3.907 \(\int \frac {e^{\tanh ^{-1}(a x)} x^7}{(c-a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=143 \[ -\frac {7 \sin ^{-1}(a x)}{2 a^8 c^3}+\frac {x^6 (a x+1)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {(35 a x+32) \sqrt {1-a^2 x^2}}{10 a^8 c^3}+\frac {x^2 (35 a x+24)}{15 a^6 c^3 \sqrt {1-a^2 x^2}}-\frac {x^4 (7 a x+6)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}} \]

[Out]

1/5*x^6*(a*x+1)/a^2/c^3/(-a^2*x^2+1)^(5/2)-1/15*x^4*(7*a*x+6)/a^4/c^3/(-a^2*x^2+1)^(3/2)-7/2*arcsin(a*x)/a^8/c
^3+1/15*x^2*(35*a*x+24)/a^6/c^3/(-a^2*x^2+1)^(1/2)+1/10*(35*a*x+32)*(-a^2*x^2+1)^(1/2)/a^8/c^3

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Rubi [A]  time = 0.18, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6148, 819, 780, 216} \[ \frac {x^6 (a x+1)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^4 (7 a x+6)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {x^2 (35 a x+24)}{15 a^6 c^3 \sqrt {1-a^2 x^2}}+\frac {(35 a x+32) \sqrt {1-a^2 x^2}}{10 a^8 c^3}-\frac {7 \sin ^{-1}(a x)}{2 a^8 c^3} \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*x^7)/(c - a^2*c*x^2)^3,x]

[Out]

(x^6*(1 + a*x))/(5*a^2*c^3*(1 - a^2*x^2)^(5/2)) - (x^4*(6 + 7*a*x))/(15*a^4*c^3*(1 - a^2*x^2)^(3/2)) + (x^2*(2
4 + 35*a*x))/(15*a^6*c^3*Sqrt[1 - a^2*x^2]) + ((32 + 35*a*x)*Sqrt[1 - a^2*x^2])/(10*a^8*c^3) - (7*ArcSin[a*x])
/(2*a^8*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 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

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]

Rubi steps

\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^7}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {x^7 (1+a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac {x^6 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {\int \frac {x^5 (6+7 a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a^2 c^3}\\ &=\frac {x^6 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^4 (6+7 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {x^3 (24+35 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 a^4 c^3}\\ &=\frac {x^6 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^4 (6+7 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {x^2 (24+35 a x)}{15 a^6 c^3 \sqrt {1-a^2 x^2}}-\frac {\int \frac {x (48+105 a x)}{\sqrt {1-a^2 x^2}} \, dx}{15 a^6 c^3}\\ &=\frac {x^6 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^4 (6+7 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {x^2 (24+35 a x)}{15 a^6 c^3 \sqrt {1-a^2 x^2}}+\frac {(32+35 a x) \sqrt {1-a^2 x^2}}{10 a^8 c^3}-\frac {7 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a^7 c^3}\\ &=\frac {x^6 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^4 (6+7 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {x^2 (24+35 a x)}{15 a^6 c^3 \sqrt {1-a^2 x^2}}+\frac {(32+35 a x) \sqrt {1-a^2 x^2}}{10 a^8 c^3}-\frac {7 \sin ^{-1}(a x)}{2 a^8 c^3}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 116, normalized size = 0.81 \[ \frac {-15 a^6 x^6-15 a^5 x^5+176 a^4 x^4+4 a^3 x^3-249 a^2 x^2-105 (a x-1)^2 (a x+1) \sqrt {1-a^2 x^2} \sin ^{-1}(a x)+9 a x+96}{30 a^8 c^3 (a x-1)^2 (a x+1) \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[a*x]*x^7)/(c - a^2*c*x^2)^3,x]

[Out]

(96 + 9*a*x - 249*a^2*x^2 + 4*a^3*x^3 + 176*a^4*x^4 - 15*a^5*x^5 - 15*a^6*x^6 - 105*(-1 + a*x)^2*(1 + a*x)*Sqr
t[1 - a^2*x^2]*ArcSin[a*x])/(30*a^8*c^3*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2])

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fricas [A]  time = 0.74, size = 221, normalized size = 1.55 \[ \frac {96 \, a^{5} x^{5} - 96 \, a^{4} x^{4} - 192 \, a^{3} x^{3} + 192 \, a^{2} x^{2} + 96 \, a x + 210 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (15 \, a^{6} x^{6} + 15 \, a^{5} x^{5} - 176 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 249 \, a^{2} x^{2} - 9 \, a x - 96\right )} \sqrt {-a^{2} x^{2} + 1} - 96}{30 \, {\left (a^{13} c^{3} x^{5} - a^{12} c^{3} x^{4} - 2 \, a^{11} c^{3} x^{3} + 2 \, a^{10} c^{3} x^{2} + a^{9} c^{3} x - a^{8} c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^7/(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

