3.2.79 \(\int e^{3 \tanh ^{-1}(a x)} (c-a c x)^3 \, dx\) [179]

Optimal. Leaf size=59 \[ \frac {3}{8} c^3 x \sqrt {1-a^2 x^2}+\frac {1}{4} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {3 c^3 \text {ArcSin}(a x)}{8 a} \]

[Out]

1/4*c^3*x*(-a^2*x^2+1)^(3/2)+3/8*c^3*arcsin(a*x)/a+3/8*c^3*x*(-a^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6262, 201, 222} \begin {gather*} \frac {1}{4} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {3}{8} c^3 x \sqrt {1-a^2 x^2}+\frac {3 c^3 \text {ArcSin}(a x)}{8 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*c^3*x*Sqrt[1 - a^2*x^2])/8 + (c^3*x*(1 - a^2*x^2)^(3/2))/4 + (3*c^3*ArcSin[a*x])/(8*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 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)^3 \, dx &=c^3 \int \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac {1}{4} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} \left (3 c^3\right ) \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {3}{8} c^3 x \sqrt {1-a^2 x^2}+\frac {1}{4} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {1}{8} \left (3 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3}{8} c^3 x \sqrt {1-a^2 x^2}+\frac {1}{4} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {3 c^3 \sin ^{-1}(a x)}{8 a}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 44, normalized size = 0.75 \begin {gather*} \frac {c^3 \left (a x \left (5-2 a^2 x^2\right ) \sqrt {1-a^2 x^2}+3 \text {ArcSin}(a x)\right )}{8 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(257\) vs. \(2(49)=98\).
time = 1.13, size = 258, normalized size = 4.37

method	result	size
risch	\(\frac {x \left (2 a^{2} x^{2}-5\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{8 \sqrt {-a^{2} x^{2}+1}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{3}}{8 \sqrt {a^{2}}}\)	\(69\)
meijerg	\(\frac {3 c^{3} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{3} x}{\sqrt {-a^{2} x^{2}+1}}+\frac {c^{3} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (-14 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{56 a^{6} \sqrt {-a^{2} x^{2}+1}}-\frac {15 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c^{3} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}\)	\(234\)
default	\(-c^{3} \left (a^{6} \left (-\frac {x^{5}}{4 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {5 x^{3}}{8 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {5 \left (\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{4 a^{2}}}{a^{2}}\right )-3 a^{4} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+3 a^{2} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )-\frac {x}{\sqrt {-a^{2} x^{2}+1}}\right )\)	\(258\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.47, size = 77, normalized size = 1.31 \begin {gather*} \frac {a^{4} c^{3} x^{5}}{4 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {7 \, a^{2} c^{3} x^{3}}{8 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {5 \, c^{3} x}{8 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, c^{3} \arcsin \left (a x\right )}{8 \, 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)^3,x, algorithm="maxima")

[Out]

1/4*a^4*c^3*x^5/sqrt(-a^2*x^2 + 1) - 7/8*a^2*c^3*x^3/sqrt(-a^2*x^2 + 1) + 5/8*c^3*x/sqrt(-a^2*x^2 + 1) + 3/8*c
^3*arcsin(a*x)/a

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Fricas [A]
time = 0.35, size = 65, normalized size = 1.10 \begin {gather*} -\frac {6 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{3} c^{3} x^{3} - 5 \, a c^{3} x\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, 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)^3,x, algorithm="fricas")

[Out]

-1/8*(6*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^3*c^3*x^3 - 5*a*c^3*x)*sqrt(-a^2*x^2 + 1))/a

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Sympy [A]
time = 5.83, size = 301, normalized size = 5.10 \begin {gather*} a^{4} c^{3} \left (\begin {cases} - \frac {i x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {i x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {3 i x}{8 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \operatorname {acosh}{\left (a x \right )}}{8 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {x^{3}}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{5}} & \text {otherwise} \end {cases}\right ) - 2 a^{2} c^{3} \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 ) + 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 ) \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)**3,x)

[Out]

a**4*c**3*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqr
t(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**
2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) - 2*a**2*c**3*Piece
wise((-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)) + c**3*Piecewise((sqrt(a**(-2))*asin(x*s
qrt(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 = 48, normalized size = 0.81 \begin {gather*} \frac {3 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, {\left | a \right |}} - \frac {1}{8} \, {\left (2 \, a^{2} c^{3} x^{2} - 5 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} x \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)^3,x, algorithm="giac")

[Out]

3/8*c^3*arcsin(a*x)*sgn(a)/abs(a) - 1/8*(2*a^2*c^3*x^2 - 5*c^3)*sqrt(-a^2*x^2 + 1)*x

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Mupad [B]
time = 0.05, size = 64, normalized size = 1.08 \begin {gather*} \frac {5\,c^3\,x\,\sqrt {1-a^2\,x^2}}{8}+\frac {3\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}-\frac {a^2\,c^3\,x^3\,\sqrt {1-a^2\,x^2}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*c^3*x*(1 - a^2*x^2)^(1/2))/8 + (3*c^3*asinh(x*(-a^2)^(1/2)))/(8*(-a^2)^(1/2)) - (a^2*c^3*x^3*(1 - a^2*x^2)^
(1/2))/4

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