Integrand size = 25, antiderivative size = 111 \[ \int \frac {e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=-a x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-p,\frac {3}{2},a^2 x^2\right )-\frac {\sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+p,\frac {3}{2}+p,1-a^2 x^2\right )}{1+2 p} \]
-a*x*(-a^2*c*x^2+c)^p*hypergeom([1/2, 1/2-p],[3/2],a^2*x^2)/((-a^2*x^2+1)^ p)-(-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 = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (-a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-p,\frac {3}{2},a^2 x^2\right )-\frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+p,\frac {3}{2}+p,1-a^2 x^2\right )}{2 \left (\frac {1}{2}+p\right )}\right ) \]
((c - a^2*c*x^2)^p*(-(a*x*Hypergeometric2F1[1/2, 1/2 - p, 3/2, a^2*x^2]) - ((1 - a^2*x^2)^(1/2 + p)*Hypergeometric2F1[1, 1/2 + p, 3/2 + p, 1 - a^2*x ^2])/(2*(1/2 + p))))/(1 - a^2*x^2)^p
Time = 0.41 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6703, 6699, 542, 237, 243, 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.
\(\displaystyle \int \frac {e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int \frac {e^{-\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p}{x}dx\) |
\(\Big \downarrow \) 6699 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int \frac {(1-a x) \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x}dx\) |
\(\Big \downarrow \) 542 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\int \frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x}dx-a \int \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx\right )\) |
\(\Big \downarrow \) 237 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\int \frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x}dx-a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-p,\frac {3}{2},a^2 x^2\right )\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {1}{2} \int \frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x^2}dx^2-a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-p,\frac {3}{2},a^2 x^2\right )\right )\) |
\(\Big \downarrow \) 75 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (-\frac {\left (1-a^2 x^2\right )^{p+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (1,p+\frac {1}{2},p+\frac {3}{2},1-a^2 x^2\right )}{2 p+1}-a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-p,\frac {3}{2},a^2 x^2\right )\right )\) |
((c - a^2*c*x^2)^p*(-(a*x*Hypergeometric2F1[1/2, 1/2 - p, 3/2, a^2*x^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)))/(1 - a^2*x^2)^p
3.13.26.3.1 Defintions of rubi rules used
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_)^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_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c Int[x^m*(a + b*x^2)^p, x], x] + Simp[d Int[x^(m + 1)*(a + b*x^2 )^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && IntegerQ[m] && !IntegerQ[2*p]
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]) && ILtQ[(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]
\[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} \sqrt {-a^{2} x^{2}+1}}{\left (a x +1\right ) x}d x\]
\[ \int \frac {e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a x + 1\right )} x} \,d x } \]
\[ \int \frac {e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p}}{x \left (a x + 1\right )}\, dx \]
\[ \int \frac {e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a x + 1\right )} x} \,d x } \]
\[ \int \frac {e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a x + 1\right )} x} \,d x } \]
Timed out. \[ \int \frac {e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,\sqrt {1-a^2\,x^2}}{x\,\left (a\,x+1\right )} \,d x \]