3.651 \(\int \frac {e^{3 \tanh ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^3} \, dx\)

Optimal. Leaf size=155 \[ -\frac {(a x+1)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {38 (a x+1)^2}{35 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {137 (a x+1)}{35 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {\sqrt {1-a^2 x^2}}{a c^3}+\frac {181 a x+245}{35 a c^3 \sqrt {1-a^2 x^2}}-\frac {3 \sin ^{-1}(a x)}{a c^3} \]

[Out]

-1/7*(a*x+1)^3/a/c^3/(-a^2*x^2+1)^(7/2)+38/35*(a*x+1)^2/a/c^3/(-a^2*x^2+1)^(5/2)-137/35*(a*x+1)/a/c^3/(-a^2*x^
2+1)^(3/2)-3*arcsin(a*x)/a/c^3+1/35*(181*a*x+245)/a/c^3/(-a^2*x^2+1)^(1/2)+(-a^2*x^2+1)^(1/2)/a/c^3

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Rubi [A]  time = 0.44, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6157, 6148, 1635, 1814, 641, 216} \[ -\frac {(a x+1)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {38 (a x+1)^2}{35 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {137 (a x+1)}{35 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {\sqrt {1-a^2 x^2}}{a c^3}+\frac {181 a x+245}{35 a c^3 \sqrt {1-a^2 x^2}}-\frac {3 \sin ^{-1}(a x)}{a c^3} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^3,x]

[Out]

-(1 + a*x)^3/(7*a*c^3*(1 - a^2*x^2)^(7/2)) + (38*(1 + a*x)^2)/(35*a*c^3*(1 - a^2*x^2)^(5/2)) - (137*(1 + a*x))
/(35*a*c^3*(1 - a^2*x^2)^(3/2)) + (245 + 181*a*x)/(35*a*c^3*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/(a*c^3) - (
3*ArcSin[a*x])/(a*c^3)

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 641

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 1635

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, -Simp[(d*f*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*
a*e*(p + 1)), x] + Dist[d/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)*Q
+ f*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p +
1/2, 0] && GtQ[m, 0]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[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[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 6148

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

Rule 6157

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

Rubi steps

\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx &=-\frac {a^6 \int \frac {e^{3 \tanh ^{-1}(a x)} x^6}{\left (1-a^2 x^2\right )^3} \, dx}{c^3}\\ &=-\frac {a^6 \int \frac {x^6 (1+a x)^3}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{c^3}\\ &=-\frac {(1+a x)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {a^6 \int \frac {(1+a x)^2 \left (\frac {3}{a^6}+\frac {7 x}{a^5}+\frac {7 x^2}{a^4}+\frac {7 x^3}{a^3}+\frac {7 x^4}{a^2}+\frac {7 x^5}{a}\right )}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^3}\\ &=-\frac {(1+a x)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {38 (1+a x)^2}{35 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {a^6 \int \frac {(1+a x) \left (\frac {61}{a^6}+\frac {140 x}{a^5}+\frac {105 x^2}{a^4}+\frac {70 x^3}{a^3}+\frac {35 x^4}{a^2}\right )}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^3}\\ &=-\frac {(1+a x)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {38 (1+a x)^2}{35 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {137 (1+a x)}{35 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^6 \int \frac {\frac {228}{a^6}+\frac {630 x}{a^5}+\frac {315 x^2}{a^4}+\frac {105 x^3}{a^3}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{105 c^3}\\ &=-\frac {(1+a x)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {38 (1+a x)^2}{35 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {137 (1+a x)}{35 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {245+181 a x}{35 a c^3 \sqrt {1-a^2 x^2}}-\frac {a^6 \int \frac {\frac {315}{a^6}+\frac {105 x}{a^5}}{\sqrt {1-a^2 x^2}} \, dx}{105 c^3}\\ &=-\frac {(1+a x)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {38 (1+a x)^2}{35 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {137 (1+a x)}{35 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {245+181 a x}{35 a c^3 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^3}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^3}\\ &=-\frac {(1+a x)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {38 (1+a x)^2}{35 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {137 (1+a x)}{35 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {245+181 a x}{35 a c^3 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^3}-\frac {3 \sin ^{-1}(a x)}{a c^3}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 96, normalized size = 0.62 \[ \frac {-35 a^5 x^5+286 a^4 x^4-368 a^3 x^3-125 a^2 x^2-105 (a x-1)^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)+423 a x-176}{35 a c^3 (a x-1)^3 \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^3,x]

