3.468 \(\int e^{3 \tanh ^{-1}(a x)} (c-\frac {c}{a x})^2 \, dx\)

Optimal. Leaf size=67 \[ -\frac {c^2 (1-a x) \sqrt {1-a^2 x^2}}{a^2 x}-\frac {c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}-\frac {c^2 \sin ^{-1}(a x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.16, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6131, 6128, 850, 813, 844, 216, 266, 63, 208} \[ -\frac {c^2 (1-a x) \sqrt {1-a^2 x^2}}{a^2 x}-\frac {c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}-\frac {c^2 \sin ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c^2*(1 - a*x)*Sqrt[1 - a^2*x^2])/(a^2*x)) - (c^2*ArcSin[a*x])/a - (c^2*ArcTanh[Sqrt[1 - a^2*x^2]])/a

Rule 63

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 208

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

Rule 216

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 266

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 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 844

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 850

Int[((x_)^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + (c*x)/e)*(a + c*x
^2)^(p - 1), x] /; FreeQ[{a, c, d, e, n, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||
!IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2]))

Rule 6128

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]

Rule 6131

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

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 82, normalized size = 1.22 \[ \frac {\sqrt {1-a^2 x^2} \left (c^2-\frac {c^2}{a x}\right )}{a}-\frac {c^2 \log \left (\sqrt {1-a^2 x^2}+1\right )}{a}+\frac {c^2 \log (a x)}{a}-\frac {c^2 \sin ^{-1}(a x)}{a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.58, size = 93, normalized size = 1.39 \[ \frac {2 \, a c^{2} x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + a c^{2} x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + a c^{2} x + {\left (a c^{2} x - c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.25, size = 139, normalized size = 2.07 \[ \frac {a^{2} c^{2} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{2 \, a^{2} x {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 133, normalized size = 1.99 \[ -\frac {c^{2} a \,x^{2}}{\sqrt {-a^{2} x^{2}+1}}+\frac {c^{2}}{a \sqrt {-a^{2} x^{2}+1}}+\frac {c^{2} x}{\sqrt {-a^{2} x^{2}+1}}-\frac {c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {c^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a}-\frac {c^{2}}{a^{2} x \sqrt {-a^{2} x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.43, size = 209, normalized size = 3.12 \[ -a^{3} c^{2} {\left (\frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{4}}\right )} + a^{2} c^{2} {\left (\frac {x}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )} - \frac {2 \, c^{2} x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {c^{2} {\left (\frac {1}{\sqrt {-a^{2} x^{2} + 1}} - \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )\right )}}{a} + \frac {{\left (\frac {2 \, a^{2} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x}\right )} c^{2}}{a^{2}} - \frac {2 \, c^{2}}{\sqrt {-a^{2} x^{2} + 1} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^3*c^2*(x^2/(sqrt(-a^2*x^2 + 1)*a^2) - 2/(sqrt(-a^2*x^2 + 1)*a^4)) + a^2*c^2*(x/(sqrt(-a^2*x^2 + 1)*a^2) - a
rcsin(a*x)/a^3) - 2*c^2*x/sqrt(-a^2*x^2 + 1) + c^2*(1/sqrt(-a^2*x^2 + 1) - log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2
/abs(x)))/a + (2*a^2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*x))*c^2/a^2 - 2*c^2/(sqrt(-a^2*x^2 + 1)*a)

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mupad [B]  time = 0.81, size = 90, normalized size = 1.34 \[ \frac {c^2\,\sqrt {1-a^2\,x^2}}{a}-\frac {c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{a^2\,x}+\frac {c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 13.95, size = 151, normalized size = 2.25 \[ - a c^{2} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) - c^{2} \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) + \frac {c^{2} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right )}{a} + \frac {c^{2} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**2,x)

[Out]

-a*c**2*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - c**2*Piecewise((sqrt(a**(-2))*a
sin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + c**2*Piecewise((-acosh(1/(a*x)
), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a + c**2*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2
) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2

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