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

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]

7/12*c^4*(-a^2*x^2+1)^(3/2)/a+7/20*c^4*(-a*x+1)*(-a^2*x^2+1)^(3/2)/a+1/5*c^4*(-a*x+1)^2*(-a^2*x^2+1)^(3/2)/a+7
/8*c^4*arcsin(a*x)/a+7/8*c^4*x*(-a^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 123, normalized size of antiderivative = 1.00, 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 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]

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 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 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]

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 = 0.11, 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]

-1/120*(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]/Sqrt[2]]))/a

________________________________________________________________________________________

fricas [A]  time = 0.75, size = 92, normalized size = 0.75 \[ -\frac {210 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (24 \, a^{4} c^{4} x^{4} - 90 \, a^{3} c^{4} x^{3} + 112 \, a^{2} c^{4} x^{2} - 15 \, a c^{4} x - 136 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, a} \]

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

________________________________________________________________________________________

giac [A]  time = 0.21, size = 78, normalized size = 0.63 \[ \frac {7 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{8 \, {\left | a \right |}} + \frac {1}{120} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {136 \, c^{4}}{a} + {\left (15 \, c^{4} - 2 \, {\left (56 \, a c^{4} + 3 \, {\left (4 \, a^{3} c^{4} x - 15 \, a^{2} c^{4}\right )} x\right )} x\right )} x\right )} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 137, normalized size = 1.11 \[ -\frac {c^{4} a^{3} x^{4} \sqrt {-a^{2} x^{2}+1}}{5}-\frac {14 c^{4} a \,x^{2} \sqrt {-a^{2} x^{2}+1}}{15}+\frac {17 c^{4} \sqrt {-a^{2} x^{2}+1}}{15 a}+\frac {3 c^{4} a^{2} x^{3} \sqrt {-a^{2} x^{2}+1}}{4}+\frac {c^{4} x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {7 c^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}} \]

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))

________________________________________________________________________________________

maxima [A]  time = 0.45, size = 118, normalized size = 0.96 \[ -\frac {1}{5} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{4} + \frac {3}{4} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x^{3} - \frac {14}{15} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} x^{2} + \frac {1}{8} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x + \frac {7 \, c^{4} \arcsin \left (a x\right )}{8 \, a} + \frac {17 \, \sqrt {-a^{2} x^{2} + 1} c^{4}}{15 \, a} \]

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*x)/a + 17/15*sqrt(-a^2*x^2 + 1)*c^4/a

________________________________________________________________________________________

mupad [B]  time = 0.79, size = 128, normalized size = 1.04 \[ \frac {c^4\,x\,\sqrt {1-a^2\,x^2}}{8}+\frac {7\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}+\frac {17\,c^4\,\sqrt {1-a^2\,x^2}}{15\,a}-\frac {14\,a\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{15}+\frac {3\,a^2\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{4}-\frac {a^3\,c^4\,x^4\,\sqrt {1-a^2\,x^2}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 8.98, size = 226, normalized size = 1.84 \[ \begin {cases} \frac {3 c^{4} \sqrt {- a^{2} x^{2} + 1} + 2 c^{4} \left (\begin {cases} - \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\operatorname {asin}{\left (a x \right )}}{2} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + 2 c^{4} \left (\begin {cases} \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} - \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) - 3 c^{4} \left (\begin {cases} \frac {a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt {- a^{2} x^{2} + 1}}{8} - \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + c^{4} \left (\begin {cases} - \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {5}{2}}}{5} + \frac {2 \left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} - \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + c^{4} \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\c^{4} x & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________