3.319 \(\int e^{\tanh ^{-1}(a x)} (c-a c x)^4 \, dx\)

Optimal. Leaf size=123 \[ \frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7}{8} c^4 x \sqrt{1-a^2 x^2}+\frac{7 c^4 \sin ^{-1}(a x)}{8 a} \]

[Out]

(7*c^4*x*Sqrt[1 - a^2*x^2])/8 + (7*c^4*(1 - a^2*x^2)^(3/2))/(12*a) + (7*c^4*(1 - a*x)*(1 - a^2*x^2)^(3/2))/(20
*a) + (c^4*(1 - a*x)^2*(1 - a^2*x^2)^(3/2))/(5*a) + (7*c^4*ArcSin[a*x])/(8*a)

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Rubi [A]  time = 0.0812557, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {6127, 671, 641, 195, 216} \[ \frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7}{8} c^4 x \sqrt{1-a^2 x^2}+\frac{7 c^4 \sin ^{-1}(a x)}{8 a} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*(c - a*c*x)^4,x]

[Out]

(7*c^4*x*Sqrt[1 - a^2*x^2])/8 + (7*c^4*(1 - a^2*x^2)^(3/2))/(12*a) + (7*c^4*(1 - a*x)*(1 - a^2*x^2)^(3/2))/(20
*a) + (c^4*(1 - a*x)^2*(1 - a^2*x^2)^(3/2))/(5*a) + (7*c^4*ArcSin[a*x])/(8*a)

Rule 6127

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]

Rule 671

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 641

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 195

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 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]

Rubi steps

\begin{align*} \int e^{\tanh ^{-1}(a x)} (c-a c x)^4 \, dx &=c \int (c-a c x)^3 \sqrt{1-a^2 x^2} \, dx\\ &=\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{5} \left (7 c^2\right ) \int (c-a c x)^2 \sqrt{1-a^2 x^2} \, dx\\ &=\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{4} \left (7 c^3\right ) \int (c-a c x) \sqrt{1-a^2 x^2} \, dx\\ &=\frac{7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{4} \left (7 c^4\right ) \int \sqrt{1-a^2 x^2} \, dx\\ &=\frac{7}{8} c^4 x \sqrt{1-a^2 x^2}+\frac{7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{8} \left (7 c^4\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{7}{8} c^4 x \sqrt{1-a^2 x^2}+\frac{7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{7 c^4 \sin ^{-1}(a x)}{8 a}\\ \end{align*}

Mathematica [A]  time = 0.0951718, size = 75, normalized size = 0.61 \[ -\frac{c^4 \left (\sqrt{1-a^2 x^2} \left (24 a^4 x^4-90 a^3 x^3+112 a^2 x^2-15 a x-136\right )+210 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{120 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c^4*(Sqrt[1 - a^2*x^2]*(-136 - 15*a*x + 112*a^2*x^2 - 90*a^3*x^3 + 24*a^4*x^4) + 210*ArcSin[Sqrt[1 - a*x]/Sq
rt[2]]))/(120*a)

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Maple [A]  time = 0.043, size = 137, normalized size = 1.1 \begin{align*} -{\frac{{c}^{4}{x}^{4}{a}^{3}}{5}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{14\,{c}^{4}a{x}^{2}}{15}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{17\,{c}^{4}}{15\,a}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{3\,{a}^{2}{c}^{4}{x}^{3}}{4}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{c}^{4}x}{8}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{7\,{c}^{4}}{8}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*c^4*a^3*x^4*(-a^2*x^2+1)^(1/2)-14/15*c^4*a*x^2*(-a^2*x^2+1)^(1/2)+17/15*c^4*(-a^2*x^2+1)^(1/2)/a+3/4*c^4*
a^2*x^3*(-a^2*x^2+1)^(1/2)+1/8*c^4*x*(-a^2*x^2+1)^(1/2)+7/8*c^4/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^
(1/2))

