3.4.79 \(\int e^{\coth ^{-1}(a x)} (c-\frac {c}{a x})^4 \, dx\) [379]

Optimal. Leaf size=114 \[ -\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )}{2 a^2}+c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x-\frac {c^4 \csc ^{-1}(a x)}{2 a}-\frac {3 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]

[Out]

-1/3*c^4*(1-1/a^2/x^2)^(3/2)/a+c^4*(1-1/a^2/x^2)^(3/2)*x-1/2*c^4*arccsc(a*x)/a-3*c^4*arctanh((1-1/a^2/x^2)^(1/
2))/a+1/2*c^4*(6*a-1/x)*(1-1/a^2/x^2)^(1/2)/a^2

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Rubi [A]
time = 0.18, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6312, 1821, 1823, 829, 858, 222, 272, 65, 214} \begin {gather*} -\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )}{2 a^2}+c^4 x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-\frac {3 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}-\frac {c^4 \csc ^{-1}(a x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[a*x]*(c - c/(a*x))^4,x]

[Out]

-1/3*(c^4*(1 - 1/(a^2*x^2))^(3/2))/a + (c^4*Sqrt[1 - 1/(a^2*x^2)]*(6*a - x^(-1)))/(2*a^2) + c^4*(1 - 1/(a^2*x^
2))^(3/2)*x - (c^4*ArcCsc[a*x])/(2*a) - (3*c^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a

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 1821

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

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 6312

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

Rubi steps

\begin {align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx &=-\left (c \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^3 \sqrt {1-\frac {x^2}{a^2}}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x+c \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}} \left (\frac {3 c^3}{a}-\frac {c^3 x}{a^2}+\frac {c^3 x^2}{a^3}\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x-\frac {1}{3} \left (a^2 c\right ) \text {Subst}\left (\int \frac {\left (-\frac {9 c^3}{a^3}+\frac {3 c^3 x}{a^4}\right ) \sqrt {1-\frac {x^2}{a^2}}}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )}{2 a^2}+c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x+\frac {1}{6} \left (a^4 c\right ) \text {Subst}\left (\int \frac {\frac {18 c^3}{a^5}-\frac {3 c^3 x}{a^6}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )}{2 a^2}+c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x-\frac {c^4 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a^2}+\frac {\left (3 c^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )}{2 a^2}+c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x-\frac {c^4 \csc ^{-1}(a x)}{2 a}+\frac {\left (3 c^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a}\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )}{2 a^2}+c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x-\frac {c^4 \csc ^{-1}(a x)}{2 a}-\left (3 a c^4\right ) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )}{2 a^2}+c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x-\frac {c^4 \csc ^{-1}(a x)}{2 a}-\frac {3 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 175, normalized size = 1.54 \begin {gather*} \frac {c^4 \left (-2+9 a x-14 a^2 x^2-15 a^3 x^3+16 a^4 x^4+6 a^5 x^5+24 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^4 \text {ArcSin}\left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )+9 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^4 \text {ArcSin}\left (\frac {1}{a x}\right )-18 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{6 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^4*(-2 + 9*a*x - 14*a^2*x^2 - 15*a^3*x^3 + 16*a^4*x^4 + 6*a^5*x^5 + 24*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^4*ArcSin[
Sqrt[1 - 1/(a*x)]/Sqrt[2]] + 9*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^4*ArcSin[1/(a*x)] - 18*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^
4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/(6*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(223\) vs. \(2(100)=200\).
time = 0.07, size = 224, normalized size = 1.96

