3.4.74 \(\int \frac {(c-a^2 c x^2)^3}{\text {ArcSin}(a x)^2} \, dx\) [374]

Optimal. Leaf size=95 \[ -\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \text {ArcSin}(a x)}-\frac {35 c^3 \text {Si}(\text {ArcSin}(a x))}{64 a}-\frac {63 c^3 \text {Si}(3 \text {ArcSin}(a x))}{64 a}-\frac {35 c^3 \text {Si}(5 \text {ArcSin}(a x))}{64 a}-\frac {7 c^3 \text {Si}(7 \text {ArcSin}(a x))}{64 a} \]

[Out]

-c^3*(-a^2*x^2+1)^(7/2)/a/arcsin(a*x)-35/64*c^3*Si(arcsin(a*x))/a-63/64*c^3*Si(3*arcsin(a*x))/a-35/64*c^3*Si(5
*arcsin(a*x))/a-7/64*c^3*Si(7*arcsin(a*x))/a

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Rubi [A]
time = 0.12, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4751, 4809, 4491, 3380} \begin {gather*} -\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \text {ArcSin}(a x)}-\frac {35 c^3 \text {Si}(\text {ArcSin}(a x))}{64 a}-\frac {63 c^3 \text {Si}(3 \text {ArcSin}(a x))}{64 a}-\frac {35 c^3 \text {Si}(5 \text {ArcSin}(a x))}{64 a}-\frac {7 c^3 \text {Si}(7 \text {ArcSin}(a x))}{64 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((c^3*(1 - a^2*x^2)^(7/2))/(a*ArcSin[a*x])) - (35*c^3*SinIntegral[ArcSin[a*x]])/(64*a) - (63*c^3*SinIntegral[
3*ArcSin[a*x]])/(64*a) - (35*c^3*SinIntegral[5*ArcSin[a*x]])/(64*a) - (7*c^3*SinIntegral[7*ArcSin[a*x]])/(64*a
)

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(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]
&& IGtQ[p, 0]

Rule 4751

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[Sqrt[1 - c^2*x^2]*(
d + e*x^2)^p*((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + Dist[c*((2*p + 1)/(b*(n + 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
, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]

Rule 4809

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(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]

Rubi steps

\begin {align*} \int \frac {\left (c-a^2 c x^2\right )^3}{\sin ^{-1}(a x)^2} \, dx &=-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \sin ^{-1}(a x)}-\left (7 a c^3\right ) \int \frac {x \left (1-a^2 x^2\right )^{5/2}}{\sin ^{-1}(a x)} \, dx\\ &=-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \sin ^{-1}(a x)}-\frac {\left (7 c^3\right ) \text {Subst}\left (\int \frac {\cos ^6(x) \sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \sin ^{-1}(a x)}-\frac {\left (7 c^3\right ) \text {Subst}\left (\int \left (\frac {5 \sin (x)}{64 x}+\frac {9 \sin (3 x)}{64 x}+\frac {5 \sin (5 x)}{64 x}+\frac {\sin (7 x)}{64 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \sin ^{-1}(a x)}-\frac {\left (7 c^3\right ) \text {Subst}\left (\int \frac {\sin (7 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a}-\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a}-\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {\sin (5 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a}-\frac {\left (63 c^3\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a}\\ &=-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \sin ^{-1}(a x)}-\frac {35 c^3 \text {Si}\left (\sin ^{-1}(a x)\right )}{64 a}-\frac {63 c^3 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{64 a}-\frac {35 c^3 \text {Si}\left (5 \sin ^{-1}(a x)\right )}{64 a}-\frac {7 c^3 \text {Si}\left (7 \sin ^{-1}(a x)\right )}{64 a}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 83, normalized size = 0.87 \begin {gather*} -\frac {c^3 \left (64 \left (1-a^2 x^2\right )^{7/2}+35 \text {ArcSin}(a x) \text {Si}(\text {ArcSin}(a x))+63 \text {ArcSin}(a x) \text {Si}(3 \text {ArcSin}(a x))+35 \text {ArcSin}(a x) \text {Si}(5 \text {ArcSin}(a x))+7 \text {ArcSin}(a x) \text {Si}(7 \text {ArcSin}(a x))\right )}{64 a \text {ArcSin}(a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/64*(c^3*(64*(1 - a^2*x^2)^(7/2) + 35*ArcSin[a*x]*SinIntegral[ArcSin[a*x]] + 63*ArcSin[a*x]*SinIntegral[3*Ar
cSin[a*x]] + 35*ArcSin[a*x]*SinIntegral[5*ArcSin[a*x]] + 7*ArcSin[a*x]*SinIntegral[7*ArcSin[a*x]]))/(a*ArcSin[
a*x])

