3.1247 \(\int e^{-2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^{7/2} \, dx\)

Optimal. Leaf size=154 \[ \frac{45}{128} c^3 x \sqrt{c-a^2 c x^2}+\frac{15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac{45 c^{7/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{128 a}+\frac{3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac{(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a} \]

[Out]

(45*c^3*x*Sqrt[c - a^2*c*x^2])/128 + (15*c^2*x*(c - a^2*c*x^2)^(3/2))/64 + (3*c*x*(c - a^2*c*x^2)^(5/2))/16 +
(9*(c - a^2*c*x^2)^(7/2))/(56*a) + ((1 - a*x)*(c - a^2*c*x^2)^(7/2))/(8*a) + (45*c^(7/2)*ArcTan[(a*Sqrt[c]*x)/
Sqrt[c - a^2*c*x^2]])/(128*a)

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Rubi [A]  time = 0.116381, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6142, 671, 641, 195, 217, 203} \[ \frac{45}{128} c^3 x \sqrt{c-a^2 c x^2}+\frac{15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac{45 c^{7/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{128 a}+\frac{3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac{(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(45*c^3*x*Sqrt[c - a^2*c*x^2])/128 + (15*c^2*x*(c - a^2*c*x^2)^(3/2))/64 + (3*c*x*(c - a^2*c*x^2)^(5/2))/16 +
(9*(c - a^2*c*x^2)^(7/2))/(56*a) + ((1 - a*x)*(c - a^2*c*x^2)^(7/2))/(8*a) + (45*c^(7/2)*ArcTan[(a*Sqrt[c]*x)/
Sqrt[c - a^2*c*x^2]])/(128*a)

Rule 6142

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

Rule 671

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*(m + p))/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx &=c \int (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=\frac{(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac{1}{8} (9 c) \int (1-a x) \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac{(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac{1}{8} (9 c) \int \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=\frac{3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac{(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac{1}{16} \left (15 c^2\right ) \int \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=\frac{15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac{3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac{(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac{1}{64} \left (45 c^3\right ) \int \sqrt{c-a^2 c x^2} \, dx\\ &=\frac{45}{128} c^3 x \sqrt{c-a^2 c x^2}+\frac{15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac{3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac{(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac{1}{128} \left (45 c^4\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{45}{128} c^3 x \sqrt{c-a^2 c x^2}+\frac{15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac{3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac{(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac{1}{128} \left (45 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )\\ &=\frac{45}{128} c^3 x \sqrt{c-a^2 c x^2}+\frac{15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac{3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac{9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac{(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac{45 c^{7/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{128 a}\\ \end{align*}

Mathematica [A]  time = 0.129328, size = 151, normalized size = 0.98 \[ -\frac{c^3 \sqrt{c-a^2 c x^2} \left (\sqrt{a x+1} \left (112 a^8 x^8-368 a^7 x^7+88 a^6 x^6+936 a^5 x^5-978 a^4 x^4-558 a^3 x^3+1349 a^2 x^2-325 a x-256\right )+630 \sqrt{1-a x} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{896 a \sqrt{1-a x} \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c^3*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(-256 - 325*a*x + 1349*a^2*x^2 - 558*a^3*x^3 - 978*a^4*x^4 + 936*a^5*
x^5 + 88*a^6*x^6 - 368*a^7*x^7 + 112*a^8*x^8) + 630*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(896*a*Sqrt[
1 - a*x]*Sqrt[1 - a^2*x^2])

