3.4.20 \(\int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^4}{x} \, dx\) [320]

Optimal. Leaf size=101 \[ \frac {1}{8} c^4 (8-13 a x) \sqrt {1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac {13}{8} c^4 \text {ArcSin}(a x)-c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]

[Out]

-c^4*(-a^2*x^2+1)^(3/2)+1/4*a*c^4*x*(-a^2*x^2+1)^(3/2)-13/8*c^4*arcsin(a*x)-c^4*arctanh((-a^2*x^2+1)^(1/2))+1/
8*c^4*(-13*a*x+8)*(-a^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6263, 1823, 829, 858, 222, 272, 65, 214} \begin {gather*} \frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{8} c^4 (8-13 a x) \sqrt {1-a^2 x^2}-c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {13}{8} c^4 \text {ArcSin}(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^4*(8 - 13*a*x)*Sqrt[1 - a^2*x^2])/8 - c^4*(1 - a^2*x^2)^(3/2) + (a*c^4*x*(1 - a^2*x^2)^(3/2))/4 - (13*c^4*A
rcSin[a*x])/8 - c^4*ArcTanh[Sqrt[1 - a^2*x^2]]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 829

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m +
2*p + 2))), x] + Dist[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*
m, 2*p])

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1823

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 6263

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

Rubi steps

\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^4}{x} \, dx &=c \int \frac {(c-a c x)^3 \sqrt {1-a^2 x^2}}{x} \, dx\\ &=\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac {c \int \frac {\sqrt {1-a^2 x^2} \left (-4 a^2 c^3+13 a^3 c^3 x-12 a^4 c^3 x^2\right )}{x} \, dx}{4 a^2}\\ &=-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {c \int \frac {\left (12 a^4 c^3-39 a^5 c^3 x\right ) \sqrt {1-a^2 x^2}}{x} \, dx}{12 a^4}\\ &=\frac {1}{8} c^4 (8-13 a x) \sqrt {1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac {c \int \frac {-24 a^6 c^3+39 a^7 c^3 x}{x \sqrt {1-a^2 x^2}} \, dx}{24 a^6}\\ &=\frac {1}{8} c^4 (8-13 a x) \sqrt {1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}+c^4 \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx-\frac {1}{8} \left (13 a c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {1}{8} c^4 (8-13 a x) \sqrt {1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac {13}{8} c^4 \sin ^{-1}(a x)+\frac {1}{2} c^4 \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{8} c^4 (8-13 a x) \sqrt {1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac {13}{8} c^4 \sin ^{-1}(a x)-\frac {c^4 \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a^2}\\ &=\frac {1}{8} c^4 (8-13 a x) \sqrt {1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac {13}{8} c^4 \sin ^{-1}(a x)-c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 142, normalized size = 1.41 \begin {gather*} \frac {c^4 \left (-11 a x+8 a^2 x^2+9 a^3 x^3-8 a^4 x^4+2 a^5 x^5+4 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)+34 \sqrt {1-a^2 x^2} \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-8 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\right )}{8 \sqrt {1-a^2 x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^4*(-11*a*x + 8*a^2*x^2 + 9*a^3*x^3 - 8*a^4*x^4 + 2*a^5*x^5 + 4*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + 34*Sqrt[1 -
a^2*x^2]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]] - 8*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]))/(8*Sqrt[1 - a^2*x^2]
)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(238\) vs. \(2(87)=174\).
time = 0.84, size = 239, normalized size = 2.37

method	result	size
default	\(c^{4} \left (a^{5} \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )-3 a^{4} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )+2 a^{3} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )-2 \sqrt {-a^{2} x^{2}+1}-\frac {3 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\)	\(239\)
meijerg	\(\frac {a \,c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 c^{4} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{2 \sqrt {\pi }}-\frac {a \,c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {c^{4} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{\sqrt {\pi }}-3 c^{4} \arcsin \left (a x \right )+\frac {c^{4} \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }\right )}{2 \sqrt {\pi }}\)	\(270\)

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,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.46, size = 105, normalized size = 1.04 \begin {gather*} -\frac {1}{4} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{3} + \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x^{2} - \frac {11}{8} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} x - \frac {13}{8} \, c^{4} \arcsin \left (a x\right ) - c^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\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,x, algorithm="maxima")

[Out]

-1/4*sqrt(-a^2*x^2 + 1)*a^3*c^4*x^3 + sqrt(-a^2*x^2 + 1)*a^2*c^4*x^2 - 11/8*sqrt(-a^2*x^2 + 1)*a*c^4*x - 13/8*
c^4*arcsin(a*x) - c^4*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x))

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Fricas [A]
time = 0.38, size = 95, normalized size = 0.94 \begin {gather*} \frac {13}{4} \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + c^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - \frac {1}{8} \, {\left (2 \, a^{3} c^{4} x^{3} - 8 \, a^{2} c^{4} x^{2} + 11 \, a c^{4} x\right )} \sqrt {-a^{2} x^{2} + 1} \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,x, algorithm="fricas")

[Out]

13/4*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + c^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 1/8*(2*a^3*c^4*x^3 - 8
*a^2*c^4*x^2 + 11*a*c^4*x)*sqrt(-a^2*x^2 + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 13.18, size = 420, normalized size = 4.16 \begin {gather*} a^{5} c^{4} \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 ) - 3 a^{4} c^{4} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{4} \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 ) + 2 a^{2} c^{4} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) - 3 a c^{4} \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 ) + c^{4} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\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,x)

[Out]

a**5*c**4*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)) - 3*a**4*c**4*Piece
wise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) + 2*a*
*3*c**4*Piecewise((-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*sq
rt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + 2*a**2*c**4*Piecewise((x*
*2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - 3*a*c**4*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2))
, a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + c**4*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**
2) > 1), (I*asin(1/(a*x)), True))

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Giac [A]
time = 0.43, size = 100, normalized size = 0.99 \begin {gather*} -\frac {13 \, a c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, {\left | a \right |}} - \frac {a c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {1}{8} \, {\left (11 \, a c^{4} + 2 \, {\left (a^{3} c^{4} x - 4 \, a^{2} c^{4}\right )} x\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)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^4/x,x, algorithm="giac")

[Out]

-13/8*a*c^4*arcsin(a*x)*sgn(a)/abs(a) - a*c^4*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/ab
s(a) - 1/8*(11*a*c^4 + 2*(a^3*c^4*x - 4*a^2*c^4)*x)*sqrt(-a^2*x^2 + 1)*x

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Mupad [B]
time = 0.78, size = 110, normalized size = 1.09 \begin {gather*} a^2\,c^4\,x^2\,\sqrt {1-a^2\,x^2}-\frac {a^3\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{4}-\frac {11\,a\,c^4\,x\,\sqrt {1-a^2\,x^2}}{8}-\frac {13\,a\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}+c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*atan((1 - a^2*x^2)^(1/2)*1i)*1i + a^2*c^4*x^2*(1 - a^2*x^2)^(1/2) - (a^3*c^4*x^3*(1 - a^2*x^2)^(1/2))/4 -
(11*a*c^4*x*(1 - a^2*x^2)^(1/2))/8 - (13*a*c^4*asinh(x*(-a^2)^(1/2)))/(8*(-a^2)^(1/2))

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