3.2.80 \(\int e^{3 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx\) [180]

Optimal. Leaf size=61 \[ \frac {1}{2} c^2 x \sqrt {1-a^2 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {c^2 \text {ArcSin}(a x)}{2 a} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6262, 679, 201, 222} \begin {gather*} -\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {1}{2} c^2 x \sqrt {1-a^2 x^2}+\frac {c^2 \text {ArcSin}(a x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 679

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + c*x^2)^p/(e
*(m + 2*p + 1))), x] - Dist[2*c*d*(p/(e^2*(m + 2*p + 1))), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && 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 &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{c-a c x} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+c^2 \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {1}{2} c^2 x \sqrt {1-a^2 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {1}{2} c^2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {1}{2} c^2 x \sqrt {1-a^2 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {c^2 \sin ^{-1}(a x)}{2 a}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 59, normalized size = 0.97 \begin {gather*} \frac {c^2 \left (\sqrt {1-a^2 x^2} \left (-2+3 a x+2 a^2 x^2\right )-6 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{6 a} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(277\) vs. \(2(51)=102\).
time = 1.18, size = 278, normalized size = 4.56

method	result	size
risch	\(-\frac {\left (2 a^{2} x^{2}+3 a x -2\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{6 a \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{2}}{2 \sqrt {a^{2}}}\)	\(75\)
default	\(c^{2} \left (a^{5} \left (-\frac {x^{4}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}}{a^{2}}\right )+a^{4} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )-2 a^{3} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )-2 a^{2} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )+\frac {1}{a \sqrt {-a^{2} x^{2}+1}}+\frac {x}{\sqrt {-a^{2} x^{2}+1}}\right )\)	\(278\)
meijerg	\(\frac {2 c^{2} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {c^{2} \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}+\frac {c^{2} x}{\sqrt {-a^{2} x^{2}+1}}-\frac {c^{2} \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 a^{4} x^{4}-8 a^{2} x^{2}+16\right )}{6 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}+\frac {c^{2} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {2 c^{2} \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}\)	\(281\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (51) = 102\).
time = 0.47, size = 118, normalized size = 1.93 \begin {gather*} -\frac {a^{3} c^{2} x^{4}}{3 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {a^{2} c^{2} x^{3}}{2 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {2 \, a c^{2} x^{2}}{3 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {c^{2} x}{2 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {c^{2} \arcsin \left (a x\right )}{2 \, a} - \frac {c^{2}}{3 \, \sqrt {-a^{2} x^{2} + 1} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 71, normalized size = 1.16 \begin {gather*} -\frac {6 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (2 \, a^{2} c^{2} x^{2} + 3 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 9.00, size = 221, normalized size = 3.62 \begin {gather*} - a^{3} c^{2} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) + 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.43, size = 54, normalized size = 0.89 \begin {gather*} \frac {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 + 3 \, c^{2}\right )} x - \frac {2 \, c^{2}}{a}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.04, size = 82, normalized size = 1.34 \begin {gather*} \frac {c^2\,x\,\sqrt {1-a^2\,x^2}}{2}+\frac {c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{3\,a}+\frac {a\,c^2\,x^2\,\sqrt {1-a^2\,x^2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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