3.767 \(\int e^{2 \coth ^{-1}(a x)} (c-a^2 c x^2)^p \, dx\)
Optimal. Leaf size=54 \[ \frac {2^{p+1} (a x+1)^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-p-1,p;p+1;\frac {1}{2} (1-a x)\right )}{a p} \]
[Out]
2^(1+p)*(-a^2*c*x^2+c)^p*hypergeom([p, -1-p],[1+p],-1/2*a*x+1/2)/a/p/((a*x+1)^p)
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Rubi [A] time = 0.09, antiderivative size = 54, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used =
{6167, 6141, 678, 69} \[ \frac {2^{p+1} (a x+1)^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-p-1,p;p+1;\frac {1}{2} (1-a x)\right )}{a p} \]
Antiderivative was successfully verified.
[In]
Int[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^p,x]
[Out]
(2^(1 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1 - p, p, 1 + p, (1 - a*x)/2])/(a*p*(1 + a*x)^p)
Rule 69
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))
Rule 678
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(d^(m - 1)*(a + c*x^2)^(p + 1))/((1
+ (e*x)/d)^(p + 1)*(a/d + (c*x)/e)^(p + 1)), Int[(1 + (e*x)/d)^(m + p)*(a/d + (c*x)/e)^p, x], x] /; FreeQ[{a,
c, d, e, m}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && (IntegerQ[m] || GtQ[d, 0]) && !(IGtQ[m, 0] && (
IntegerQ[3*p] || IntegerQ[4*p]))
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 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]
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx\\ &=-\left (c \int (1+a x)^2 \left (c-a^2 c x^2\right )^{-1+p} \, dx\right )\\ &=-\left (\left (c (1+a x)^{-p} (c-a c x)^{-p} \left (c-a^2 c x^2\right )^p\right ) \int (1+a x)^{1+p} (c-a c x)^{-1+p} \, dx\right )\\ &=\frac {2^{1+p} (1+a x)^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-1-p,p;1+p;\frac {1}{2} (1-a x)\right )}{a p}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 67, normalized size = 1.24 \[ \frac {2^{p+1} (1-a x)^p \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-p-1,p;p+1;\frac {1}{2} (1-a x)\right )}{a p} \]
Antiderivative was successfully verified.
[In]
Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^p,x]
[Out]
(2^(1 + p)*(1 - a*x)^p*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1 - p, p, 1 + p, (1 - a*x)/2])/(a*p*(1 - a^2*x^2)^
p)
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{a x - 1}, x\right ) \]
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)^p,x, algorithm="fricas")
[Out]
integral((a*x + 1)*(-a^2*c*x^2 + c)^p/(a*x - 1), x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{a x - 1}\,{d x} \]
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)^p,x, algorithm="giac")
[Out]
integrate((a*x + 1)*(-a^2*c*x^2 + c)^p/(a*x - 1), x)
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maple [F] time = 0.53, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right ) \left (-a^{2} c \,x^{2}+c \right )^{p}}{a x -1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a*x+1)/(a*x-1)*(-a^2*c*x^2+c)^p,x)
[Out]
int((a*x+1)/(a*x-1)*(-a^2*c*x^2+c)^p,x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{a x - 1}\,{d x} \]
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)^p,x, algorithm="maxima")
[Out]
integrate((a*x + 1)*(-a^2*c*x^2 + c)^p/(a*x - 1), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,\left (a\,x+1\right )}{a\,x-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((c - a^2*c*x^2)^p*(a*x + 1))/(a*x - 1),x)
[Out]
int(((c - a^2*c*x^2)^p*(a*x + 1))/(a*x - 1), x)
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sympy [C] time = 19.97, size = 651, normalized size = 12.06 \[ \text {result too large to display} \]
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)**p,x)
[Out]
a*Piecewise((0**p*x/a - 0**p*log(1/(a**2*x**2))/(2*a**2) + 0**p*log(-1 + 1/(a**2*x**2))/(2*a**2) - 0**p*acoth(
1/(a*x))/a**2 + c**p*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*g
amma(-p)*gamma(p + 1)) - a**(2*p)*c**p*p*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p, -p - 1/
2), (1/2 - p,), 1/(a**2*x**2))/(2*a*gamma(1/2 - p)*gamma(p + 1)), 1/Abs(a**2*x**2) > 1), (0**p*x/a - 0**p*log(
1/(a**2*x**2))/(2*a**2) + 0**p*log(1 - 1/(a**2*x**2))/(2*a**2) - 0**p*atanh(1/(a*x))/a**2 + c**p*x**2*gamma(p)
*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(-p)*gamma(p + 1)) - a**(2*p)*
c**p*p*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p, -p - 1/2), (1/2 - p,), 1/(a**2*x**2))/(2*
a*gamma(1/2 - p)*gamma(p + 1)), True)) + Piecewise((0**p*log(a**2*x**2 - 1)/(2*a) - 0**p*acoth(a*x)/a + a*c**p
*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(-p)*gamma(p + 1
)) - a**(2*p)*c**p*p*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(1/2 - p)*hyper((1 - p, 1/2 - p), (3/2 - p,), 1/(a**2*
x**2))/(2*a**2*x*gamma(3/2 - p)*gamma(p + 1)), Abs(a**2*x**2) > 1), (0**p*log(-a**2*x**2 + 1)/(2*a) - 0**p*ata
nh(a*x)/a + a*c**p*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*gam
ma(-p)*gamma(p + 1)) - a**(2*p)*c**p*p*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(1/2 - p)*hyper((1 - p, 1/2 - p), (3
/2 - p,), 1/(a**2*x**2))/(2*a**2*x*gamma(3/2 - p)*gamma(p + 1)), True))
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