3.429 \(\int e^{-3 \coth ^{-1}(a x)} (c-\frac {c}{a x})^4 \, dx\)

Optimal. Leaf size=164 \[ \frac {64 c^4 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+c^4 x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {68 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a}-\frac {7 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}-\frac {7 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}+\frac {91 c^4 \csc ^{-1}(a x)}{2 a} \]

[Out]

91/2*c^4*arccsc(a*x)/a-7*c^4*arctanh((1-1/a^2/x^2)^(1/2))/a+64*c^4*(a-1/x)/a^2/(1-1/a^2/x^2)^(1/2)+68/3*c^4*(1
-1/a^2/x^2)^(1/2)/a+1/3*c^4*(1-1/a^2/x^2)^(1/2)/a^3/x^2-7/2*c^4*(1-1/a^2/x^2)^(1/2)/a^2/x+c^4*x*(1-1/a^2/x^2)^
(1/2)

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Rubi [A]  time = 0.55, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6177, 1805, 1807, 1809, 844, 216, 266, 63, 208} \[ \frac {64 c^4 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+c^4 x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {68 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a}-\frac {7 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}-\frac {7 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+\frac {91 c^4 \csc ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[(c - c/(a*x))^4/E^(3*ArcCoth[a*x]),x]

[Out]

(68*c^4*Sqrt[1 - 1/(a^2*x^2)])/(3*a) + (64*c^4*(a - x^(-1)))/(a^2*Sqrt[1 - 1/(a^2*x^2)]) + (c^4*Sqrt[1 - 1/(a^
2*x^2)])/(3*a^3*x^2) - (7*c^4*Sqrt[1 - 1/(a^2*x^2)])/(2*a^2*x) + c^4*Sqrt[1 - 1/(a^2*x^2)]*x + (91*c^4*ArcCsc[
a*x])/(2*a) - (7*c^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/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 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 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 844

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 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 1809

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 6177

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^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^7}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {64 c^4 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\operatorname {Subst}\left (\int \frac {-c^7+\frac {7 c^7 x}{a}+\frac {42 c^7 x^2}{a^2}-\frac {22 c^7 x^3}{a^3}+\frac {7 c^7 x^4}{a^4}-\frac {c^7 x^5}{a^5}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {64 c^4 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {\operatorname {Subst}\left (\int \frac {-\frac {7 c^7}{a}-\frac {42 c^7 x}{a^2}+\frac {22 c^7 x^2}{a^3}-\frac {7 c^7 x^3}{a^4}+\frac {c^7 x^4}{a^5}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {64 c^4 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {a^2 \operatorname {Subst}\left (\int \frac {\frac {21 c^7}{a^3}+\frac {126 c^7 x}{a^4}-\frac {68 c^7 x^2}{a^5}+\frac {21 c^7 x^3}{a^6}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{3 c^3}\\ &=\frac {64 c^4 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}-\frac {7 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {a^4 \operatorname {Subst}\left (\int \frac {-\frac {42 c^7}{a^5}-\frac {273 c^7 x}{a^6}+\frac {136 c^7 x^2}{a^7}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{6 c^3}\\ &=\frac {68 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a}+\frac {64 c^4 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}-\frac {7 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {a^6 \operatorname {Subst}\left (\int \frac {\frac {42 c^7}{a^7}+\frac {273 c^7 x}{a^8}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{6 c^3}\\ &=\frac {68 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a}+\frac {64 c^4 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}-\frac {7 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {\left (91 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a^2}+\frac {\left (7 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {68 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a}+\frac {64 c^4 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}-\frac {7 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {91 c^4 \csc ^{-1}(a x)}{2 a}+\frac {\left (7 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a}\\ &=\frac {68 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a}+\frac {64 c^4 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}-\frac {7 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {91 c^4 \csc ^{-1}(a x)}{2 a}-\left (7 a c^4\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 {68 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a}+\frac {64 c^4 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}-\frac {7 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {91 c^4 \csc ^{-1}(a x)}{2 a}-\frac {7 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\\ \end {align*}

