3.3.18 \(\int e^{-3 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx\) [218]

Optimal. Leaf size=131 \[ -\frac {2 c^2 (1-a x)^4}{a \sqrt {1-a^2 x^2}}-\frac {35 c^2 \sqrt {1-a^2 x^2}}{2 a}-\frac {35 c^2 (1-a x) \sqrt {1-a^2 x^2}}{6 a}-\frac {7 c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}-\frac {35 c^2 \text {ArcSin}(a x)}{2 a} \]

[Out]

-35/2*c^2*arcsin(a*x)/a-2*c^2*(-a*x+1)^4/a/(-a^2*x^2+1)^(1/2)-35/2*c^2*(-a^2*x^2+1)^(1/2)/a-35/6*c^2*(-a*x+1)*
(-a^2*x^2+1)^(1/2)/a-7/3*c^2*(-a*x+1)^2*(-a^2*x^2+1)^(1/2)/a

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Rubi [A]
time = 0.07, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6262, 683, 685, 655, 222} \begin {gather*} -\frac {2 c^2 (1-a x)^4}{a \sqrt {1-a^2 x^2}}-\frac {7 c^2 \sqrt {1-a^2 x^2} (1-a x)^2}{3 a}-\frac {35 c^2 \sqrt {1-a^2 x^2} (1-a x)}{6 a}-\frac {35 c^2 \sqrt {1-a^2 x^2}}{2 a}-\frac {35 c^2 \text {ArcSin}(a x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c^2*(1 - a*x)^4)/(a*Sqrt[1 - a^2*x^2]) - (35*c^2*Sqrt[1 - a^2*x^2])/(2*a) - (35*c^2*(1 - a*x)*Sqrt[1 - a^2
*x^2])/(6*a) - (7*c^2*(1 - a*x)^2*Sqrt[1 - a^2*x^2])/(3*a) - (35*c^2*ArcSin[a*x])/(2*a)

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 655

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

Rule 683

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

Rule 685

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

Rule 6262

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

Rubi steps

\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx &=\frac {\int \frac {(c-a c x)^5}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac {2 c^2 (1-a x)^4}{a \sqrt {1-a^2 x^2}}-\frac {7 \int \frac {(c-a c x)^3}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {2 c^2 (1-a x)^4}{a \sqrt {1-a^2 x^2}}-\frac {7 c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}-\frac {35}{3} \int \frac {(c-a c x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 c^2 (1-a x)^4}{a \sqrt {1-a^2 x^2}}-\frac {35 c^2 (1-a x) \sqrt {1-a^2 x^2}}{6 a}-\frac {7 c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}-\frac {1}{2} (35 c) \int \frac {c-a c x}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 c^2 (1-a x)^4}{a \sqrt {1-a^2 x^2}}-\frac {35 c^2 \sqrt {1-a^2 x^2}}{2 a}-\frac {35 c^2 (1-a x) \sqrt {1-a^2 x^2}}{6 a}-\frac {7 c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}-\frac {1}{2} \left (35 c^2\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 c^2 (1-a x)^4}{a \sqrt {1-a^2 x^2}}-\frac {35 c^2 \sqrt {1-a^2 x^2}}{2 a}-\frac {35 c^2 (1-a x) \sqrt {1-a^2 x^2}}{6 a}-\frac {7 c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}-\frac {35 c^2 \sin ^{-1}(a x)}{2 a}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.02, size = 45, normalized size = 0.34 \begin {gather*} -\frac {c^2 (1-a x)^{9/2} \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {1}{2} (1-a x)\right )}{9 \sqrt {2} a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*(c^2*(1 - a*x)^(9/2)*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - a*x)/2])/(Sqrt[2]*a)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(461\) vs. \(2(115)=230\).
time = 1.20, size = 462, normalized size = 3.53

