3.4.13 \(\int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^3}{x^4} \, dx\) [313]

Optimal. Leaf size=88 \[ \frac {a c^3 (1-a x) \sqrt {1-a^2 x^2}}{x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-a^3 c^3 \text {ArcSin}(a x)-a^3 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]

[Out]

-1/3*c^3*(-a^2*x^2+1)^(3/2)/x^3-a^3*c^3*arcsin(a*x)-a^3*c^3*arctanh((-a^2*x^2+1)^(1/2))+a*c^3*(-a*x+1)*(-a^2*x
^2+1)^(1/2)/x^2

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Rubi [A]
time = 0.12, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6263, 1821, 825, 858, 222, 272, 65, 214} \begin {gather*} -a^3 c^3 \text {ArcSin}(a x)+\frac {a c^3 (1-a x) \sqrt {1-a^2 x^2}}{x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-a^3 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*(c - a*c*x)^3)/x^4,x]

[Out]

(a*c^3*(1 - a*x)*Sqrt[1 - a^2*x^2])/x^2 - (c^3*(1 - a^2*x^2)^(3/2))/(3*x^3) - a^3*c^3*ArcSin[a*x] - a^3*c^3*Ar
cTanh[Sqrt[1 - a^2*x^2]]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 825

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/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*
p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^
2 + a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p +
1) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e
^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 858

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

Rule 1821

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 6263

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

Rubi steps

\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^3}{x^4} \, dx &=c \int \frac {(c-a c x)^2 \sqrt {1-a^2 x^2}}{x^4} \, dx\\ &=-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-\frac {1}{3} c \int \frac {\left (6 a c^2-3 a^2 c^2 x\right ) \sqrt {1-a^2 x^2}}{x^3} \, dx\\ &=\frac {a c^3 (1-a x) \sqrt {1-a^2 x^2}}{x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac {1}{12} c \int \frac {12 a^3 c^2-12 a^4 c^2 x}{x \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {a c^3 (1-a x) \sqrt {1-a^2 x^2}}{x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\left (a^3 c^3\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx-\left (a^4 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {a c^3 (1-a x) \sqrt {1-a^2 x^2}}{x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-a^3 c^3 \sin ^{-1}(a x)+\frac {1}{2} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac {a c^3 (1-a x) \sqrt {1-a^2 x^2}}{x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-a^3 c^3 \sin ^{-1}(a x)-\left (a c^3\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )\\ &=\frac {a c^3 (1-a x) \sqrt {1-a^2 x^2}}{x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-a^3 c^3 \sin ^{-1}(a x)-a^3 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 156, normalized size = 1.77 \begin {gather*} \frac {c^3 \left (-2+6 a x-2 a^2 x^2-6 a^3 x^3+4 a^4 x^4+3 a^3 x^3 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)+18 a^3 x^3 \sqrt {1-a^2 x^2} \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-6 a^3 x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\right )}{6 x^3 \sqrt {1-a^2 x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^ArcTanh[a*x]*(c - a*c*x)^3)/x^4,x]

[Out]

(c^3*(-2 + 6*a*x - 2*a^2*x^2 - 6*a^3*x^3 + 4*a^4*x^4 + 3*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + 18*a^3*x^3*Sq
rt[1 - a^2*x^2]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]] - 6*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]))/(6*x^
3*Sqrt[1 - a^2*x^2])

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Maple [A]
time = 1.02, size = 130, normalized size = 1.48

method	result	size
risch	\(\frac {\left (2 a^{4} x^{4}-3 a^{3} x^{3}-a^{2} x^{2}+3 a x -1\right ) c^{3}}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}-\left (\frac {a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+a^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) c^{3}\)	\(104\)
default	\(-c^{3} \left (\frac {a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}+\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}+2 a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )+2 a^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\)	\(130\)
meijerg	\(-a^{3} c^{3} \arcsin \left (a x \right )+\frac {a^{3} c^{3} \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }\right )}{\sqrt {\pi }}+\frac {a^{3} c^{3} \left (-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }}{x^{2} a^{2}}\right )}{\sqrt {\pi }}-\frac {c^{3} \left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{3 x^{3}}\)	\(202\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.46, size = 110, normalized size = 1.25 \begin {gather*} -a^{3} c^{3} \arcsin \left (a x\right ) - a^{3} c^{3} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3}}{3 \, x} + \frac {\sqrt {-a^{2} x^{2} + 1} a c^{3}}{x^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x^4,x, algorithm="maxima")

[Out]

-a^3*c^3*arcsin(a*x) - a^3*c^3*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 2/3*sqrt(-a^2*x^2 + 1)*a^2*c^3/x
+ sqrt(-a^2*x^2 + 1)*a*c^3/x^2 - 1/3*sqrt(-a^2*x^2 + 1)*c^3/x^3

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Fricas [A]
time = 0.35, size = 105, normalized size = 1.19 \begin {gather*} \frac {6 \, a^{3} c^{3} x^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 3 \, a^{3} c^{3} x^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (2 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x + c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x^4,x, algorithm="fricas")

[Out]

1/3*(6*a^3*c^3*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 3*a^3*c^3*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (2
*a^2*c^3*x^2 - 3*a*c^3*x + c^3)*sqrt(-a^2*x^2 + 1))/x^3

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Sympy [C] Result contains complex when optimal does not.
time = 3.97, size = 277, normalized size = 3.15 \begin {gather*} - a^{4} c^{3} \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) + 2 a^{3} c^{3} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) - 2 a c^{3} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\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)*(-a*c*x+c)**3/x**4,x)

[Out]

-a**4*c**3*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 <
0)) + 2*a**3*c**3*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) - 2*a*c**3*Piec
ewise((-a**2*acosh(1/(a*x))/2 + a/(2*x*sqrt(-1 + 1/(a**2*x**2))) - 1/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Ab
s(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True)) + c**3*Piecewise((-2*I*
a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2
+ 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (80) = 160\).
time = 0.44, size = 250, normalized size = 2.84 \begin {gather*} -\frac {a^{4} c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {a^{4} c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {{\left (a^{4} c^{3} - \frac {6 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2} c^{3}}{x} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} - \frac {\frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{3}}{x} - \frac {6 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{3}}{x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{x^{3}}}{24 \, a^{2} {\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)*(-a*c*x+c)^3/x^4,x, algorithm="giac")

[Out]

-a^4*c^3*arcsin(a*x)*sgn(a)/abs(a) - a^4*c^3*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs
(a) + 1/24*(a^4*c^3 - 6*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^2*c^3/x + 9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^3/x^
2)*a^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*abs(a)) - 1/24*(9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4*c^3/x - 6*
(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^2*c^3/x^2 + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^3/x^3)/(a^2*abs(a))

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Mupad [B]
time = 0.04, size = 114, normalized size = 1.30 \begin {gather*} \frac {a\,c^3\,\sqrt {1-a^2\,x^2}}{x^2}-\frac {c^3\,\sqrt {1-a^2\,x^2}}{3\,x^3}-\frac {a^4\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {2\,a^2\,c^3\,\sqrt {1-a^2\,x^2}}{3\,x}+a^3\,c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - a*c*x)^3*(a*x + 1))/(x^4*(1 - a^2*x^2)^(1/2)),x)

[Out]

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

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