3.217 \(\int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx\)

Optimal. Leaf size=152 \[ -\frac {67}{8} a c^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+30 c^3 x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {315 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a}+2 a^2 c^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {1}{4} a^3 c^3 x^4 \sqrt {1-\frac {1}{a^2 x^2}} \]

[Out]

-315/8*c^3*arctanh((1-1/a^2/x^2)^(1/2))/a+32*c^3*(a-1/x)/a^2/(1-1/a^2/x^2)^(1/2)+30*c^3*x*(1-1/a^2/x^2)^(1/2)-
67/8*a*c^3*x^2*(1-1/a^2/x^2)^(1/2)+2*a^2*c^3*x^3*(1-1/a^2/x^2)^(1/2)-1/4*a^3*c^3*x^4*(1-1/a^2/x^2)^(1/2)

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Rubi [A]  time = 0.44, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6175, 6178, 1805, 1807, 807, 266, 63, 208} \[ -\frac {1}{4} a^3 c^3 x^4 \sqrt {1-\frac {1}{a^2 x^2}}+2 a^2 c^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {67}{8} a c^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+30 c^3 x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {315 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a} \]

Antiderivative was successfully verified.

[In]

Int[(c - a*c*x)^3/E^(3*ArcCoth[a*x]),x]

[Out]

(32*c^3*(a - x^(-1)))/(a^2*Sqrt[1 - 1/(a^2*x^2)]) + 30*c^3*Sqrt[1 - 1/(a^2*x^2)]*x - (67*a*c^3*Sqrt[1 - 1/(a^2
*x^2)]*x^2)/8 + 2*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^3 - (a^3*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^4)/4 - (315*c^3*ArcTanh
[Sqrt[1 - 1/(a^2*x^2)]])/(8*a)

Rule 63

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 208

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

Rule 266

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 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(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]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1807

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 6175

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

Rule 6178

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> -Dist[c^n, Subst[Int[((c
+ d*x)^(p - n)*(1 - x^2/a^2)^(n/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] &&
IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && In
tegerQ[2*p]

Rubi steps

\begin {align*} \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx &=-\left (\left (a^3 c^3\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^3 x^3 \, dx\right )\\ &=\left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^6}{x^5 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {-1+\frac {6 x}{a}-\frac {16 x^2}{a^2}+\frac {26 x^3}{a^3}-\frac {31 x^4}{a^4}}{x^5 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {1}{4} \left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {-\frac {24}{a}+\frac {67 x}{a^2}-\frac {104 x^2}{a^3}+\frac {124 x^3}{a^4}}{x^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+2 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {1}{12} \left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {-\frac {201}{a^2}+\frac {360 x}{a^3}-\frac {372 x^2}{a^4}}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {67}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {1}{24} \left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {-\frac {720}{a^3}+\frac {945 x}{a^4}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+30 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {67}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {\left (315 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{8 a}\\ &=\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+30 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {67}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {\left (315 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{16 a}\\ &=\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+30 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {67}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {1}{8} \left (315 a c^3\right ) \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )\\ &=\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+30 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {67}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {315 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a}\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 86, normalized size = 0.57 \[ \frac {1}{8} c^3 \left (\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left (-2 a^4 x^4+14 a^3 x^3-51 a^2 x^2+173 a x+496\right )}{a x+1}-\frac {315 \log \left (a x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )}{a}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - a*c*x)^3/E^(3*ArcCoth[a*x]),x]

[Out]

(c^3*((Sqrt[1 - 1/(a^2*x^2)]*x*(496 + 173*a*x - 51*a^2*x^2 + 14*a^3*x^3 - 2*a^4*x^4))/(1 + a*x) - (315*Log[a*(
1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a))/8

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fricas [A]  time = 0.49, size = 114, normalized size = 0.75 \[ -\frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (2 \, a^{4} c^{3} x^{4} - 14 \, a^{3} c^{3} x^{3} + 51 \, a^{2} c^{3} x^{2} - 173 \, a c^{3} x - 496 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{8 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^3*((a*x-1)/(a*x+1))^(3/2),x, algorithm="fricas")