method	result	size
risch	\(\frac {\left (2 a^{2} x^{2}-15 a x +70\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{6 a \sqrt {-a^{2} x^{2}+1}}-\left (\frac {35 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}+\frac {16 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a^{2} \left (x +\frac {1}{a}\right )}\right ) c^{2}\)	\(113\)
default	\(c^{2} \left (\frac {\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )}{a}+\frac {-\frac {4 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-8 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{3}}-\frac {4 \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{2}}\right )\)	\(462\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.48, size = 196, normalized size = 1.50 \begin {gather*} \frac {1}{2} \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} c^{2} x + \frac {4 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}{a^{3} x^{2} + 2 \, a^{2} x + a} - \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}{a^{2} x + a} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}{3 \, a} - \frac {i \, c^{2} \arcsin \left (a x + 2\right )}{2 \, a} - \frac {18 \, c^{2} \arcsin \left (a x\right )}{a} - \frac {24 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{a^{2} x + a} + \frac {\sqrt {a^{2} x^{2} + 4 \, a x + 3} c^{2}}{a} - \frac {6 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(a^2*x^2 + 4*a*x + 3)*c^2*x + 4*(-a^2*x^2 + 1)^(3/2)*c^2/(a^3*x^2 + 2*a^2*x + a) - 2*(-a^2*x^2 + 1)^(3
/2)*c^2/(a^2*x + a) + 1/3*(-a^2*x^2 + 1)^(3/2)*c^2/a - 1/2*I*c^2*arcsin(a*x + 2)/a - 18*c^2*arcsin(a*x)/a - 24
*sqrt(-a^2*x^2 + 1)*c^2/(a^2*x + a) + sqrt(a^2*x^2 + 4*a*x + 3)*c^2/a - 6*sqrt(-a^2*x^2 + 1)*c^2/a

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Fricas [A]
time = 0.34, size = 106, normalized size = 0.81 \begin {gather*} -\frac {166 \, a c^{2} x + 166 \, c^{2} - 210 \, {\left (a c^{2} x + c^{2}\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{3} c^{2} x^{3} - 13 \, a^{2} c^{2} x^{2} + 55 \, a c^{2} x + 166 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{2} x + a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(166*a*c^2*x + 166*c^2 - 210*(a*c^2*x + c^2)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^3*c^2*x^3 - 13
*a^2*c^2*x^2 + 55*a*c^2*x + 166*c^2)*sqrt(-a^2*x^2 + 1))/(a^2*x + a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*(Integral(sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-2*a*x*sqrt(-a**2*x**
2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(2*a**3*x**3*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a*
*2*x**2 + 3*a*x + 1), x) + Integral(-a**4*x**4*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x))

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Giac [A]
time = 0.44, size = 91, normalized size = 0.69 \begin {gather*} -\frac {35 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, {\left | a \right |}} - \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a c^{2} x - 15 \, c^{2}\right )} x + \frac {70 \, c^{2}}{a}\right )} + \frac {32 \, c^{2}}{{\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*c*x+c)^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac")

[Out]

-35/2*c^2*arcsin(a*x)*sgn(a)/abs(a) - 1/6*sqrt(-a^2*x^2 + 1)*((2*a*c^2*x - 15*c^2)*x + 70*c^2/a) + 32*c^2/(((s
qrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a))

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Mupad [B]
time = 0.06, size = 150, normalized size = 1.15 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {11\,a\,c^2}{\sqrt {-a^2}}+\frac {5\,c^2\,x\,\sqrt {-a^2}}{2}-\frac {2\,a^3\,c^2}{3\,{\left (-a^2\right )}^{3/2}}-\frac {a^5\,c^2\,x^2}{3\,{\left (-a^2\right )}^{3/2}}\right )}{\sqrt {-a^2}}-\frac {35\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}+\frac {16\,c^2\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((1 - a^2*x^2)^(1/2)*((11*a*c^2)/(-a^2)^(1/2) + (5*c^2*x*(-a^2)^(1/2))/2 - (2*a^3*c^2)/(3*(-a^2)^(3/2)) - (a^5
*c^2*x^2)/(3*(-a^2)^(3/2))))/(-a^2)^(1/2) - (35*c^2*asinh(x*(-a^2)^(1/2)))/(2*(-a^2)^(1/2)) + (16*c^2*(1 - a^2
*x^2)^(1/2))/((x*(-a^2)^(1/2) + (-a^2)^(1/2)/a)*(-a^2)^(1/2))

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