### 3.623 $$\int e^{2 \coth ^{-1}(a x)} (c-a^2 c x^2)^{9/2} \, dx$$

Optimal. Leaf size=176 $-\frac{77}{256} c^4 x \sqrt{c-a^2 c x^2}-\frac{77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac{77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac{77 c^{9/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{256 a}-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{(a x+1) \left (c-a^2 c x^2\right )^{9/2}}{10 a}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}$

[Out]

(-77*c^4*x*Sqrt[c - a^2*c*x^2])/256 - (77*c^3*x*(c - a^2*c*x^2)^(3/2))/384 - (77*c^2*x*(c - a^2*c*x^2)^(5/2))/
480 - (11*c*x*(c - a^2*c*x^2)^(7/2))/80 + (11*(c - a^2*c*x^2)^(9/2))/(90*a) + ((1 + a*x)*(c - a^2*c*x^2)^(9/2)
)/(10*a) - (77*c^(9/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(256*a)

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Rubi [A]  time = 0.165263, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.292, Rules used = {6167, 6141, 671, 641, 195, 217, 203} $-\frac{77}{256} c^4 x \sqrt{c-a^2 c x^2}-\frac{77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac{77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac{77 c^{9/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{256 a}-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{(a x+1) \left (c-a^2 c x^2\right )^{9/2}}{10 a}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(9/2),x]

[Out]

(-77*c^4*x*Sqrt[c - a^2*c*x^2])/256 - (77*c^3*x*(c - a^2*c*x^2)^(3/2))/384 - (77*c^2*x*(c - a^2*c*x^2)^(5/2))/
480 - (11*c*x*(c - a^2*c*x^2)^(7/2))/80 + (11*(c - a^2*c*x^2)^(9/2))/(90*a) + ((1 + a*x)*(c - a^2*c*x^2)^(9/2)
)/(10*a) - (77*c^(9/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(256*a)

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6141

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[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]) && IGt
Q[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 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx\\ &=-\left (c \int (1+a x)^2 \left (c-a^2 c x^2\right )^{7/2} \, dx\right )\\ &=\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{1}{10} (11 c) \int (1+a x) \left (c-a^2 c x^2\right )^{7/2} \, dx\\ &=\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{1}{10} (11 c) \int \left (c-a^2 c x^2\right )^{7/2} \, dx\\ &=-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{1}{80} \left (77 c^2\right ) \int \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=-\frac{77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{1}{96} \left (77 c^3\right ) \int \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\frac{77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac{77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{1}{128} \left (77 c^4\right ) \int \sqrt{c-a^2 c x^2} \, dx\\ &=-\frac{77}{256} c^4 x \sqrt{c-a^2 c x^2}-\frac{77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac{77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{1}{256} \left (77 c^5\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{77}{256} c^4 x \sqrt{c-a^2 c x^2}-\frac{77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac{77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{1}{256} \left (77 c^5\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{77}{256} c^4 x \sqrt{c-a^2 c x^2}-\frac{77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac{77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac{11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac{11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac{(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac{77 c^{9/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{256 a}\\ \end{align*}

Mathematica [A]  time = 0.15354, size = 167, normalized size = 0.95 $\frac{c^4 \sqrt{c-a^2 c x^2} \left (\sqrt{a x+1} \left (-1152 a^{10} x^{10}-1408 a^9 x^9+5584 a^8 x^8+7216 a^7 x^7-10552 a^6 x^6-15048 a^5 x^5+9210 a^4 x^4+16390 a^3 x^3-2185 a^2 x^2-10615 a x+2560\right )+6930 \sqrt{1-a x} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{11520 a \sqrt{1-a x} \sqrt{1-a^2 x^2}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(9/2),x]

[Out]

(c^4*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(2560 - 10615*a*x - 2185*a^2*x^2 + 16390*a^3*x^3 + 9210*a^4*x^4 - 1504
8*a^5*x^5 - 10552*a^6*x^6 + 7216*a^7*x^7 + 5584*a^8*x^8 - 1408*a^9*x^9 - 1152*a^10*x^10) + 6930*Sqrt[1 - a*x]*
ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(11520*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])

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Maple [B]  time = 0.059, size = 350, normalized size = 2. \begin{align*}{\frac{x}{10} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{9}{2}}}}+{\frac{9\,cx}{80} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{21\,x{c}^{2}}{160} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{21\,{c}^{3}x}{128} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{63\,{c}^{4}x}{256}\sqrt{-{a}^{2}c{x}^{2}+c}}+{\frac{63\,{c}^{5}}{256}\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}{9\,a} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{9}{2}}}}-{\frac{cx}{4} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{7}{2}}}}-{\frac{7\,x{c}^{2}}{24} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{35\,{c}^{3}x}{96} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{35\,{c}^{4}x}{64}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}-{\frac{35\,{c}^{5}}{64}\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*x+1)/(a*x-1)*(-a^2*c*x^2+c)^(9/2),x)

