3.2.91 \(\int e^{4 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx\) [191]

Optimal. Leaf size=17 \[ \frac {c^2 (1+a x)^3}{3 a} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6264, 32} \begin {gather*} \frac {c^2 (a x+1)^3}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcTanh[a*x])*(c - a*c*x)^2,x]

[Out]

(c^2*(1 + a*x)^3)/(3*a)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 6264

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

Rubi steps

\begin {align*} \int e^{4 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx &=c^2 \int (1+a x)^2 \, dx\\ &=\frac {c^2 (1+a x)^3}{3 a}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 27, normalized size = 1.59 \begin {gather*} c^2 x+a c^2 x^2+\frac {1}{3} a^2 c^2 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcTanh[a*x])*(c - a*c*x)^2,x]

[Out]

c^2*x + a*c^2*x^2 + (a^2*c^2*x^3)/3

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Maple [A]
time = 1.14, size = 16, normalized size = 0.94

method	result	size
default	\(\frac {c^{2} \left (a x +1\right )^{3}}{3 a}\)	\(16\)
gosper	\(\frac {x \left (a^{2} x^{2}+3 a x +3\right ) c^{2}}{3}\)	\(20\)
risch	\(\frac {a^{2} c^{2} x^{3}}{3}+a \,c^{2} x^{2}+x \,c^{2}+\frac {c^{2}}{3 a}\)	\(34\)
norman	\(\frac {-a \,c^{2} x^{2}+a^{3} c^{2} x^{4}-x \,c^{2}+\frac {2}{3} a^{2} c^{2} x^{3}+\frac {1}{3} a^{4} c^{2} x^{5}}{a^{2} x^{2}-1}\)	\(61\)
meijerg	\(-\frac {c^{2} \left (\frac {x \left (-a^{2}\right )^{\frac {7}{2}} \left (-14 a^{4} x^{4}-70 a^{2} x^{2}+105\right )}{21 a^{6} \left (-a^{2} x^{2}+1\right )}-\frac {5 \left (-a^{2}\right )^{\frac {7}{2}} \arctanh \left (a x \right )}{a^{7}}\right )}{2 \sqrt {-a^{2}}}-\frac {c^{2} \left (-\frac {a^{2} x^{2} \left (-3 a^{2} x^{2}+6\right )}{3 \left (-a^{2} x^{2}+1\right )}-2 \ln \left (-a^{2} x^{2}+1\right )\right )}{a}-\frac {c^{2} \left (\frac {x \left (-a^{2}\right )^{\frac {5}{2}} \left (-10 a^{2} x^{2}+15\right )}{5 a^{4} \left (-a^{2} x^{2}+1\right )}-\frac {3 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}\right )}{2 \sqrt {-a^{2}}}-\frac {2 c^{2} \left (\frac {a^{2} x^{2}}{-a^{2} x^{2}+1}+\ln \left (-a^{2} x^{2}+1\right )\right )}{a}+\frac {c^{2} \left (\frac {x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \left (-a^{2} x^{2}+1\right )}-\frac {\left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}+\frac {a \,c^{2} x^{2}}{-a^{2} x^{2}+1}+\frac {c^{2} \left (\frac {2 x \sqrt {-a^{2}}}{-2 a^{2} x^{2}+2}+\frac {\sqrt {-a^{2}}\, \arctanh \left (a x \right )}{a}\right )}{2 \sqrt {-a^{2}}}\)	\(352\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 25, normalized size = 1.47 \begin {gather*} \frac {1}{3} \, a^{2} c^{2} x^{3} + a c^{2} x^{2} + c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*c^2*x^3 + a*c^2*x^2 + c^2*x

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Fricas [A]
time = 0.40, size = 25, normalized size = 1.47 \begin {gather*} \frac {1}{3} \, a^{2} c^{2} x^{3} + a c^{2} x^{2} + c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*c^2*x^3 + a*c^2*x^2 + c^2*x

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Sympy [A]
time = 0.03, size = 24, normalized size = 1.41 \begin {gather*} \frac {a^{2} c^{2} x^{3}}{3} + a c^{2} x^{2} + c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a*c*x+c)**2,x)

[Out]

a**2*c**2*x**3/3 + a*c**2*x**2 + c**2*x

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Giac [A]
time = 0.41, size = 25, normalized size = 1.47 \begin {gather*} \frac {1}{3} \, a^{2} c^{2} x^{3} + a c^{2} x^{2} + c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*c^2*x^3 + a*c^2*x^2 + c^2*x

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Mupad [B]
time = 0.04, size = 19, normalized size = 1.12 \begin {gather*} \frac {c^2\,x\,\left (a^2\,x^2+3\,a\,x+3\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*x*(3*a*x + a^2*x^2 + 3))/3

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