\(\int \frac {x}{\arcsin (a x)^{7/2}} \, dx\) [116]

Optimal result

Integrand size = 10, antiderivative size = 119 \[ \int \frac {x}{\arcsin (a x)^{7/2}} \, dx=-\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {4}{15 a^2 \arcsin (a x)^{3/2}}+\frac {8 x^2}{15 \arcsin (a x)^{3/2}}+\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\arcsin (a x)}}-\frac {32 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{15 a^2} \]

[Out]

-4/15/a^2/arcsin(a*x)^(3/2)+8/15*x^2/arcsin(a*x)^(3/2)-32/15*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a
^2-2/5*x*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(5/2)+32/15*x*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4729, 4807, 4727, 3385, 3433, 4737} \[ \int \frac {x}{\arcsin (a x)^{7/2}} \, dx=-\frac {32 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{15 a^2}+\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\arcsin (a x)}}-\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {4}{15 a^2 \arcsin (a x)^{3/2}}+\frac {8 x^2}{15 \arcsin (a x)^{3/2}} \]

[In]

Int[x/ArcSin[a*x]^(7/2),x]

[Out]

(-2*x*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2)) - 4/(15*a^2*ArcSin[a*x]^(3/2)) + (8*x^2)/(15*ArcSin[a*x]^(3/2
)) + (32*x*Sqrt[1 - a^2*x^2])/(15*a*Sqrt[ArcSin[a*x]]) - (32*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]]
)/(15*a^2)

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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]

Rule 3433

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]

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]

Rule 4729

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[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcSin[c*x])^(n + 1)/
Sqrt[1 - c^2*x^2]), x], x] - Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2
]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4807

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}+\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2} \arcsin (a x)^{5/2}} \, dx}{5 a}-\frac {1}{5} (4 a) \int \frac {x^2}{\sqrt {1-a^2 x^2} \arcsin (a x)^{5/2}} \, dx \\ & = -\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {4}{15 a^2 \arcsin (a x)^{3/2}}+\frac {8 x^2}{15 \arcsin (a x)^{3/2}}-\frac {16}{15} \int \frac {x}{\arcsin (a x)^{3/2}} \, dx \\ & = -\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {4}{15 a^2 \arcsin (a x)^{3/2}}+\frac {8 x^2}{15 \arcsin (a x)^{3/2}}+\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\arcsin (a x)}}-\frac {32 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{15 a^2} \\ & = -\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {4}{15 a^2 \arcsin (a x)^{3/2}}+\frac {8 x^2}{15 \arcsin (a x)^{3/2}}+\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\arcsin (a x)}}-\frac {64 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{15 a^2} \\ & = -\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {4}{15 a^2 \arcsin (a x)^{3/2}}+\frac {8 x^2}{15 \arcsin (a x)^{3/2}}+\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\arcsin (a x)}}-\frac {32 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{15 a^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.23 \[ \int \frac {x}{\arcsin (a x)^{7/2}} \, dx=-\frac {\arcsin (a x) \left (2 e^{2 i \arcsin (a x)} (1+4 i \arcsin (a x))+8 \sqrt {2} (-i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},-2 i \arcsin (a x)\right )+e^{-2 i \arcsin (a x)} \left (2-8 i \arcsin (a x)+8 \sqrt {2} e^{2 i \arcsin (a x)} (i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},2 i \arcsin (a x)\right )\right )\right )+3 \sin (2 \arcsin (a x))}{15 a^2 \arcsin (a x)^{5/2}} \]

[In]

Integrate[x/ArcSin[a*x]^(7/2),x]

[Out]

-1/15*(ArcSin[a*x]*(2*E^((2*I)*ArcSin[a*x])*(1 + (4*I)*ArcSin[a*x]) + 8*Sqrt[2]*((-I)*ArcSin[a*x])^(3/2)*Gamma
[1/2, (-2*I)*ArcSin[a*x]] + (2 - (8*I)*ArcSin[a*x] + 8*Sqrt[2]*E^((2*I)*ArcSin[a*x])*(I*ArcSin[a*x])^(3/2)*Gam
ma[1/2, (2*I)*ArcSin[a*x]])/E^((2*I)*ArcSin[a*x])) + 3*Sin[2*ArcSin[a*x]])/(a^2*ArcSin[a*x]^(5/2))

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.61

method	result	size
default	\(\frac {-32 \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}+16 \sin \left (2 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )^{2}-4 \arcsin \left (a x \right ) \cos \left (2 \arcsin \left (a x \right )\right )-3 \sin \left (2 \arcsin \left (a x \right )\right )}{15 a^{2} \arcsin \left (a x \right )^{\frac {5}{2}}}\)	\(73\)

[In]

int(x/arcsin(a*x)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/15/a^2*(-32*Pi^(1/2)*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))*arcsin(a*x)^(5/2)+16*sin(2*arcsin(a*x))*arcsin(a
*x)^2-4*arcsin(a*x)*cos(2*arcsin(a*x))-3*sin(2*arcsin(a*x)))/arcsin(a*x)^(5/2)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x}{\arcsin (a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x/arcsin(a*x)^(7/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {x}{\arcsin (a x)^{7/2}} \, dx=\int \frac {x}{\operatorname {asin}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]

[In]

integrate(x/asin(a*x)**(7/2),x)

[Out]

Integral(x/asin(a*x)**(7/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x}{\arcsin (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x/arcsin(a*x)^(7/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {x}{\arcsin (a x)^{7/2}} \, dx=\int { \frac {x}{\arcsin \left (a x\right )^{\frac {7}{2}}} \,d x } \]

[In]

integrate(x/arcsin(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(x/arcsin(a*x)^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\arcsin (a x)^{7/2}} \, dx=\int \frac {x}{{\mathrm {asin}\left (a\,x\right )}^{7/2}} \,d x \]

[In]

int(x/asin(a*x)^(7/2),x)

[Out]

int(x/asin(a*x)^(7/2), x)