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Maxima [A]  time = 1.44511, size = 171, normalized size = 1.39 \begin{align*} -\frac{1}{5} \, \sqrt{-a^{2} x^{2} + 1} a^{3} c^{4} x^{4} + \frac{3}{4} \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{4} x^{3} - \frac{14}{15} \, \sqrt{-a^{2} x^{2} + 1} a c^{4} x^{2} + \frac{1}{8} \, \sqrt{-a^{2} x^{2} + 1} c^{4} x + \frac{7 \, c^{4} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{8 \, \sqrt{a^{2}}} + \frac{17 \, \sqrt{-a^{2} x^{2} + 1} c^{4}}{15 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*sqrt(-a^2*x^2 + 1)*a^3*c^4*x^4 + 3/4*sqrt(-a^2*x^2 + 1)*a^2*c^4*x^3 - 14/15*sqrt(-a^2*x^2 + 1)*a*c^4*x^2
+ 1/8*sqrt(-a^2*x^2 + 1)*c^4*x + 7/8*c^4*arcsin(a^2*x/sqrt(a^2))/sqrt(a^2) + 17/15*sqrt(-a^2*x^2 + 1)*c^4/a

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Fricas [A]  time = 1.58102, size = 209, normalized size = 1.7 \begin{align*} -\frac{210 \, c^{4} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (24 \, a^{4} c^{4} x^{4} - 90 \, a^{3} c^{4} x^{3} + 112 \, a^{2} c^{4} x^{2} - 15 \, a c^{4} x - 136 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{120 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*(210*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (24*a^4*c^4*x^4 - 90*a^3*c^4*x^3 + 112*a^2*c^4*x^2 -
15*a*c^4*x - 136*c^4)*sqrt(-a^2*x^2 + 1))/a

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Sympy [A]  time = 8.73666, size = 226, normalized size = 1.84 \begin{align*} \begin{cases} \frac{3 c^{4} \sqrt{- a^{2} x^{2} + 1} + 2 c^{4} \left (\begin{cases} - \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (a x \right )}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) + 2 c^{4} \left (\begin{cases} \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{3} - \sqrt{- a^{2} x^{2} + 1} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) - 3 c^{4} \left (\begin{cases} \frac{a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt{- a^{2} x^{2} + 1}}{8} - \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{3 \operatorname{asin}{\left (a x \right )}}{8} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) + c^{4} \left (\begin{cases} - \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{5}{2}}}{5} + \frac{2 \left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{3} - \sqrt{- a^{2} x^{2} + 1} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) + c^{4} \operatorname{asin}{\left (a x \right )}}{a} & \text{for}\: a \neq 0 \\c^{4} x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((3*c**4*sqrt(-a**2*x**2 + 1) + 2*c**4*Piecewise((-a*x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/2, (a*x >
-1) & (a*x < 1))) + 2*c**4*Piecewise(((-a**2*x**2 + 1)**(3/2)/3 - sqrt(-a**2*x**2 + 1), (a*x > -1) & (a*x < 1)
)) - 3*c**4*Piecewise((a*x*(-2*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/8 - a*x*sqrt(-a**2*x**2 + 1)/2 + 3*asin(a*x
)/8, (a*x > -1) & (a*x < 1))) + c**4*Piecewise((-(-a**2*x**2 + 1)**(5/2)/5 + 2*(-a**2*x**2 + 1)**(3/2)/3 - sqr
t(-a**2*x**2 + 1), (a*x > -1) & (a*x < 1))) + c**4*asin(a*x))/a, Ne(a, 0)), (c**4*x, True))

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Giac [A]  time = 1.25185, size = 105, normalized size = 0.85 \begin{align*} \frac{7 \, c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \,{\left | a \right |}} + \frac{1}{120} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{136 \, c^{4}}{a} +{\left (15 \, c^{4} - 2 \,{\left (56 \, a c^{4} + 3 \,{\left (4 \, a^{3} c^{4} x - 15 \, a^{2} c^{4}\right )} x\right )} x\right )} x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/8*c^4*arcsin(a*x)*sgn(a)/abs(a) + 1/120*sqrt(-a^2*x^2 + 1)*(136*c^4/a + (15*c^4 - 2*(56*a*c^4 + 3*(4*a^3*c^4
*x - 15*a^2*c^4)*x)*x)*x)