3.8.13 \(\int \frac {e^{3 \tanh ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^{7/2}} \, dx\) [713]

Optimal. Leaf size=363 \[ -\frac {\left (1-a^2 x^2\right )^{7/2}}{a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}+\frac {\left (1-a^2 x^2\right )^{7/2}}{16 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)^4}-\frac {\left (1-a^2 x^2\right )^{7/2}}{2 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)^3}+\frac {59 \left (1-a^2 x^2\right )^{7/2}}{32 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)^2}-\frac {75 \left (1-a^2 x^2\right )^{7/2}}{16 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{7/2}}{32 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1+a x)}-\frac {201 \left (1-a^2 x^2\right )^{7/2} \log (1-a x)}{64 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}+\frac {9 \left (1-a^2 x^2\right )^{7/2} \log (1+a x)}{64 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7} \]

[Out]

-(-a^2*x^2+1)^(7/2)/a^7/(c-c/a^2/x^2)^(7/2)/x^6+1/16*(-a^2*x^2+1)^(7/2)/a^8/(c-c/a^2/x^2)^(7/2)/x^7/(-a*x+1)^4
-1/2*(-a^2*x^2+1)^(7/2)/a^8/(c-c/a^2/x^2)^(7/2)/x^7/(-a*x+1)^3+59/32*(-a^2*x^2+1)^(7/2)/a^8/(c-c/a^2/x^2)^(7/2
)/x^7/(-a*x+1)^2-75/16*(-a^2*x^2+1)^(7/2)/a^8/(c-c/a^2/x^2)^(7/2)/x^7/(-a*x+1)+1/32*(-a^2*x^2+1)^(7/2)/a^8/(c-
c/a^2/x^2)^(7/2)/x^7/(a*x+1)-201/64*(-a^2*x^2+1)^(7/2)*ln(-a*x+1)/a^8/(c-c/a^2/x^2)^(7/2)/x^7+9/64*(-a^2*x^2+1
)^(7/2)*ln(a*x+1)/a^8/(c-c/a^2/x^2)^(7/2)/x^7

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Rubi [A]
time = 0.17, antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6295, 6285, 90} \begin {gather*} -\frac {75 \left (1-a^2 x^2\right )^{7/2}}{16 a^8 x^7 (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {\left (1-a^2 x^2\right )^{7/2}}{32 a^8 x^7 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {59 \left (1-a^2 x^2\right )^{7/2}}{32 a^8 x^7 (1-a x)^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {\left (1-a^2 x^2\right )^{7/2}}{2 a^8 x^7 (1-a x)^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {\left (1-a^2 x^2\right )^{7/2}}{16 a^8 x^7 (1-a x)^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {201 \left (1-a^2 x^2\right )^{7/2} \log (1-a x)}{64 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {9 \left (1-a^2 x^2\right )^{7/2} \log (a x+1)}{64 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {\left (1-a^2 x^2\right )^{7/2}}{a^7 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^(7/2),x]

[Out]

-((1 - a^2*x^2)^(7/2)/(a^7*(c - c/(a^2*x^2))^(7/2)*x^6)) + (1 - a^2*x^2)^(7/2)/(16*a^8*(c - c/(a^2*x^2))^(7/2)
*x^7*(1 - a*x)^4) - (1 - a^2*x^2)^(7/2)/(2*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 - a*x)^3) + (59*(1 - a^2*x^2)^(7
/2))/(32*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 - a*x)^2) - (75*(1 - a^2*x^2)^(7/2))/(16*a^8*(c - c/(a^2*x^2))^(7/
2)*x^7*(1 - a*x)) + (1 - a^2*x^2)^(7/2)/(32*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 + a*x)) - (201*(1 - a^2*x^2)^(7
/2)*Log[1 - a*x])/(64*a^8*(c - c/(a^2*x^2))^(7/2)*x^7) + (9*(1 - a^2*x^2)^(7/2)*Log[1 + a*x])/(64*a^8*(c - c/(
a^2*x^2))^(7/2)*x^7)

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 6285

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p
] || GtQ[c, 0])

Rule 6295

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbol] :> Dist[x^(2*p)*((c + d/x^2)^p/(
1 + c*(x^2/d))^p), Int[(u/x^(2*p))*(1 + c*(x^2/d))^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] &
& EqQ[c + a^2*d, 0] &&  !IntegerQ[p] &&  !IntegerQ[n/2]

