3.9.87 \(\int \frac {e^{\tanh ^{-1}(a x)} x^4}{c-a^2 c x^2} \, dx\) [887]

Optimal. Leaf size=101 \[ \frac {x^3 (1+a x)}{a^2 c \sqrt {1-a^2 x^2}}+\frac {4 x^2 \sqrt {1-a^2 x^2}}{3 a^3 c}+\frac {(16+9 a x) \sqrt {1-a^2 x^2}}{6 a^5 c}-\frac {3 \text {ArcSin}(a x)}{2 a^5 c} \]

[Out]

-3/2*arcsin(a*x)/a^5/c+x^3*(a*x+1)/a^2/c/(-a^2*x^2+1)^(1/2)+4/3*x^2*(-a^2*x^2+1)^(1/2)/a^3/c+1/6*(9*a*x+16)*(-
a^2*x^2+1)^(1/2)/a^5/c

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Rubi [A]
time = 0.10, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6283, 833, 847, 794, 222} \begin {gather*} -\frac {3 \text {ArcSin}(a x)}{2 a^5 c}+\frac {x^3 (a x+1)}{a^2 c \sqrt {1-a^2 x^2}}+\frac {(9 a x+16) \sqrt {1-a^2 x^2}}{6 a^5 c}+\frac {4 x^2 \sqrt {1-a^2 x^2}}{3 a^3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*x^4)/(c - a^2*c*x^2),x]

[Out]

(x^3*(1 + a*x))/(a^2*c*Sqrt[1 - a^2*x^2]) + (4*x^2*Sqrt[1 - a^2*x^2])/(3*a^3*c) + ((16 + 9*a*x)*Sqrt[1 - a^2*x
^2])/(6*a^5*c) - (3*ArcSin[a*x])/(2*a^5*c)

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
- 1)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 847

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^
m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 6283

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || Gt
Q[c, 0]) && IGtQ[(n + 1)/2, 0] &&  !IntegerQ[p - n/2]

Rubi steps

\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^4}{c-a^2 c x^2} \, dx &=\frac {\int \frac {x^4 (1+a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=\frac {x^3 (1+a x)}{a^2 c \sqrt {1-a^2 x^2}}-\frac {\int \frac {x^2 (3+4 a x)}{\sqrt {1-a^2 x^2}} \, dx}{a^2 c}\\ &=\frac {x^3 (1+a x)}{a^2 c \sqrt {1-a^2 x^2}}+\frac {4 x^2 \sqrt {1-a^2 x^2}}{3 a^3 c}+\frac {\int \frac {x \left (-8 a-9 a^2 x\right )}{\sqrt {1-a^2 x^2}} \, dx}{3 a^4 c}\\ &=\frac {x^3 (1+a x)}{a^2 c \sqrt {1-a^2 x^2}}+\frac {4 x^2 \sqrt {1-a^2 x^2}}{3 a^3 c}+\frac {(16+9 a x) \sqrt {1-a^2 x^2}}{6 a^5 c}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a^4 c}\\ &=\frac {x^3 (1+a x)}{a^2 c \sqrt {1-a^2 x^2}}+\frac {4 x^2 \sqrt {1-a^2 x^2}}{3 a^3 c}+\frac {(16+9 a x) \sqrt {1-a^2 x^2}}{6 a^5 c}-\frac {3 \sin ^{-1}(a x)}{2 a^5 c}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 74, normalized size = 0.73 \begin {gather*} -\frac {-16-9 a x+8 a^2 x^2+3 a^3 x^3+2 a^4 x^4+9 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{6 a^5 c \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[a*x]*x^4)/(c - a^2*c*x^2),x]

[Out]

-1/6*(-16 - 9*a*x + 8*a^2*x^2 + 3*a^3*x^3 + 2*a^4*x^4 + 9*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(a^5*c*Sqrt[1 - a^2*x
^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(187\) vs. \(2(89)=178\).
time = 0.08, size = 188, normalized size = 1.86

method	result	size
risch	\(-\frac {\left (2 a^{2} x^{2}+3 a x +10\right ) \left (a^{2} x^{2}-1\right )}{6 a^{5} \sqrt {-a^{2} x^{2}+1}\, c}-\frac {\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{4} \sqrt {a^{2}}}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a^{6} \left (x -\frac {1}{a}\right )}}{c}\)	\(121\)
default	\(-\frac {\frac {-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}}{a}+\frac {-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{5}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2}}}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a^{6} \left (x -\frac {1}{a}\right )}}{c}\)	\(188\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a^2*c*x^2+c),x,method=_RETURNVERBOSE)

