3.2.58 \(\int e^{\tanh ^{-1}(a x)} (c-a c x)^4 \, dx\) [158]

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

[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)

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Rubi [A]
time = 0.06, 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 = {6262, 685, 655, 201, 222} \begin {gather*} \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 \text {ArcSin}(a x)}{8 a} \end {gather*}

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 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 655

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 685

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 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^{\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*}

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Mathematica [A]
time = 0.09, size = 75, normalized size = 0.61 \begin {gather*} -\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (-136-15 a x+112 a^2 x^2-90 a^3 x^3+24 a^4 x^4\right )+210 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{120 a} \end {gather*}

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(293\) vs. \(2(105)=210\).
time = 0.83, size = 294, normalized size = 2.39

method	result	size
risch	\(\frac {\left (24 a^{4} x^{4}-90 a^{3} x^{3}+112 a^{2} x^{2}-15 a x -136\right ) \left (a^{2} x^{2}-1\right ) c^{4}}{120 a \sqrt {-a^{2} x^{2}+1}}+\frac {7 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{4}}{8 \sqrt {a^{2}}}\)	\(91\)
meijerg	\(-\frac {c^{4} \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 a^{4} x^{4}+8 a^{2} x^{2}+16\right ) \sqrt {-a^{2} x^{2}+1}}{15}\right )}{2 a \sqrt {\pi }}-\frac {3 c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{4} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{a \sqrt {\pi }}-\frac {c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c^{4} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}+\frac {c^{4} \arcsin \left (a x \right )}{a}\)	\(277\)
default	\(c^{4} \left (a^{5} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )-3 a^{4} \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+2 a^{3} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )+2 a^{2} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )+\frac {3 \sqrt {-a^{2} x^{2}+1}}{a}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right )\)	\(294\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.46, size = 118, normalized size = 0.96 \begin {gather*} -\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} \end {gather*}

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

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Fricas [A]
time = 0.38, size = 92, normalized size = 0.75 \begin {gather*} -\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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (104) = 208\).
time = 5.35, size = 226, normalized size = 1.84 \begin {gather*} \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 {gather*}

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 = 0.43, size = 78, normalized size = 0.63 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.79, size = 128, normalized size = 1.04 \begin {gather*} \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} \end {gather*}

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

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