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Maple [A]
time = 0.24, size = 105, normalized size = 1.11

method	result	size
derivativedivides	\(-\frac {c^{3} \left (35 \sinIntegral \left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+63 \sinIntegral \left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+35 \sinIntegral \left (5 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+7 \sinIntegral \left (7 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+35 \sqrt {-a^{2} x^{2}+1}+21 \cos \left (3 \arcsin \left (a x \right )\right )+7 \cos \left (5 \arcsin \left (a x \right )\right )+\cos \left (7 \arcsin \left (a x \right )\right )\right )}{64 a \arcsin \left (a x \right )}\)	\(105\)
default	\(-\frac {c^{3} \left (35 \sinIntegral \left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+63 \sinIntegral \left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+35 \sinIntegral \left (5 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+7 \sinIntegral \left (7 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+35 \sqrt {-a^{2} x^{2}+1}+21 \cos \left (3 \arcsin \left (a x \right )\right )+7 \cos \left (5 \arcsin \left (a x \right )\right )+\cos \left (7 \arcsin \left (a x \right )\right )\right )}{64 a \arcsin \left (a x \right )}\)	\(105\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^3/arcsin(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

-1/64/a*c^3*(35*Si(arcsin(a*x))*arcsin(a*x)+63*Si(3*arcsin(a*x))*arcsin(a*x)+35*Si(5*arcsin(a*x))*arcsin(a*x)+
7*Si(7*arcsin(a*x))*arcsin(a*x)+35*(-a^2*x^2+1)^(1/2)+21*cos(3*arcsin(a*x))+7*cos(5*arcsin(a*x))+cos(7*arcsin(
a*x)))/arcsin(a*x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))*integrate(7*(a^5*c^3*x^5 - 2*a^3*c^3*x^3 + a*c^3*x)*sqrt(a*x +
1)*sqrt(-a*x + 1)/arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)), x) - (a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^
2 - c^3)*sqrt(a*x + 1)*sqrt(-a*x + 1))/(a*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)))

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)/arcsin(a*x)^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {3 a^{4} x^{4}}{\operatorname {asin}^{2}{\left (a x \right )}}\right )\, dx + \int \frac {a^{6} x^{6}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {asin}^{2}{\left (a x \right )}}\right )\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**3/asin(a*x)**2,x)

[Out]

-c**3*(Integral(3*a**2*x**2/asin(a*x)**2, x) + Integral(-3*a**4*x**4/asin(a*x)**2, x) + Integral(a**6*x**6/asi
n(a*x)**2, x) + Integral(-1/asin(a*x)**2, x))

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Giac [A]
time = 0.48, size = 95, normalized size = 1.00 \begin {gather*} \frac {{\left (a^{2} x^{2} - 1\right )}^{3} \sqrt {-a^{2} x^{2} + 1} c^{3}}{a \arcsin \left (a x\right )} - \frac {7 \, c^{3} \operatorname {Si}\left (7 \, \arcsin \left (a x\right )\right )}{64 \, a} - \frac {35 \, c^{3} \operatorname {Si}\left (5 \, \arcsin \left (a x\right )\right )}{64 \, a} - \frac {63 \, c^{3} \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{64 \, a} - \frac {35 \, c^{3} \operatorname {Si}\left (\arcsin \left (a x\right )\right )}{64 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*x^2 - 1)^3*sqrt(-a^2*x^2 + 1)*c^3/(a*arcsin(a*x)) - 7/64*c^3*sin_integral(7*arcsin(a*x))/a - 35/64*c^3*si
n_integral(5*arcsin(a*x))/a - 63/64*c^3*sin_integral(3*arcsin(a*x))/a - 35/64*c^3*sin_integral(arcsin(a*x))/a

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c-a^2\,c\,x^2\right )}^3}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - a^2*c*x^2)^3/asin(a*x)^2,x)

[Out]

int((c - a^2*c*x^2)^3/asin(a*x)^2, x)

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