Rubi steps

\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx &=\frac {\left (1-a^2 x^2\right )^{7/2} \int \frac {e^{3 \tanh ^{-1}(a x)} x^7}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {\left (1-a^2 x^2\right )^{7/2} \int \frac {x^7}{(1-a x)^5 (1+a x)^2} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {\left (1-a^2 x^2\right )^{7/2} \int \left (-\frac {1}{a^7}-\frac {1}{4 a^7 (-1+a x)^5}-\frac {3}{2 a^7 (-1+a x)^4}-\frac {59}{16 a^7 (-1+a x)^3}-\frac {75}{16 a^7 (-1+a x)^2}-\frac {201}{64 a^7 (-1+a x)}-\frac {1}{32 a^7 (1+a x)^2}+\frac {9}{64 a^7 (1+a x)}\right ) \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac {\left (1-a^2 x^2\right )^{7/2}}{a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}+\frac {\left (1-a^2 x^2\right )^{7/2}}{16 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)^4}-\frac {\left (1-a^2 x^2\right )^{7/2}}{2 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)^3}+\frac {59 \left (1-a^2 x^2\right )^{7/2}}{32 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)^2}-\frac {75 \left (1-a^2 x^2\right )^{7/2}}{16 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{7/2}}{32 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1+a x)}-\frac {201 \left (1-a^2 x^2\right )^{7/2} \log (1-a x)}{64 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}+\frac {9 \left (1-a^2 x^2\right )^{7/2} \log (1+a x)}{64 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 146, normalized size = 0.40 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (2 \left (104-207 a x-59 a^2 x^2+309 a^3 x^3-87 a^4 x^4-96 a^5 x^5+32 a^6 x^6\right )+201 (-1+a x)^4 (1+a x) \log (1-a x)-9 (-1+a x)^4 (1+a x) \log (1+a x)\right )}{64 a^2 c^3 \sqrt {c-\frac {c}{a^2 x^2}} x (-1+a x)^4 (1+a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^(7/2),x]

[Out]

(Sqrt[1 - a^2*x^2]*(2*(104 - 207*a*x - 59*a^2*x^2 + 309*a^3*x^3 - 87*a^4*x^4 - 96*a^5*x^5 + 32*a^6*x^6) + 201*
(-1 + a*x)^4*(1 + a*x)*Log[1 - a*x] - 9*(-1 + a*x)^4*(1 + a*x)*Log[1 + a*x]))/(64*a^2*c^3*Sqrt[c - c/(a^2*x^2)
]*x*(-1 + a*x)^4*(1 + a*x))

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Maple [A]
time = 0.80, size = 248, normalized size = 0.68

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a x +1\right )^{2} \left (-64 a^{6} x^{6}+9 \ln \left (a x +1\right ) a^{5} x^{5}-201 \ln \left (a x -1\right ) a^{5} x^{5}+192 a^{5} x^{5}-27 \ln \left (a x +1\right ) a^{4} x^{4}+603 \ln \left (a x -1\right ) a^{4} x^{4}+174 a^{4} x^{4}+18 \ln \left (a x +1\right ) a^{3} x^{3}-402 \ln \left (a x -1\right ) a^{3} x^{3}-618 a^{3} x^{3}+18 a^{2} x^{2} \ln \left (a x +1\right )-402 \ln \left (a x -1\right ) a^{2} x^{2}+118 a^{2} x^{2}-27 a x \ln \left (a x +1\right )+603 \ln \left (a x -1\right ) a x +414 a x +9 \ln \left (a x +1\right )-201 \ln \left (a x -1\right )-208\right )}{64 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}} x^{7} a^{8} \left (a x -1\right )}\)	\(248\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/64*(-a^2*x^2+1)^(1/2)*(a*x+1)^2*(-64*a^6*x^6+9*ln(a*x+1)*a^5*x^5-201*ln(a*x-1)*a^5*x^5+192*a^5*x^5-27*ln(a*
x+1)*a^4*x^4+603*ln(a*x-1)*a^4*x^4+174*a^4*x^4+18*ln(a*x+1)*a^3*x^3-402*ln(a*x-1)*a^3*x^3-618*a^3*x^3+18*a^2*x
^2*ln(a*x+1)-402*ln(a*x-1)*a^2*x^2+118*a^2*x^2-27*a*x*ln(a*x+1)+603*ln(a*x-1)*a*x+414*a*x+9*ln(a*x+1)-201*ln(a
*x-1)-208)/(c*(a^2*x^2-1)/a^2/x^2)^(7/2)/x^7/a^8/(a*x-1)

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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*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm="maxima")

[Out]

integrate((a*x + 1)^3/((-a^2*x^2 + 1)^(3/2)*(c - c/(a^2*x^2))^(7/2)), 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*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*x^2 + 1)*a^8*x^8*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^9*c^4*x^9 - 3*a^8*c^4*x^8 + 8*a^6*c^4*x
^6 - 6*a^5*c^4*x^5 - 6*a^4*c^4*x^4 + 8*a^3*c^4*x^3 - 3*a*c^4*x + c^4), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**(7/2),x)

[Out]

Integral((a*x + 1)**3/((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(7/2)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,x+1\right )}^3}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + 1)^3/((c - c/(a^2*x^2))^(7/2)*(1 - a^2*x^2)^(3/2)),x)

[Out]

int((a*x + 1)^3/((c - c/(a^2*x^2))^(7/2)*(1 - a^2*x^2)^(3/2)), x)

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