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Maple [B]  time = 0.037, size = 276, normalized size = 1.8 \begin{align*} -{\frac{x}{8} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{7\,cx}{48} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{35\,x{c}^{2}}{192} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{35\,{c}^{3}x}{128}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{35\,{c}^{4}}{128}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}+{\frac{2}{7\,a} \left ( -c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{7}{2}}}}+{\frac{cx}{3} \left ( -c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{5\,x{c}^{2}}{12} \left ( -c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{c}^{3}x}{8}\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }}+{\frac{5\,{c}^{4}}{8}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*x*(-a^2*c*x^2+c)^(7/2)-7/48*c*x*(-a^2*c*x^2+c)^(5/2)-35/192*c^2*x*(-a^2*c*x^2+c)^(3/2)-35/128*c^3*x*(-a^2
*c*x^2+c)^(1/2)-35/128*c^4/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))+2/7/a*(-c*a^2*(x+1/a)^2+
2*a*c*(x+1/a))^(7/2)+1/3*c*(-c*a^2*(x+1/a)^2+2*a*c*(x+1/a))^(5/2)*x+5/12*c^2*(-c*a^2*(x+1/a)^2+2*a*c*(x+1/a))^
(3/2)*x+5/8*c^3*(-c*a^2*(x+1/a)^2+2*a*c*(x+1/a))^(1/2)*x+5/8*c^4/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-c*a^2*
(x+1/a)^2+2*a*c*(x+1/a))^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.50984, size = 655, normalized size = 4.25 \begin{align*} \left [\frac{315 \, \sqrt{-c} c^{3} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) + 2 \,{\left (112 \, a^{7} c^{3} x^{7} - 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} + 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} - 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x + 256 \, c^{3}\right )} \sqrt{-a^{2} c x^{2} + c}}{1792 \, a}, -\frac{315 \, c^{\frac{7}{2}} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) -{\left (112 \, a^{7} c^{3} x^{7} - 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} + 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} - 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x + 256 \, c^{3}\right )} \sqrt{-a^{2} c x^{2} + c}}{896 \, a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/1792*(315*sqrt(-c)*c^3*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(112*a^7*c^3*x^7 - 25
6*a^6*c^3*x^6 - 168*a^5*c^3*x^5 + 768*a^4*c^3*x^4 - 210*a^3*c^3*x^3 - 768*a^2*c^3*x^2 + 581*a*c^3*x + 256*c^3)
*sqrt(-a^2*c*x^2 + c))/a, -1/896*(315*c^(7/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - (112*
a^7*c^3*x^7 - 256*a^6*c^3*x^6 - 168*a^5*c^3*x^5 + 768*a^4*c^3*x^4 - 210*a^3*c^3*x^3 - 768*a^2*c^3*x^2 + 581*a*
c^3*x + 256*c^3)*sqrt(-a^2*c*x^2 + c))/a]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.42158, size = 562, normalized size = 3.65 \begin{align*} -\frac{{\left (80640 \, a^{9} c^{\frac{7}{2}} \arctan \left (\frac{\sqrt{-c + \frac{2 \, c}{a x + 1}}}{\sqrt{c}}\right ) \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) - \frac{{\left (315 \, a^{9}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{7} c^{4} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) - 2415 \, a^{9}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{6} c^{5} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 8043 \, a^{9}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{5} c^{6} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 17609 \, a^{9}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{4} c^{7} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) - 15159 \, a^{9}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{3} c^{8} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 8043 \, a^{9}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{2} c^{9} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 315 \, a^{9} c^{11} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 2415 \, a^{9} c^{10}{\left (-c + \frac{2 \, c}{a x + 1}\right )}^{\frac{3}{2}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right )\right )}{\left (a x + 1\right )}^{8}}{c^{8}}\right )}{\left | a \right |}}{114688 \, a^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/114688*(80640*a^9*c^(7/2)*arctan(sqrt(-c + 2*c/(a*x + 1))/sqrt(c))*sgn(1/(a*x + 1))*sgn(a) - (315*a^9*(c -
2*c/(a*x + 1))^7*c^4*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) - 2415*a^9*(c - 2*c/(a*x + 1))^6*c^5*sqr
t(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) + 8043*a^9*(c - 2*c/(a*x + 1))^5*c^6*sqrt(-c + 2*c/(a*x + 1))*sg
n(1/(a*x + 1))*sgn(a) + 17609*a^9*(c - 2*c/(a*x + 1))^4*c^7*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) -
15159*a^9*(c - 2*c/(a*x + 1))^3*c^8*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) + 8043*a^9*(c - 2*c/(a*x
+ 1))^2*c^9*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) + 315*a^9*c^11*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a
*x + 1))*sgn(a) + 2415*a^9*c^10*(-c + 2*c/(a*x + 1))^(3/2)*sgn(1/(a*x + 1))*sgn(a))*(a*x + 1)^8/c^8)*abs(a)/a^
11