3.59 \(\int \frac {\sqrt {d-c^2 d x^2} (a+b \sin ^{-1}(c x))}{x^4} \, dx\)

Optimal. Leaf size=111 \[ -\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^3}-\frac {b c \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}-\frac {b c^3 \log (x) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}} \]

[Out]

-1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))/d/x^3-1/6*b*c*(-c^2*d*x^2+d)^(1/2)/x^2/(-c^2*x^2+1)^(1/2)-1/3*b*c^
3*ln(x)*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {4681, 14} \[ -\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^3}-\frac {b c \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}-\frac {b c^3 \log (x) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x^4,x]

[Out]

-(b*c*Sqrt[d - c^2*d*x^2])/(6*x^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*d*x^3) -
(b*c^3*Sqrt[d - c^2*d*x^2]*Log[x])/(3*Sqrt[1 - c^2*x^2])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 4681

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(d*f*(m + 1)), x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x
^2)^FracPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSi
n[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p
+ 3, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^3}+\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {1-c^2 x^2}{x^3} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^3}+\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {1}{x^3}-\frac {c^2}{x}\right ) \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^3}-\frac {b c^3 \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 134, normalized size = 1.21 \[ \frac {\sqrt {d-c^2 d x^2} \left (2 a \left (c^2 x^2-1\right )^2+b c x \left (1-3 c^2 x^2\right ) \sqrt {1-c^2 x^2}+2 b \left (c^2 x^2-1\right )^2 \sin ^{-1}(c x)\right )}{6 x^3 \left (c^2 x^2-1\right )}-\frac {b c^3 \log (x) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x^4,x]

[Out]

(Sqrt[d - c^2*d*x^2]*(b*c*x*(1 - 3*c^2*x^2)*Sqrt[1 - c^2*x^2] + 2*a*(-1 + c^2*x^2)^2 + 2*b*(-1 + c^2*x^2)^2*Ar
cSin[c*x]))/(6*x^3*(-1 + c^2*x^2)) - (b*c^3*Sqrt[d - c^2*d*x^2]*Log[x])/(3*Sqrt[1 - c^2*x^2])

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fricas [B]  time = 0.89, size = 414, normalized size = 3.73 \[ \left [\frac {{\left (b c^{5} x^{5} - b c^{3} x^{3}\right )} \sqrt {d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{4} - 1\right )} \sqrt {d} - d}{c^{2} x^{4} - x^{2}}\right ) - \sqrt {-c^{2} d x^{2} + d} {\left (b c x^{3} - b c x\right )} \sqrt {-c^{2} x^{2} + 1} + 2 \, {\left (a c^{4} x^{4} - 2 \, a c^{2} x^{2} + {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \arcsin \left (c x\right ) + a\right )} \sqrt {-c^{2} d x^{2} + d}}{6 \, {\left (c^{2} x^{5} - x^{3}\right )}}, -\frac {2 \, {\left (b c^{5} x^{5} - b c^{3} x^{3}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{2} + 1\right )} \sqrt {-d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) + \sqrt {-c^{2} d x^{2} + d} {\left (b c x^{3} - b c x\right )} \sqrt {-c^{2} x^{2} + 1} - 2 \, {\left (a c^{4} x^{4} - 2 \, a c^{2} x^{2} + {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \arcsin \left (c x\right ) + a\right )} \sqrt {-c^{2} d x^{2} + d}}{6 \, {\left (c^{2} x^{5} - x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^4,x, algorithm="fricas")

[Out]

