Integrand size = 25, antiderivative size = 187 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx=\frac {4 a \left (c-a^2 c x^2\right )^p}{(1-2 p) \sqrt {1-a^2 x^2}}-\frac {\left (c-a^2 c x^2\right )^p}{x \sqrt {1-a^2 x^2}}+a^2 (5-2 p) x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}-p,\frac {3}{2},a^2 x^2\right )-\frac {3 a \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+p,\frac {3}{2}+p,1-a^2 x^2\right )}{1+2 p} \]
a^2*(5-2*p)*x*(-a^2*c*x^2+c)^p*hypergeom([1/2, 3/2-p],[3/2],a^2*x^2)/((-a^ 2*x^2+1)^p)+4*a*(-a^2*c*x^2+c)^p/(1-2*p)/(-a^2*x^2+1)^(1/2)-(-a^2*c*x^2+c) ^p/x/(-a^2*x^2+1)^(1/2)-3*a*(-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)
Time = 0.17 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.71 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx=\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{2}-p,\frac {1}{2},a^2 x^2\right )}{x}+a \left (3 a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}-p,\frac {3}{2},a^2 x^2\right )+\frac {\left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \left (1+3 \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2}+p,\frac {1}{2}+p,1-a^2 x^2\right )\right )}{1-2 p}\right )\right ) \]
((c - a^2*c*x^2)^p*(-(Hypergeometric2F1[-1/2, 3/2 - p, 1/2, a^2*x^2]/x) + a*(3*a*x*Hypergeometric2F1[1/2, 3/2 - p, 3/2, a^2*x^2] + ((1 - a^2*x^2)^(- 1/2 + p)*(1 + 3*Hypergeometric2F1[1, -1/2 + p, 1/2 + p, 1 - a^2*x^2]))/(1 - 2*p))))/(1 - a^2*x^2)^p
Time = 0.51 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.85, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6703, 6698, 543, 354, 27, 88, 75, 359, 237}
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.
\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int \frac {e^{3 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^p}{x^2}dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int \frac {(a x+1)^3 \left (1-a^2 x^2\right )^{p-\frac {3}{2}}}{x^2}dx\) |
\(\Big \downarrow \) 543 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\int \frac {\left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 a^2 x^2+1\right )}{x^2}dx+\int \frac {\left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (x^2 a^3+3 a\right )}{x}dx\right )\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {1}{2} \int \frac {a \left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (a^2 x^2+3\right )}{x^2}dx^2+\int \frac {\left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 a^2 x^2+1\right )}{x^2}dx\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {1}{2} a \int \frac {\left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (a^2 x^2+3\right )}{x^2}dx^2+\int \frac {\left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 a^2 x^2+1\right )}{x^2}dx\right )\) |
\(\Big \downarrow \) 88 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {1}{2} a \left (3 \int \frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x^2}dx^2+\frac {8 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{1-2 p}\right )+\int \frac {\left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 a^2 x^2+1\right )}{x^2}dx\right )\) |
\(\Big \downarrow \) 75 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\int \frac {\left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 a^2 x^2+1\right )}{x^2}dx+\frac {1}{2} a \left (\frac {8 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{1-2 p}-\frac {6 \left (1-a^2 x^2\right )^{p+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (1,p+\frac {1}{2},p+\frac {3}{2},1-a^2 x^2\right )}{2 p+1}\right )\right )\) |
\(\Big \downarrow \) 359 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (a^2 (5-2 p) \int \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx+\frac {1}{2} a \left (\frac {8 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{1-2 p}-\frac {6 \left (1-a^2 x^2\right )^{p+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (1,p+\frac {1}{2},p+\frac {3}{2},1-a^2 x^2\right )}{2 p+1}\right )-\frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x}\right )\) |
\(\Big \downarrow \) 237 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (a^2 (5-2 p) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}-p,\frac {3}{2},a^2 x^2\right )+\frac {1}{2} a \left (\frac {8 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{1-2 p}-\frac {6 \left (1-a^2 x^2\right )^{p+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (1,p+\frac {1}{2},p+\frac {3}{2},1-a^2 x^2\right )}{2 p+1}\right )-\frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x}\right )\) |
((c - a^2*c*x^2)^p*(-((1 - a^2*x^2)^(-1/2 + p)/x) + a^2*(5 - 2*p)*x*Hyperg eometric2F1[1/2, 3/2 - p, 3/2, a^2*x^2] + (a*((8*(1 - a^2*x^2)^(-1/2 + p)) /(1 - 2*p) - (6*(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
3.12.80.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
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])
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]
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]
\[\int \frac {\left (a x +1\right )^{3} \left (-a^{2} c \,x^{2}+c \right )^{p}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} x^{2}}d x\]
\[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (a x + 1\right )}^{3} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
\[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx=\int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} \left (a x + 1\right )^{3}}{x^{2} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (a x + 1\right )}^{3} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
\[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (a x + 1\right )}^{3} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,{\left (a\,x+1\right )}^3}{x^2\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]