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Mathematica [C]  time = 1.19, size = 567, normalized size = 3.46 \[ \frac {c^4 \left (2772 \sqrt {2} a^3 x^3 (a x+1) (a x-1)^3 \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )+1980 \sqrt {2} a^2 x^2 (a x+1) (a x-1)^4 \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )+35 \left (396 a^8 x^8 \sqrt {\frac {1}{a x}+1}-50160 a^7 x^7 \sqrt {\frac {1}{a x}+1}+66726 a^7 x^7 \sqrt {1-\frac {1}{a x}} \sin ^{-1}\left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )-1980 a^7 x^7 \sqrt {1-\frac {1}{a x}} \sin ^{-1}\left (\frac {1}{a x}\right )+29403 a^6 x^6 \sqrt {\frac {1}{a x}+1}+66726 a^6 x^6 \sqrt {1-\frac {1}{a x}} \sin ^{-1}\left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )-1980 a^6 x^6 \sqrt {1-\frac {1}{a x}} \sin ^{-1}\left (\frac {1}{a x}\right )+26268 a^5 x^5 \sqrt {\frac {1}{a x}+1}-7425 a^4 x^4 \sqrt {\frac {1}{a x}+1}+1716 a^3 x^3 \sqrt {\frac {1}{a x}+1}-198 a^2 x^2 \sqrt {\frac {1}{a x}+1}-2772 a^7 x^7 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {\frac {1}{a x}+1} \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )+44 \sqrt {2} a x (a x-1)^5 (a x+1) \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )+36 \sqrt {2} (a x-1)^6 (a x+1) \, _2F_1\left (\frac {3}{2},\frac {11}{2};\frac {13}{2};\frac {1}{2} \left (1-\frac {1}{a x}\right )\right )\right )\right )}{13860 a^7 x^6 \sqrt {1-\frac {1}{a x}} (a x+1)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - c/(a*x))^4/E^(3*ArcCoth[a*x]),x]

[Out]

(c^4*(2772*Sqrt[2]*a^3*x^3*(-1 + a*x)^3*(1 + a*x)*Hypergeometric2F1[3/2, 5/2, 7/2, (1 - 1/(a*x))/2] + 1980*Sqr
t[2]*a^2*x^2*(-1 + a*x)^4*(1 + a*x)*Hypergeometric2F1[3/2, 7/2, 9/2, (1 - 1/(a*x))/2] + 35*(-198*a^2*Sqrt[1 +
1/(a*x)]*x^2 + 1716*a^3*Sqrt[1 + 1/(a*x)]*x^3 - 7425*a^4*Sqrt[1 + 1/(a*x)]*x^4 + 26268*a^5*Sqrt[1 + 1/(a*x)]*x
^5 + 29403*a^6*Sqrt[1 + 1/(a*x)]*x^6 - 50160*a^7*Sqrt[1 + 1/(a*x)]*x^7 + 396*a^8*Sqrt[1 + 1/(a*x)]*x^8 + 66726
*a^6*Sqrt[1 - 1/(a*x)]*x^6*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] + 66726*a^7*Sqrt[1 - 1/(a*x)]*x^7*ArcSin[Sqrt[1 -
1/(a*x)]/Sqrt[2]] - 1980*a^6*Sqrt[1 - 1/(a*x)]*x^6*ArcSin[1/(a*x)] - 1980*a^7*Sqrt[1 - 1/(a*x)]*x^7*ArcSin[1/
(a*x)] - 2772*a^7*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[1 + 1/(a*x)]*x^7*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]] + 44*Sqrt[2]*a*x*
(-1 + a*x)^5*(1 + a*x)*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] + 36*Sqrt[2]*(-1 + a*x)^6*(1 + a*x)*
Hypergeometric2F1[3/2, 11/2, 13/2, (1 - 1/(a*x))/2])))/(13860*a^7*Sqrt[1 - 1/(a*x)]*x^6*(1 + a*x))

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fricas [A]  time = 0.58, size = 157, normalized size = 0.96 \[ -\frac {546 \, a^{3} c^{4} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 42 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 42 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (6 \, a^{4} c^{4} x^{4} + 526 \, a^{3} c^{4} x^{3} + 115 \, a^{2} c^{4} x^{2} - 19 \, a c^{4} x + 2 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a/x)^4*((a*x-1)/(a*x+1))^(3/2),x, algorithm="fricas")

[Out]

-1/6*(546*a^3*c^4*x^3*arctan(sqrt((a*x - 1)/(a*x + 1))) + 42*a^3*c^4*x^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1) -
42*a^3*c^4*x^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (6*a^4*c^4*x^4 + 526*a^3*c^4*x^3 + 115*a^2*c^4*x^2 - 19*a*
c^4*x + 2*c^4)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*x^3)

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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((c-c/a/x)^4*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac")