method	result	size
risch	\(\frac {\left (a x -1\right ) \left (16 a^{2} x^{2}-9 a x +2\right ) c^{4}}{6 x^{3} a^{4} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (a^{3} \sqrt {\left (a x +1\right ) \left (a x -1\right )}-\frac {3 a^{4} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}-\frac {a^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}\right ) c^{4} \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a^{4} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(157\)
default	\(-\frac {\left (a x -1\right ) c^{4} \left (-18 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+18 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+18 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+3 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-9 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x +2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{6 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{4} x^{3} \sqrt {a^{2}}}\)	\(224\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(a*x-1)*c^4*(-18*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*a^4*x^4+18*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*a^2*x^2+3*(a^2*x^
2-1)^(1/2)*(a^2)^(1/2)*a^3*x^3+18*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a^4*x^3+3*a^3*x^3*(a^2
)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))-9*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*a*x+2*(a^2*x^2-1)^(3/2)*(a^2)^(1/2))/((a*x
-1)/(a*x+1))^(1/2)/((a*x+1)*(a*x-1))^(1/2)/a^4/x^3/(a^2)^(1/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (100) = 200\).
time = 0.49, size = 224, normalized size = 1.96 \begin {gather*} \frac {1}{3} \, {\left (\frac {3 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {9 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {9 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {21 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 17 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 37 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 15 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {2 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + a^{2}}\right )} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*c^4*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 - 9*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 + 9*c^4*log(sq
rt((a*x - 1)/(a*x + 1)) - 1)/a^2 - (21*c^4*((a*x - 1)/(a*x + 1))^(7/2) - 17*c^4*((a*x - 1)/(a*x + 1))^(5/2) -
37*c^4*((a*x - 1)/(a*x + 1))^(3/2) - 15*c^4*sqrt((a*x - 1)/(a*x + 1)))/(2*(a*x - 1)*a^2/(a*x + 1) - 2*(a*x - 1
)^3*a^2/(a*x + 1)^3 - (a*x - 1)^4*a^2/(a*x + 1)^4 + a^2))*a

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Fricas [A]
time = 0.35, size = 156, normalized size = 1.37 \begin {gather*} \frac {6 \, a^{3} c^{4} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 18 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 18 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (6 \, a^{4} c^{4} x^{4} + 22 \, a^{3} c^{4} x^{3} + 7 \, a^{2} c^{4} x^{2} - 7 \, a c^{4} x + 2 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a^{4} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*a^3*c^4*x^3*arctan(sqrt((a*x - 1)/(a*x + 1))) - 18*a^3*c^4*x^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1) + 18*
a^3*c^4*x^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (6*a^4*c^4*x^4 + 22*a^3*c^4*x^3 + 7*a^2*c^4*x^2 - 7*a*c^4*x +
2*c^4)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*x^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {c^{4} \left (\int \frac {a^{4}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \frac {1}{x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {4 a}{x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {6 a^{2}}{x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {4 a^{3}}{x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx\right )}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**4*(Integral(a**4/sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(1/(x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))
), x) + Integral(-4*a/(x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))), x) + Integral(6*a**2/(x**2*sqrt(a*x/(a*x + 1)
- 1/(a*x + 1))), x) + Integral(-4*a**3/(x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))), x))/a**4

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (100) = 200\).
time = 0.42, size = 248, normalized size = 2.18 \begin {gather*} \frac {c^{4} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {3 \, c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c^{4}}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} c^{4} {\left | a \right |} + 12 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a c^{4} + 36 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{4} - 9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{4} {\left | a \right |} + 16 \, a c^{4}}{3 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{3} a {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))/(a*sgn(a*x + 1)) + 3*c^4*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(ab
s(a)*sgn(a*x + 1)) + sqrt(a^2*x^2 - 1)*c^4/(a*sgn(a*x + 1)) + 1/3*(9*(x*abs(a) - sqrt(a^2*x^2 - 1))^5*c^4*abs(
a) + 12*(x*abs(a) - sqrt(a^2*x^2 - 1))^4*a*c^4 + 36*(x*abs(a) - sqrt(a^2*x^2 - 1))^2*a*c^4 - 9*(x*abs(a) - sqr
t(a^2*x^2 - 1))*c^4*abs(a) + 16*a*c^4)/(((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)^3*a*abs(a)*sgn(a*x + 1))

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Mupad [B]
time = 1.32, size = 183, normalized size = 1.61 \begin {gather*} \frac {5\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {37\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+\frac {17\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{3}-7\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{a+\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}-\frac {2\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}}+\frac {c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {6\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*c^4*((a*x - 1)/(a*x + 1))^(1/2) + (37*c^4*((a*x - 1)/(a*x + 1))^(3/2))/3 + (17*c^4*((a*x - 1)/(a*x + 1))^(5
/2))/3 - 7*c^4*((a*x - 1)/(a*x + 1))^(7/2))/(a + (2*a*(a*x - 1))/(a*x + 1) - (2*a*(a*x - 1)^3)/(a*x + 1)^3 - (
a*(a*x - 1)^4)/(a*x + 1)^4) + (c^4*atan(((a*x - 1)/(a*x + 1))^(1/2)))/a - (6*c^4*atanh(((a*x - 1)/(a*x + 1))^(
1/2)))/a

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