3.12.79 \(\int \frac {e^{3 \text {arctanh}(a x)} (c-a^2 c x^2)^p}{x} \, dx\) [1179]

3.12.79.1 Optimal result

Integrand size = 25, antiderivative size = 193 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\frac {4 \left (c-a^2 c x^2\right )^p}{(1-2 p) \sqrt {1-a^2 x^2}}-\frac {a x \left (c-a^2 c x^2\right )^p}{2 p \sqrt {1-a^2 x^2}}+\frac {a (1+6 p) x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}-p,\frac {3}{2},a^2 x^2\right )}{2 p}-\frac {\sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+p,\frac {3}{2}+p,1-a^2 x^2\right )}{1+2 p} \]

output

1/2*a*(1+6*p)*x*(-a^2*c*x^2+c)^p*hypergeom([1/2, 3/2-p],[3/2],a^2*x^2)/p/( 
(-a^2*x^2+1)^p)+4*(-a^2*c*x^2+c)^p/(1-2*p)/(-a^2*x^2+1)^(1/2)-1/2*a*x*(-a^ 
2*c*x^2+c)^p/p/(-a^2*x^2+1)^(1/2)-(-a^2*c*x^2+c)^p*hypergeom([1, 1/2+p],[3 
/2+p],-a^2*x^2+1)*(-a^2*x^2+1)^(1/2)/(1+2*p)
 

3.12.79.2 Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.82 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {3 \left (1-a^2 x^2\right )^{-\frac {1}{2}+p}}{1-2 p}+3 a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}-p,\frac {3}{2},a^2 x^2\right )-\frac {\left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2}+p,\frac {1}{2}+p,1-a^2 x^2\right )}{2 \left (-\frac {1}{2}+p\right )}+\frac {1}{3} a^3 x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {3}{2}-p,\frac {5}{2},a^2 x^2\right )\right ) \]

input

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

output

((c - a^2*c*x^2)^p*((3*(1 - a^2*x^2)^(-1/2 + p))/(1 - 2*p) + 3*a*x*Hyperge 
ometric2F1[1/2, 3/2 - p, 3/2, a^2*x^2] - ((1 - a^2*x^2)^(-1/2 + p)*Hyperge 
ometric2F1[1, -1/2 + p, 1/2 + p, 1 - a^2*x^2])/(2*(-1/2 + p)) + (a^3*x^3*H 
ypergeometric2F1[3/2, 3/2 - p, 5/2, a^2*x^2])/3))/(1 - a^2*x^2)^p
 

3.12.79.3 Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.86, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {6703, 6698, 543, 299, 237, 354, 88, 75}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx\)

\(\Big \downarrow \) 6703

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int \frac {e^{3 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^p}{x}dx\)

\(\Big \downarrow \) 6698

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int \frac {(a x+1)^3 \left (1-a^2 x^2\right )^{p-\frac {3}{2}}}{x}dx\)

\(\Big \downarrow \) 543

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\int \frac {\left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 a^2 x^2+1\right )}{x}dx+\int \left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (x^2 a^3+3 a\right )dx\right )\)

\(\Big \downarrow \) 299

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {a (6 p+1) \int \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx}{2 p}+\int \frac {\left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 a^2 x^2+1\right )}{x}dx-\frac {a x \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{2 p}\right )\)

\(\Big \downarrow \) 237

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\int \frac {\left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 a^2 x^2+1\right )}{x}dx+\frac {a (6 p+1) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}-p,\frac {3}{2},a^2 x^2\right )}{2 p}-\frac {a x \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{2 p}\right )\)

\(\Big \downarrow \) 354

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {1}{2} \int \frac {\left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 a^2 x^2+1\right )}{x^2}dx^2+\frac {a (6 p+1) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}-p,\frac {3}{2},a^2 x^2\right )}{2 p}-\frac {a x \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{2 p}\right )\)

\(\Big \downarrow \) 88

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {1}{2} \left (\int \frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x^2}dx^2+\frac {8 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{1-2 p}\right )+\frac {a (6 p+1) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}-p,\frac {3}{2},a^2 x^2\right )}{2 p}-\frac {a x \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{2 p}\right )\)

\(\Big \downarrow \) 75

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {a (6 p+1) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}-p,\frac {3}{2},a^2 x^2\right )}{2 p}+\frac {1}{2} \left (\frac {8 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{1-2 p}-\frac {2 \left (1-a^2 x^2\right )^{p+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (1,p+\frac {1}{2},p+\frac {3}{2},1-a^2 x^2\right )}{2 p+1}\right )-\frac {a x \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{2 p}\right )\)

input

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

output

((c - a^2*c*x^2)^p*(-1/2*(a*x*(1 - a^2*x^2)^(-1/2 + p))/p + (a*(1 + 6*p)*x 
*Hypergeometric2F1[1/2, 3/2 - p, 3/2, a^2*x^2])/(2*p) + ((8*(1 - a^2*x^2)^ 
(-1/2 + p))/(1 - 2*p) - (2*(1 - a^2*x^2)^(1/2 + p)*Hypergeometric2F1[1, 1/ 
2 + p, 3/2 + p, 1 - a^2*x^2])/(1 + 2*p))/2))/(1 - a^2*x^2)^p
 

3.12.79.3.1 Defintions of rubi rules used

rule 75

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x 
)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + 
 d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

rule 88

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], 
 x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimpl 
erQ[p, 1]
 

rule 237

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[- 
p, 1/2, 1/2 + 1, (-b)*(x^2/a)], x] /; FreeQ[{a, b, p}, x] &&  !IntegerQ[2*p 
] && GtQ[a, 0]
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 543

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Module[{k}, Int[x^m*Sum[Binomial[n, 2*k]*c^(n - 2*k)*d^(2*k)*x^(2*k), 
 {k, 0, n/2}]*(a + b*x^2)^p, x] + Int[x^(m + 1)*Sum[Binomial[n, 2*k + 1]*c^ 
(n - 2*k - 1)*d^(2*k + 1)*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2)^p, x]] /; 
 FreeQ[{a, b, c, d, p}, x] && IGtQ[n, 1] && IntegerQ[m] &&  !IntegerQ[2*p] 
&&  !(EqQ[m, 1] && EqQ[b*c^2 + a*d^2, 0])
 

rule 6698

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[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] || GtQ[c, 
 0]) && IGtQ[(n + 1)/2, 0] &&  !IntegerQ[p - n/2]
 

rule 6703

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar 
t[p])   Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, 
d, m, n, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0]) &&  !I 
ntegerQ[n/2]
 

3.12.79.4 Maple [F]

\[\int \frac {\left (a x +1\right )^{3} \left (-a^{2} c \,x^{2}+c \right )^{p}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} x}d x\]

input

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

output

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

3.12.79.5 Fricas [F]

\[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\int { \frac {{\left (a x + 1\right )}^{3} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x} \,d x } \]

input

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

output

integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*(-a^2*c*x^2 + c)^p/(a^2*x^3 - 2*a*x^ 
2 + x), x)
 

3.12.79.6 Sympy [F]

\[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} \left (a x + 1\right )^{3}}{x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

input

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

output

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

3.12.79.7 Maxima [F]

\[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\int { \frac {{\left (a x + 1\right )}^{3} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x} \,d x } \]

input

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

output

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

3.12.79.8 Giac [F]

\[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\int { \frac {{\left (a x + 1\right )}^{3} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x} \,d x } \]

input

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

output

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

3.12.79.9 Mupad [F(-1)]

Timed out. \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,{\left (a\,x+1\right )}^3}{x\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]

input

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

output

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