[Out]

undef

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maple [B]  time = 0.06, size = 672, normalized size = 4.10 \[ -\frac {\left (-138 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{6} a^{6}+138 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}-549 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{5} a^{5}-273 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}\, x^{5} a^{5}+138 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{5} a^{6}+96 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{5} a^{5}-96 \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^{5} a^{6}+255 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{3} a^{3}-684 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{4} a^{4}-546 a^{4} x^{4} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+276 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{4} a^{5}+192 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{3} a^{3}+192 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} a^{4}-192 \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^{4} a^{5}+98 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}-273 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}-273 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+138 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}+96 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}-96 \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^{3} a^{4}-17 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right ) c^{4} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{6 a^{4} \sqrt {a^{2}}\, x^{3} \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((c-c/a/x)^4*((a*x-1)/(a*x+1))^(3/2),x)

[Out]

-1/6*(-138*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^6*a^6+138*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^4*a^4-549*(a^2*x^2-1)^(1/
2)*(a^2)^(1/2)*x^5*a^5-273*arctan(1/(a^2*x^2-1)^(1/2))*(a^2)^(1/2)*x^5*a^5+138*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^
2)^(1/2))/(a^2)^(1/2))*x^5*a^6+96*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^5*a^5-96*ln((a^2*x+((a*x-1)*(a*x+1))^(
1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^5*a^6+255*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^3*a^3-684*(a^2*x^2-1)^(1/2)*(a^2)^(
1/2)*x^4*a^4-546*a^4*x^4*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))+276*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/
(a^2)^(1/2))*x^4*a^5+192*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^3*a^3+192*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x
^4*a^4-192*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^4*a^5+98*(a^2*x^2-1)^(3/2)*(a^2)^(1/2
)*x^2*a^2-273*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^3*a^3-273*a^3*x^3*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))+138*ln
((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^3*a^4+96*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a^3-96*
ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^3*a^4-17*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x*a+2*(a^
2*x^2-1)^(3/2)*(a^2)^(1/2))/a^4*c^4*((a*x-1)/(a*x+1))^(3/2)/(a^2)^(1/2)/x^3/((a*x-1)*(a*x+1))^(1/2)/(a*x-1)

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maxima [A]  time = 0.43, size = 246, normalized size = 1.50 \[ -\frac {1}{3} \, {\left (\frac {273 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {21 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {21 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {192 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2}} + \frac {153 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 91 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 169 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 123 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {2 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + a^{2}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a/x)^4*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")

[Out]

-1/3*(273*c^4*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 21*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 21*c^4*l
og(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - 192*c^4*sqrt((a*x - 1)/(a*x + 1))/a^2 + (153*c^4*((a*x - 1)/(a*x + 1))
^(7/2) + 91*c^4*((a*x - 1)/(a*x + 1))^(5/2) - 169*c^4*((a*x - 1)/(a*x + 1))^(3/2) - 123*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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mupad [B]  time = 0.14, size = 211, normalized size = 1.29 \[ \frac {41\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {169\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}-\frac {91\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{3}-51\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{a+\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}-\frac {2\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}}+\frac {64\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a}-\frac {91\,c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,14{}\mathrm {i}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - c/(a*x))^4*((a*x - 1)/(a*x + 1))^(3/2),x)

[Out]

(41*c^4*((a*x - 1)/(a*x + 1))^(1/2) + (169*c^4*((a*x - 1)/(a*x + 1))^(3/2))/3 - (91*c^4*((a*x - 1)/(a*x + 1))^
(5/2))/3 - 51*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) + (64*c^4*((a*x - 1)/(a*x + 1))^(1/2))/a - (91*c^4*atan(((a*x - 1)/(a*x + 1))^(
1/2)))/a + (c^4*atan(((a*x - 1)/(a*x + 1))^(1/2)*1i)*14i)/a

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {c^{4} \left (\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{5} + x^{4}}\right )\, dx + \int \frac {5 a \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{4} + x^{3}}\, dx + \int \left (- \frac {10 a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{3} + x^{2}}\right )\, dx + \int \frac {10 a^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{2} + x}\, dx + \int \left (- \frac {5 a^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{5} x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a/x)**4*((a*x-1)/(a*x+1))**(3/2),x)

[Out]

c**4*(Integral(-sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**5 + x**4), x) + Integral(5*a*sqrt(a*x/(a*x + 1) - 1/(a
*x + 1))/(a*x**4 + x**3), x) + Integral(-10*a**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**3 + x**2), x) + Integ
ral(10*a**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**2 + x), x) + Integral(-5*a**4*sqrt(a*x/(a*x + 1) - 1/(a*x
+ 1))/(a*x + 1), x) + Integral(a**5*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x))/a**4

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