[Out]

(-176 + 423*a*x - 125*a^2*x^2 - 368*a^3*x^3 + 286*a^4*x^4 - 35*a^5*x^5 - 105*(-1 + a*x)^3*Sqrt[1 - a^2*x^2]*Ar
cSin[a*x])/(35*a*c^3*(-1 + a*x)^3*Sqrt[1 - a^2*x^2])

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fricas [A]  time = 0.53, size = 212, normalized size = 1.37 \[ \frac {176 \, a^{5} x^{5} - 528 \, a^{4} x^{4} + 352 \, a^{3} x^{3} + 352 \, a^{2} x^{2} - 528 \, a x + 210 \, {\left (a^{5} x^{5} - 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (35 \, a^{5} x^{5} - 286 \, a^{4} x^{4} + 368 \, a^{3} x^{3} + 125 \, a^{2} x^{2} - 423 \, a x + 176\right )} \sqrt {-a^{2} x^{2} + 1} + 176}{35 \, {\left (a^{6} c^{3} x^{5} - 3 \, a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} + 2 \, a^{3} c^{3} x^{2} - 3 \, a^{2} c^{3} x + a c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35*(176*a^5*x^5 - 528*a^4*x^4 + 352*a^3*x^3 + 352*a^2*x^2 - 528*a*x + 210*(a^5*x^5 - 3*a^4*x^4 + 2*a^3*x^3 +
2*a^2*x^2 - 3*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (35*a^5*x^5 - 286*a^4*x^4 + 368*a^3*x^3 + 125
*a^2*x^2 - 423*a*x + 176)*sqrt(-a^2*x^2 + 1) + 176)/(a^6*c^3*x^5 - 3*a^5*c^3*x^4 + 2*a^4*c^3*x^3 + 2*a^3*c^3*x
^2 - 3*a^2*c^3*x + a*c^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 256, normalized size = 1.65 \[ -\frac {a \,x^{2}}{c^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{a \,c^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {13 x}{c^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{3} \sqrt {a^{2}}}+\frac {38}{35 a^{3} c^{3} \left (x -\frac {1}{a}\right )^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {137}{35 a^{2} c^{3} \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {274 x}{35 c^{3} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {1}{7 a^{4} c^{3} \left (x -\frac {1}{a}\right )^{3} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/c^3*x^2/(-a^2*x^2+1)^(1/2)+8/a/c^3/(-a^2*x^2+1)^(1/2)+13*x/c^3/(-a^2*x^2+1)^(1/2)-3/c^3/(a^2)^(1/2)*arctan(
(a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+38/35/a^3/c^3/(x-1/a)^2/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+137/35/a^2/c^3/(x
-1/a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-274/35/c^3/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)*x+1/7/a^4/c^3/(x-1/a)^3
/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.61, size = 1548, normalized size = 9.99 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(71*(1 - a^2*x^2)^(1/2))/(280*(-a^2)^(1/2)*(c^3*1i + 3*c^3*x*(-a^2)^(1/2) + a^2*c^3*x^2*3i + a^2*c^3*x^3*(-a^2
)^(1/2))) - (3*a^7*(1 - a^2*x^2)^(1/2))/(112*(a^8*c^3 + c^3*x*(-a^2)^(9/2)*4i + 6*a^10*c^3*x^2 + a^12*c^3*x^4
