9 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7}-\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7}+\frac {\sqrt {\frac {7 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {14}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7} \]

[Out]

-5/32*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^7+9/32*FresnelS(6^(1/2)/Pi^(1/2)*arcsin(
a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^7-5/32*FresnelS(10^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*10^(1/2)*Pi^(1/2)/a^7+1/32
*FresnelS(14^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*14^(1/2)*Pi^(1/2)/a^7-2*x^6*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(1
/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4727, 3386, 3432} \[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=-\frac {5 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7}+\frac {9 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7}-\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7}+\frac {\sqrt {\frac {7 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {14}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7}-\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}} \]

[In]

Int[x^6/ArcSin[a*x]^(3/2),x]

[Out]

(-2*x^6*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]]) - (5*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(16*a
^7) + (9*Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(16*a^7) - (5*Sqrt[(5*Pi)/2]*FresnelS[Sqrt[10/
Pi]*Sqrt[ArcSin[a*x]]])/(16*a^7) + (Sqrt[(7*Pi)/2]*FresnelS[Sqrt[14/Pi]*Sqrt[ArcSin[a*x]]])/(16*a^7)

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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]

Rule 3432

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]

Rule 4727

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 - c^2*x^2]*((a + b*ArcSin
[c*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[
-a/b + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x]
&& IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}+\frac {2 \text {Subst}\left (\int \left (-\frac {5 \sin (x)}{64 \sqrt {x}}+\frac {27 \sin (3 x)}{64 \sqrt {x}}-\frac {25 \sin (5 x)}{64 \sqrt {x}}+\frac {7 \sin (7 x)}{64 \sqrt {x}}\right ) \, dx,x,\arcsin (a x)\right )}{a^7} \\ & = -\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}-\frac {5 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{32 a^7}+\frac {7 \text {Subst}\left (\int \frac {\sin (7 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{32 a^7}-\frac {25 \text {Subst}\left (\int \frac {\sin (5 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{32 a^7}+\frac {27 \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{32 a^7} \\ & = -\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}-\frac {5 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{16 a^7}+\frac {7 \text {Subst}\left (\int \sin \left (7 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{16 a^7}-\frac {25 \text {Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{16 a^7}+\frac {27 \text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{16 a^7} \\ & = -\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}-\frac {5 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7}+\frac {9 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7}-\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7}+\frac {\sqrt {\frac {7 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {14}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.50 \[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=\frac {-\frac {5 \left (e^{i \arcsin (a x)}-\sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-i \arcsin (a x)\right )\right )}{64 \sqrt {\arcsin (a x)}}-\frac {5 \left (e^{-i \arcsin (a x)}-\sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},i \arcsin (a x)\right )\right )}{64 \sqrt {\arcsin (a x)}}+\frac {9 \left (e^{3 i \arcsin (a x)}-\sqrt {3} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-3 i \arcsin (a x)\right )\right )}{64 \sqrt {\arcsin (a x)}}+\frac {9 \left (e^{-3 i \arcsin (a x)}-\sqrt {3} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},3 i \arcsin (a x)\right )\right )}{64 \sqrt {\arcsin (a x)}}-\frac {5 \left (e^{5 i \arcsin (a x)}-\sqrt {5} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-5 i \arcsin (a x)\right )\right )}{64 \sqrt {\arcsin (a x)}}-\frac {5 \left (e^{-5 i \arcsin (a x)}-\sqrt {5} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},5 i \arcsin (a x)\right )\right )}{64 \sqrt {\arcsin (a x)}}+\frac {e^{7 i \arcsin (a x)}-\sqrt {7} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-7 i \arcsin (a x)\right )}{64 \sqrt {\arcsin (a x)}}+\frac {e^{-7 i \arcsin (a x)}-\sqrt {7} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},7 i \arcsin (a x)\right )}{64 \sqrt {\arcsin (a x)}}}{a^7} \]

[In]

Integrate[x^6/ArcSin[a*x]^(3/2),x]

[Out]

((-5*(E^(I*ArcSin[a*x]) - Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-I)*ArcSin[a*x]]))/(64*Sqrt[ArcSin[a*x]]) - (5*(E
^((-I)*ArcSin[a*x]) - Sqrt[I*ArcSin[a*x]]*Gamma[1/2, I*ArcSin[a*x]]))/(64*Sqrt[ArcSin[a*x]]) + (9*(E^((3*I)*Ar
cSin[a*x]) - Sqrt[3]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-3*I)*ArcSin[a*x]]))/(64*Sqrt[ArcSin[a*x]]) + (9*(E^((
-3*I)*ArcSin[a*x]) - Sqrt[3]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (3*I)*ArcSin[a*x]]))/(64*Sqrt[ArcSin[a*x]]) - (5*(
E^((5*I)*ArcSin[a*x]) - Sqrt[5]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-5*I)*ArcSin[a*x]]))/(64*Sqrt[ArcSin[a*x]])
- (5*(E^((-5*I)*ArcSin[a*x]) - Sqrt[5]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (5*I)*ArcSin[a*x]]))/(64*Sqrt[ArcSin[a*
x]]) + (E^((7*I)*ArcSin[a*x]) - Sqrt[7]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-7*I)*ArcSin[a*x]])/(64*Sqrt[ArcSin
[a*x]]) + (E^((-7*I)*ArcSin[a*x]) - Sqrt[7]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (7*I)*ArcSin[a*x]])/(64*Sqrt[ArcSin
[a*x]]))/a^7

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.08

method	result	size
default	\(-\frac {-9 \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {3}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+5 \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {5}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }-\sqrt {2}\, \sqrt {\pi }\, \sqrt {7}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {7}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arcsin \left (a x \right )}+5 \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+5 \sqrt {-a^{2} x^{2}+1}-9 \cos \left (3 \arcsin \left (a x \right )\right )+5 \cos \left (5 \arcsin \left (a x \right )\right )-\cos \left (7 \arcsin \left (a x \right )\right )}{32 a^{7} \sqrt {\arcsin \left (a x \right )}}\)	\(184\)

[In]

int(x^6/arcsin(a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/32/a^7*(-9*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))*3^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)+
5*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)*arcsin(a*x)^(1/2))*5^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)-2^(1/2)*Pi^(
1/2)*7^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*7^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(1/2)+5*FresnelS(2^(1/2)/Pi^(1/2
)*arcsin(a*x)^(1/2))*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)+5*(-a^2*x^2+1)^(1/2)-9*cos(3*arcsin(a*x))+5*cos(5*arcs
in(a*x))-cos(7*arcsin(a*x)))/arcsin(a*x)^(1/2)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^6/arcsin(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=\int \frac {x^{6}}{\operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

[In]

integrate(x**6/asin(a*x)**(3/2),x)

[Out]

Integral(x**6/asin(a*x)**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^6/arcsin(a*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F]

\[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=\int { \frac {x^{6}}{\arcsin \left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^6/arcsin(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(x^6/arcsin(a*x)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=\int \frac {x^6}{{\mathrm {asin}\left (a\,x\right )}^{3/2}} \,d x \]

[In]

int(x^6/asin(a*x)^(3/2),x)

[Out]

int(x^6/asin(a*x)^(3/2), x)