3.8.70 \(\int e^{-2 \coth ^{-1}(a x)} (c-a^2 c x^2)^p \, dx\) [770]

Optimal. Leaf size=55 \[ -\frac {2^{1+p} (1-a x)^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-1-p,p;1+p;\frac {1}{2} (1+a x)\right )}{a p} \]

[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 = 0.06, antiderivative size = 55, 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 = {6302, 6277, 692, 71} \begin {gather*} -\frac {2^{p+1} (1-a x)^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-p-1,p;p+1;\frac {1}{2} (a x+1)\right )}{a p} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^p/E^(2*ArcCoth[a*x]),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 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], 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 692

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/e)*x)^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 6277

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/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]) && I
LtQ[n/2, 0]

Rule 6302

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 = 73, normalized size = 1.33 \begin {gather*} \frac {2^{-1+p} (1-a x)^{2+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (1-p,2+p;3+p;\frac {1}{2} (1-a x)\right )}{a (2+p)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} \left (a x -1\right )}{a x +1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 11.93, size = 651, normalized size = 11.84 \begin {gather*} a \left (\begin {cases} \frac {0^{p} x}{a} + \frac {0^{p} \log {\left (\frac {1}{a^{2} x^{2}} \right )}}{2 a^{2}} - \frac {0^{p} \log {\left (-1 + \frac {1}{a^{2} x^{2}} \right )}}{2 a^{2}} - \frac {0^{p} \operatorname {acoth}{\left (\frac {1}{a x} \right )}}{a^{2}} - \frac {c^{p} x^{2} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} p x x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 a \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {0^{p} x}{a} + \frac {0^{p} \log {\left (\frac {1}{a^{2} x^{2}} \right )}}{2 a^{2}} - \frac {0^{p} \log {\left (1 - \frac {1}{a^{2} x^{2}} \right )}}{2 a^{2}} - \frac {0^{p} \operatorname {atanh}{\left (\frac {1}{a x} \right )}}{a^{2}} - \frac {c^{p} x^{2} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} p x x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 a \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases}\right ) - \begin {cases} \frac {0^{p} \log {\left (a^{2} x^{2} - 1 \right )}}{2 a} + \frac {0^{p} \operatorname {acoth}{\left (a x \right )}}{a} + \frac {a c^{p} x^{2} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} + \frac {a^{2 p} c^{p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 a^{2} x \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {0^{p} \log {\left (- a^{2} x^{2} + 1 \right )}}{2 a} + \frac {0^{p} \operatorname {atanh}{\left (a x \right )}}{a} + \frac {a c^{p} x^{2} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} + \frac {a^{2 p} c^{p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 a^{2} x \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**p*(a*x-1)/(a*x+1),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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} \int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,\left (a\,x-1\right )}{a\,x+1} \,d x \end {gather*}

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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