[Out]

-1/c*(1/a*(-1/3*x^2*(-a^2*x^2+1)^(1/2)/a^2-2/3*(-a^2*x^2+1)^(1/2)/a^4)+1/a^2*(-1/2*x*(-a^2*x^2+1)^(1/2)/a^2+1/
2/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2)))-1/a^5*(-a^2*x^2+1)^(1/2)+1/a^4/(a^2)^(1/2)*arctan(
(a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+1/a^6/(x-1/a)*(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(1/2))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (89) = 178\).
time = 0.54, size = 204, normalized size = 2.02 \begin {gather*} -\frac {1}{6} \, a {\left (\frac {3 \, \sqrt {-a^{2} x^{2} + 1} c}{a^{7} c^{2} x + a^{6} c^{2}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} c}{a^{7} c^{2} x - a^{6} c^{2}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1}}{a^{7} c x + a^{6} c} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1}}{a^{7} c x - a^{6} c} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{4} c} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{a^{5} c} + \frac {9 \, \arcsin \left (a x\right )}{a^{6} c} - \frac {10 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6} c}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*a*(3*sqrt(-a^2*x^2 + 1)*c/(a^7*c^2*x + a^6*c^2) + 3*sqrt(-a^2*x^2 + 1)*c/(a^7*c^2*x - a^6*c^2) - 3*sqrt(-
a^2*x^2 + 1)/(a^7*c*x + a^6*c) + 3*sqrt(-a^2*x^2 + 1)/(a^7*c*x - a^6*c) - 2*sqrt(-a^2*x^2 + 1)*x^2/(a^4*c) - 3
*sqrt(-a^2*x^2 + 1)*x/(a^5*c) + 9*arcsin(a*x)/(a^6*c) - 10*sqrt(-a^2*x^2 + 1)/(a^6*c))

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Fricas [A]
time = 0.35, size = 86, normalized size = 0.85 \begin {gather*} \frac {16 \, a x + 18 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{3} x^{3} + a^{2} x^{2} + 7 \, a x - 16\right )} \sqrt {-a^{2} x^{2} + 1} - 16}{6 \, {\left (a^{6} c x - a^{5} c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(16*a*x + 18*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^3*x^3 + a^2*x^2 + 7*a*x - 16)*sqrt(-a
^2*x^2 + 1) - 16)/(a^6*c*x - a^5*c)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {x^{4}}{- a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{5}}{- a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(x**4/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**5/(-a**2*x**2*sqrt
(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c

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Giac [A]
time = 0.43, size = 102, normalized size = 1.01 \begin {gather*} \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left (x {\left (\frac {2 \, x}{a^{3} c} + \frac {3}{a^{4} c}\right )} + \frac {10}{a^{5} c}\right )} - \frac {3 \, \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, a^{4} c {\left | a \right |}} + \frac {2}{a^{4} c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(-a^2*x^2 + 1)*(x*(2*x/(a^3*c) + 3/(a^4*c)) + 10/(a^5*c)) - 3/2*arcsin(a*x)*sgn(a)/(a^4*c*abs(a)) + 2/
(a^4*c*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)*abs(a))

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Mupad [B]
time = 0.89, size = 140, normalized size = 1.39 \begin {gather*} \frac {5\,\sqrt {1-a^2\,x^2}}{3\,a^5\,c}-\frac {\sqrt {1-a^2\,x^2}}{\sqrt {-a^2}\,\left (a^3\,c\,\sqrt {-a^2}-a^4\,c\,x\,\sqrt {-a^2}\right )}+\frac {x\,\sqrt {1-a^2\,x^2}}{2\,a^4\,c}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^4\,c\,\sqrt {-a^2}}+\frac {x^2\,\sqrt {1-a^2\,x^2}}{3\,a^3\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*(1 - a^2*x^2)^(1/2))/(3*a^5*c) - (1 - a^2*x^2)^(1/2)/((-a^2)^(1/2)*(a^3*c*(-a^2)^(1/2) - a^4*c*x*(-a^2)^(1/
2))) + (x*(1 - a^2*x^2)^(1/2))/(2*a^4*c) - (3*asinh(x*(-a^2)^(1/2)))/(2*a^4*c*(-a^2)^(1/2)) + (x^2*(1 - a^2*x^
2)^(1/2))/(3*a^3*c)

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