[1/6*((b*c^5*x^5 - b*c^3*x^3)*sqrt(d)*log((c^2*d*x^6 + c^2*d*x^2 - d*x^4 + sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2
+ 1)*(x^4 - 1)*sqrt(d) - d)/(c^2*x^4 - x^2)) - sqrt(-c^2*d*x^2 + d)*(b*c*x^3 - b*c*x)*sqrt(-c^2*x^2 + 1) + 2*(
a*c^4*x^4 - 2*a*c^2*x^2 + (b*c^4*x^4 - 2*b*c^2*x^2 + b)*arcsin(c*x) + a)*sqrt(-c^2*d*x^2 + d))/(c^2*x^5 - x^3)
, -1/6*(2*(b*c^5*x^5 - b*c^3*x^3)*sqrt(-d)*arctan(sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1)*(x^2 + 1)*sqrt(-d)/(
c^2*d*x^4 - (c^2 + 1)*d*x^2 + d)) + sqrt(-c^2*d*x^2 + d)*(b*c*x^3 - b*c*x)*sqrt(-c^2*x^2 + 1) - 2*(a*c^4*x^4 -
2*a*c^2*x^2 + (b*c^4*x^4 - 2*b*c^2*x^2 + b)*arcsin(c*x) + a)*sqrt(-c^2*d*x^2 + d))/(c^2*x^5 - x^3)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^4,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [C]  time = 0.40, size = 1117, normalized size = 10.06 \[ -\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 d \,x^{3}}-\frac {2 i b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) c^{3}}{3 c^{2} x^{2}-3}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{4} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{7}}{\left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} \arcsin \left (c x \right ) c^{8}}{\left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} c^{8}}{6 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \left (-c^{2} x^{2}+1\right ) c^{4}}{6 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \,c^{4}}{6 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \arcsin \left (c x \right ) c^{6}}{\left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} c^{6}}{3 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3}}{3 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2} \sqrt {-c^{2} x^{2}+1}\, c^{5}}{2 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{5}}{\left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {10 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \arcsin \left (c x \right ) c^{4}}{3 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \left (-c^{2} x^{2}+1\right ) c^{6}}{6 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3} \sqrt {-c^{2} x^{2}+1}}{2 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) c^{2}}{3 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) x \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c}{6 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) x^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{3 \left (3 c^{4} x^{4}-3 c^{2} x^{2}+1\right ) x^{3} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c^{3}}{3 c^{2} x^{2}-3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^4,x)

[Out]

-1/3*a/d/x^3*(-c^2*d*x^2+d)^(3/2)-2*I*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*arcsin(c*x)*c^3/(3*c^2*x^2-3
)+I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^4/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^7+b*(-d*
(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^5/(c^2*x^2-1)*arcsin(c*x)*c^8-1/6*I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c
^4*x^4-3*c^2*x^2+1)*x^5/(c^2*x^2-1)*c^8+1/6*I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x/(c^2*x^2-1)*(
-c^2*x^2+1)*c^4-1/6*I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x/(c^2*x^2-1)*c^4-3*b*(-d*(c^2*x^2-1))^
(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)*c^6+1/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*
x^2+1)*x^3/(c^2*x^2-1)*c^6+1/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/(c^2*x^2-1)*(-c^2*x^2+1)^(1/
2)*arcsin(c*x)*c^3+1/2*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^5
-I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^5+10/3*b*
(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x/(c^2*x^2-1)*arcsin(c*x)*c^4-1/6*I*b*(-d*(c^2*x^2-1))^(1/2)/(3
*c^4*x^4-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*(-c^2*x^2+1)*c^6-1/2*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/(c
^2*x^2-1)*c^3*(-c^2*x^2+1)^(1/2)-5/3*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x/(c^2*x^2-1)*arcsin(c*x
)*c^2+1/6*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c+1/3*b*(-d*(c^2
*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x^3/(c^2*x^2-1)*arcsin(c*x)+1/3*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(
1/2)/(c^2*x^2-1)*ln((I*c*x+(-c^2*x^2+1)^(1/2))^2-1)*c^3

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maxima [A]  time = 0.44, size = 137, normalized size = 1.23 \[ \frac {{\left (\left (-1\right )^{-2 \, c^{2} d x^{2} + 2 \, d} c^{2} d^{\frac {3}{2}} \log \left (-2 \, c^{2} d + \frac {2 \, d}{x^{2}}\right ) + c^{2} d^{\frac {3}{2}} \log \left (x^{2} - \frac {1}{c^{2}}\right ) - \frac {\sqrt {c^{4} d x^{4} - 2 \, c^{2} d x^{2} + d} d}{x^{2}}\right )} b c}{6 \, d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b \arcsin \left (c x\right )}{3 \, d x^{3}} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a}{3 \, d x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^4,x, algorithm="maxima")

[Out]

1/6*((-1)^(-2*c^2*d*x^2 + 2*d)*c^2*d^(3/2)*log(-2*c^2*d + 2*d/x^2) + c^2*d^(3/2)*log(x^2 - 1/c^2) - sqrt(c^4*d
*x^4 - 2*c^2*d*x^2 + d)*d/x^2)*b*c/d - 1/3*(-c^2*d*x^2 + d)^(3/2)*b*arcsin(c*x)/(d*x^3) - 1/3*(-c^2*d*x^2 + d)
^(3/2)*a/(d*x^3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*asin(c*x))*(d - c^2*d*x^2)^(1/2))/x^4,x)

[Out]

int(((a + b*asin(c*x))*(d - c^2*d*x^2)^(1/2))/x^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(1/2)*(a+b*asin(c*x))/x**4,x)

[Out]

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))/x**4, x)

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