1/30*(96*a^5*x^5 - 96*a^4*x^4 - 192*a^3*x^3 + 192*a^2*x^2 + 96*a*x + 210*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^
2*x^2 + a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (15*a^6*x^6 + 15*a^5*x^5 - 176*a^4*x^4 - 4*a^3*x^3 +
249*a^2*x^2 - 9*a*x - 96)*sqrt(-a^2*x^2 + 1) - 96)/(a^13*c^3*x^5 - a^12*c^3*x^4 - 2*a^11*c^3*x^3 + 2*a^10*c^3
*x^2 + a^9*c^3*x - a^8*c^3)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^7/(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.05, size = 283, normalized size = 1.98 \[ \frac {x \sqrt {-a^{2} x^{2}+1}}{2 c^{3} a^{7}}-\frac {7 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 c^{3} a^{7} \sqrt {a^{2}}}+\frac {\sqrt {-a^{2} x^{2}+1}}{c^{3} a^{8}}-\frac {7 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{15 c^{3} a^{10} \left (x -\frac {1}{a}\right )^{2}}-\frac {773 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{240 c^{3} a^{9} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{20 c^{3} a^{11} \left (x -\frac {1}{a}\right )^{3}}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{24 c^{3} a^{10} \left (x +\frac {1}{a}\right )^{2}}-\frac {31 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{48 c^{3} a^{9} \left (x +\frac {1}{a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^7/(-a^2*c*x^2+c)^3,x)

[Out]

1/2/c^3/a^7*x*(-a^2*x^2+1)^(1/2)-7/2/c^3/a^7/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+1/c^3/a^8*(-
a^2*x^2+1)^(1/2)-7/15/c^3/a^10/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-773/240/c^3/a^9/(x-1/a)*(-a^2*(x-1
/a)^2-2*a*(x-1/a))^(1/2)-1/20/c^3/a^11/(x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/24/c^3/a^10/(x+1/a)^2*(-
a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)-31/48/c^3/a^9/(x+1/a)*(-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 \[ -a \int \frac {x^{8}}{{\left (a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}\,{d x} + \frac {\frac {5 \, \sqrt {-a^{2} x^{2} + 1}}{c^{3}} + \frac {5 \, a^{2} x^{2} + 15 \, {\left (a^{2} x^{2} - 1\right )}^{2} - 4}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} c^{3}}}{5 \, a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^7/(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

-a*integrate(x^8/((a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)*sqrt(a*x + 1)*sqrt(-a*x + 1)), x) + 1/5*
(5*sqrt(-a^2*x^2 + 1)/c^3 + (5*a^2*x^2 + 15*(a^2*x^2 - 1)^2 - 4)/((-a^2*x^2 + 1)^(5/2)*c^3))/a^8

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mupad [B]  time = 0.10, size = 352, normalized size = 2.46 \[ \frac {\sqrt {1-a^2\,x^2}}{24\,\left (a^{10}\,c^3\,x^2+2\,a^9\,c^3\,x+a^8\,c^3\right )}-\frac {7\,\sqrt {1-a^2\,x^2}}{15\,\left (a^{10}\,c^3\,x^2-2\,a^9\,c^3\,x+a^8\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (a^6\,c^3\,\sqrt {-a^2}+3\,a^8\,c^3\,x^2\,\sqrt {-a^2}-a^9\,c^3\,x^3\,\sqrt {-a^2}-3\,a^7\,c^3\,x\,\sqrt {-a^2}\right )}+\frac {31\,\sqrt {1-a^2\,x^2}}{48\,\left (a^6\,c^3\,\sqrt {-a^2}+a^7\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {773\,\sqrt {1-a^2\,x^2}}{240\,\left (a^6\,c^3\,\sqrt {-a^2}-a^7\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}}{a^8\,c^3}+\frac {x\,\sqrt {1-a^2\,x^2}}{2\,a^7\,c^3}-\frac {7\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^7\,c^3\,\sqrt {-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^7*(a*x + 1))/((c - a^2*c*x^2)^3*(1 - a^2*x^2)^(1/2)),x)

[Out]

(1 - a^2*x^2)^(1/2)/(24*(a^8*c^3 + 2*a^9*c^3*x + a^10*c^3*x^2)) - (7*(1 - a^2*x^2)^(1/2))/(15*(a^8*c^3 - 2*a^9
*c^3*x + a^10*c^3*x^2)) - (1 - a^2*x^2)^(1/2)/(20*(-a^2)^(1/2)*(a^6*c^3*(-a^2)^(1/2) + 3*a^8*c^3*x^2*(-a^2)^(1
/2) - a^9*c^3*x^3*(-a^2)^(1/2) - 3*a^7*c^3*x*(-a^2)^(1/2))) + (31*(1 - a^2*x^2)^(1/2))/(48*(a^6*c^3*(-a^2)^(1/
2) + a^7*c^3*x*(-a^2)^(1/2))*(-a^2)^(1/2)) - (773*(1 - a^2*x^2)^(1/2))/(240*(a^6*c^3*(-a^2)^(1/2) - a^7*c^3*x*
(-a^2)^(1/2))*(-a^2)^(1/2)) + (1 - a^2*x^2)^(1/2)/(a^8*c^3) + (x*(1 - a^2*x^2)^(1/2))/(2*a^7*c^3) - (7*asinh(x
*(-a^2)^(1/2)))/(2*a^7*c^3*(-a^2)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{7}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{8}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**7/(-a**2*c*x**2+c)**3,x)

[Out]

(Integral(x**7/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x*
*2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**8/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**
2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3

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