+ a^2*c^3*x^3*(-a^2)^(9/2)*4i)) - (a^9*(1 - a^2*x^2)^(1/2))/(112*(a^10*c^3 - c^3*x*(-a^2)^(11/2)*4i + 6*a^12*c
^3*x^2 + a^14*c^3*x^4 - a^2*c^3*x^3*(-a^2)^(11/2)*4i)) - (a^9*(1 - a^2*x^2)^(1/2))/(112*(a^10*c^3 + c^3*x*(-a^
2)^(11/2)*4i + 6*a^12*c^3*x^2 + a^14*c^3*x^4 + a^2*c^3*x^3*(-a^2)^(11/2)*4i)) - (537*a^7*(1 - a^2*x^2)^(1/2))/
(1120*(a^8*c^3 - c^3*x*(-a^2)^(9/2)*2i + a^10*c^3*x^2)) - (537*a^7*(1 - a^2*x^2)^(1/2))/(1120*(a^8*c^3 + c^3*x
*(-a^2)^(9/2)*2i + a^10*c^3*x^2)) - (417*a^9*(1 - a^2*x^2)^(1/2))/(1120*(a^10*c^3 - c^3*x*(-a^2)^(11/2)*2i + a
^12*c^3*x^2)) - (417*a^9*(1 - a^2*x^2)^(1/2))/(1120*(a^10*c^3 + c^3*x*(-a^2)^(11/2)*2i + a^12*c^3*x^2)) - (3*a
sinh(x*(-a^2)^(1/2)))/(c^3*(-a^2)^(1/2)) - (3*a^7*(1 - a^2*x^2)^(1/2))/(112*(a^8*c^3 - c^3*x*(-a^2)^(9/2)*4i +
6*a^10*c^3*x^2 + a^12*c^3*x^4 - a^2*c^3*x^3*(-a^2)^(9/2)*4i)) - (71*(1 - a^2*x^2)^(1/2))/(280*(-a^2)^(1/2)*(c
^3*1i - 3*c^3*x*(-a^2)^(1/2) + a^2*c^3*x^2*3i - a^2*c^3*x^3*(-a^2)^(1/2))) + (a^7*(1 - a^2*x^2)^(1/2)*6i)/(35*
(a^8*c^3*1i + 3*c^3*x*(-a^2)^(9/2) + a^10*c^3*x^2*3i + a^2*c^3*x^3*(-a^2)^(9/2))) + (a^7*(1 - a^2*x^2)^(1/2)*6
i)/(35*(a^8*c^3*1i - 3*c^3*x*(-a^2)^(9/2) + a^10*c^3*x^2*3i - a^2*c^3*x^3*(-a^2)^(9/2))) + (a^9*(1 - a^2*x^2)^
(1/2)*23i)/(280*(a^10*c^3*1i + 3*c^3*x*(-a^2)^(11/2) + a^12*c^3*x^2*3i + a^2*c^3*x^3*(-a^2)^(11/2))) + (a^9*(1
- a^2*x^2)^(1/2)*23i)/(280*(a^10*c^3*1i - 3*c^3*x*(-a^2)^(11/2) + a^12*c^3*x^2*3i - a^2*c^3*x^3*(-a^2)^(11/2)
)) + (181*(1 - a^2*x^2)^(1/2))/(70*(c^3*1i + c^3*x*(-a^2)^(1/2))*(-a^2)^(1/2)) - (181*(1 - a^2*x^2)^(1/2))/(70
*(c^3*1i - c^3*x*(-a^2)^(1/2))*(-a^2)^(1/2)) + (a^7*(1 - a^2*x^2)^(1/2)*1143i)/(1120*(a^8*c^3*1i + c^3*x*(-a^2
)^(9/2))) + (a^7*(1 - a^2*x^2)^(1/2)*1143i)/(1120*(a^8*c^3*1i - c^3*x*(-a^2)^(9/2))) + (a^9*(1 - a^2*x^2)^(1/2
)*1823i)/(1120*(a^10*c^3*1i + c^3*x*(-a^2)^(11/2))) + (a^9*(1 - a^2*x^2)^(1/2)*1823i)/(1120*(a^10*c^3*1i - c^3
*x*(-a^2)^(11/2))) + (1 - a^2*x^2)^(1/2)/(a*c^3) + ((1 - a^2*x^2)^(1/2)*1i)/(28*(-a^2)^(1/2)*(c^3 - c^3*x*(-a^
2)^(1/2)*4i + 6*a^2*c^3*x^2 + a^4*c^3*x^4 - a^2*c^3*x^3*(-a^2)^(1/2)*4i)) - ((1 - a^2*x^2)^(1/2)*1i)/(28*(-a^2
)^(1/2)*(c^3 + c^3*x*(-a^2)^(1/2)*4i + 6*a^2*c^3*x^2 + a^4*c^3*x^4 + a^2*c^3*x^3*(-a^2)^(1/2)*4i)) + ((1 - a^2
*x^2)^(1/2)*477i)/(560*(-a^2)^(1/2)*(c^3 - c^3*x*(-a^2)^(1/2)*2i + a^2*c^3*x^2)) - ((1 - a^2*x^2)^(1/2)*477i)/
(560*(-a^2)^(1/2)*(c^3 + c^3*x*(-a^2)^(1/2)*2i + a^2*c^3*x^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{6} \int \frac {x^{6}}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**6*Integral(x**6/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**
2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x)/c**3

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