[Out]

-1/8*(315*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 315*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (2*a^4*c^3*x^4
- 14*a^3*c^3*x^3 + 51*a^2*c^3*x^2 - 173*a*c^3*x - 496*c^3)*sqrt((a*x - 1)/(a*x + 1)))/a

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^3*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac")

[Out]

undef

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maple [B]  time = 0.06, size = 542, normalized size = 3.57 \[ \frac {\left (-2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{3} a^{3}+16 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{2} a^{2}-4 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}-69 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}+32 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a +384 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}-2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a -138 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{2} a^{2}+69 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-384 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-112 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+768 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a -69 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a +138 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}-768 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}+384 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+69 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -384 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right ) c^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{8 a \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*c*x+c)^3*((a*x-1)/(a*x+1))^(3/2),x)

[Out]

1/8*(-2*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^3*a^3+16*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^2*a^2-4*(a^2*x^2-1)^(3/
2)*(a^2)^(1/2)*x^2*a^2-69*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^3*a^3+32*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x*a+384
*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x^2*a^2-2*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x*a-138*(a^2*x^2-1)^(1/2)*(a^2)^(
1/2)*x^2*a^2+69*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a^3-384*ln((a^2*x+((a*x-1)*(a*x+1))^
(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a^3-112*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)+768*((a*x-1)*(a*x+1))^(1/2)*(a
^2)^(1/2)*x*a-69*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x*a+138*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x
*a^2-768*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x*a^2+384*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(
1/2)+69*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a-384*a*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^
(1/2))/(a^2)^(1/2)))/a*c^3*((a*x-1)/(a*x+1))^(3/2)/(a^2)^(1/2)/((a*x-1)*(a*x+1))^(1/2)/(a*x-1)

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maxima [A]  time = 0.32, size = 244, normalized size = 1.61 \[ -\frac {1}{8} \, {\left (\frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {256 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2}} - \frac {2 \, {\left (325 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 765 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 643 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 187 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {4 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {6 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} - a^{2}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^3*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")

[Out]

-1/8*(315*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 315*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - 256*c^
3*sqrt((a*x - 1)/(a*x + 1))/a^2 - 2*(325*c^3*((a*x - 1)/(a*x + 1))^(7/2) - 765*c^3*((a*x - 1)/(a*x + 1))^(5/2)
+ 643*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 187*c^3*sqrt((a*x - 1)/(a*x + 1)))/(4*(a*x - 1)*a^2/(a*x + 1) - 6*(a*
x - 1)^2*a^2/(a*x + 1)^2 + 4*(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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mupad [B]  time = 0.09, size = 199, normalized size = 1.31 \[ \frac {\frac {187\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}-\frac {643\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{4}+\frac {765\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{4}-\frac {325\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{4}}{a-\frac {4\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {6\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {4\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}}+\frac {32\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a}-\frac {315\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - a*c*x)^3*((a*x - 1)/(a*x + 1))^(3/2),x)

[Out]

((187*c^3*((a*x - 1)/(a*x + 1))^(1/2))/4 - (643*c^3*((a*x - 1)/(a*x + 1))^(3/2))/4 + (765*c^3*((a*x - 1)/(a*x
+ 1))^(5/2))/4 - (325*c^3*((a*x - 1)/(a*x + 1))^(7/2))/4)/(a - (4*a*(a*x - 1))/(a*x + 1) + (6*a*(a*x - 1)^2)/(
a*x + 1)^2 - (4*a*(a*x - 1)^3)/(a*x + 1)^3 + (a*(a*x - 1)^4)/(a*x + 1)^4) + (32*c^3*((a*x - 1)/(a*x + 1))^(1/2
))/a - (315*c^3*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(4*a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)**3*((a*x-1)/(a*x+1))**(3/2),x)

[Out]

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

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