[Out]

1/10*x*(-a^2*c*x^2+c)^(9/2)+9/80*c*x*(-a^2*c*x^2+c)^(7/2)+21/160*c^2*x*(-a^2*c*x^2+c)^(5/2)+21/128*c^3*x*(-a^2
*c*x^2+c)^(3/2)+63/256*c^4*x*(-a^2*c*x^2+c)^(1/2)+63/256*c^5/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+
c)^(1/2))+2/9/a*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(9/2)-1/4*c*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(7/2)*x-7/24*c^2
*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(5/2)*x-35/96*c^3*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(3/2)*x-35/64*c^4*(-c*a^2
*(x-1/a)^2-2*a*c*(x-1/a))^(1/2)*x-35/64*c^5/(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(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.91798, size = 790, normalized size = 4.49 \begin{align*} \left [\frac{3465 \, \sqrt{-c} c^{4} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) + 2 \,{\left (1152 \, a^{9} c^{4} x^{9} + 2560 \, a^{8} c^{4} x^{8} - 3024 \, a^{7} c^{4} x^{7} - 10240 \, a^{6} c^{4} x^{6} + 312 \, a^{5} c^{4} x^{5} + 15360 \, a^{4} c^{4} x^{4} + 6150 \, a^{3} c^{4} x^{3} - 10240 \, a^{2} c^{4} x^{2} - 8055 \, a c^{4} x + 2560 \, c^{4}\right )} \sqrt{-a^{2} c x^{2} + c}}{23040 \, a}, \frac{3465 \, c^{\frac{9}{2}} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) +{\left (1152 \, a^{9} c^{4} x^{9} + 2560 \, a^{8} c^{4} x^{8} - 3024 \, a^{7} c^{4} x^{7} - 10240 \, a^{6} c^{4} x^{6} + 312 \, a^{5} c^{4} x^{5} + 15360 \, a^{4} c^{4} x^{4} + 6150 \, a^{3} c^{4} x^{3} - 10240 \, a^{2} c^{4} x^{2} - 8055 \, a c^{4} x + 2560 \, c^{4}\right )} \sqrt{-a^{2} c x^{2} + c}}{11520 \, a}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(9/2),x, algorithm="fricas")

[Out]

[1/23040*(3465*sqrt(-c)*c^4*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(1152*a^9*c^4*x^9 +
2560*a^8*c^4*x^8 - 3024*a^7*c^4*x^7 - 10240*a^6*c^4*x^6 + 312*a^5*c^4*x^5 + 15360*a^4*c^4*x^4 + 6150*a^3*c^4*
x^3 - 10240*a^2*c^4*x^2 - 8055*a*c^4*x + 2560*c^4)*sqrt(-a^2*c*x^2 + c))/a, 1/11520*(3465*c^(9/2)*arctan(sqrt(
-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + (1152*a^9*c^4*x^9 + 2560*a^8*c^4*x^8 - 3024*a^7*c^4*x^7 - 10240
*a^6*c^4*x^6 + 312*a^5*c^4*x^5 + 15360*a^4*c^4*x^4 + 6150*a^3*c^4*x^3 - 10240*a^2*c^4*x^2 - 8055*a*c^4*x + 256
0*c^4)*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(1/(a*x-1)*(a*x+1)*(-a**2*c*x**2+c)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.17853, size = 221, normalized size = 1.26 \begin{align*} \frac{77 \, c^{5} \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{256 \, \sqrt{-c}{\left | a \right |}} + \frac{1}{11520} \, \sqrt{-a^{2} c x^{2} + c}{\left (\frac{2560 \, c^{4}}{a} -{\left (8055 \, c^{4} + 2 \,{\left (5120 \, a c^{4} -{\left (3075 \, a^{2} c^{4} + 4 \,{\left (1920 \, a^{3} c^{4} +{\left (39 \, a^{4} c^{4} - 2 \,{\left (640 \, a^{5} c^{4} +{\left (189 \, a^{6} c^{4} - 8 \,{\left (9 \, a^{8} c^{4} x + 20 \, a^{7} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(9/2),x, algorithm="giac")

[Out]

77/256*c^5*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/(sqrt(-c)*abs(a)) + 1/11520*sqrt(-a^2*c*x^2 + c)*(
2560*c^4/a - (8055*c^4 + 2*(5120*a*c^4 - (3075*a^2*c^4 + 4*(1920*a^3*c^4 + (39*a^4*c^4 - 2*(640*a^5*c^4 + (189
*a^6*c^4 - 8*(9*a^8*c^4*x + 20*a^7*c^4)*x)*x)*x)*x)*x)*x)*x)*x)