3.6.64 \(\int e^{2 \coth ^{-1}(a x)} (c-a^2 c x^2)^5 \, dx\) [564]

Optimal. Leaf size=84 \[ -\frac {16 c^5 (1+a x)^7}{7 a}+\frac {4 c^5 (1+a x)^8}{a}-\frac {8 c^5 (1+a x)^9}{3 a}+\frac {4 c^5 (1+a x)^{10}}{5 a}-\frac {c^5 (1+a x)^{11}}{11 a} \]

[Out]

-16/7*c^5*(a*x+1)^7/a+4*c^5*(a*x+1)^8/a-8/3*c^5*(a*x+1)^9/a+4/5*c^5*(a*x+1)^10/a-1/11*c^5*(a*x+1)^11/a

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Rubi [A]
time = 0.07, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6302, 6275, 45} \begin {gather*} -\frac {c^5 (a x+1)^{11}}{11 a}+\frac {4 c^5 (a x+1)^{10}}{5 a}-\frac {8 c^5 (a x+1)^9}{3 a}+\frac {4 c^5 (a x+1)^8}{a}-\frac {16 c^5 (a x+1)^7}{7 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^5,x]

[Out]

(-16*c^5*(1 + a*x)^7)/(7*a) + (4*c^5*(1 + a*x)^8)/a - (8*c^5*(1 + a*x)^9)/(3*a) + (4*c^5*(1 + a*x)^10)/(5*a) -
(c^5*(1 + a*x)^11)/(11*a)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6275

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 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 )^5 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx\\ &=-\left (c^5 \int (1-a x)^4 (1+a x)^6 \, dx\right )\\ &=-\left (c^5 \int \left (16 (1+a x)^6-32 (1+a x)^7+24 (1+a x)^8-8 (1+a x)^9+(1+a x)^{10}\right ) \, dx\right )\\ &=-\frac {16 c^5 (1+a x)^7}{7 a}+\frac {4 c^5 (1+a x)^8}{a}-\frac {8 c^5 (1+a x)^9}{3 a}+\frac {4 c^5 (1+a x)^{10}}{5 a}-\frac {c^5 (1+a x)^{11}}{11 a}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 47, normalized size = 0.56 \begin {gather*} -\frac {c^5 (1+a x)^7 \left (281-812 a x+938 a^2 x^2-504 a^3 x^3+105 a^4 x^4\right )}{1155 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^5,x]

[Out]

-1/1155*(c^5*(1 + a*x)^7*(281 - 812*a*x + 938*a^2*x^2 - 504*a^3*x^3 + 105*a^4*x^4))/a

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Maple [A]
time = 0.22, size = 85, normalized size = 1.01

method	result	size
gosper	\(-\frac {c^{5} x \left (105 a^{10} x^{10}+231 a^{9} x^{9}-385 a^{8} x^{8}-1155 a^{7} x^{7}+330 a^{6} x^{6}+2310 a^{5} x^{5}+462 a^{4} x^{4}-2310 a^{3} x^{3}-1155 a^{2} x^{2}+1155 a x +1155\right )}{1155}\)	\(85\)
default	\(c^{5} \left (-\frac {1}{11} a^{10} x^{11}-\frac {1}{5} a^{9} x^{10}+\frac {1}{3} a^{8} x^{9}+a^{7} x^{8}-\frac {2}{7} a^{6} x^{7}-2 a^{5} x^{6}-\frac {2}{5} a^{4} x^{5}+2 a^{3} x^{4}+a^{2} x^{3}-a \,x^{2}-x \right )\)	\(85\)
norman	\(a^{7} c^{5} x^{8}+c^{5} a^{2} x^{3}-c^{5} x -a \,c^{5} x^{2}-\frac {2}{5} a^{4} c^{5} x^{5}-2 a^{5} c^{5} x^{6}-\frac {2}{7} a^{6} c^{5} x^{7}+\frac {1}{3} a^{8} c^{5} x^{9}-\frac {1}{5} a^{9} c^{5} x^{10}-\frac {1}{11} a^{10} c^{5} x^{11}+2 c^{5} a^{3} x^{4}\)	\(114\)
risch	\(a^{7} c^{5} x^{8}+c^{5} a^{2} x^{3}-c^{5} x -a \,c^{5} x^{2}-\frac {2}{5} a^{4} c^{5} x^{5}-2 a^{5} c^{5} x^{6}-\frac {2}{7} a^{6} c^{5} x^{7}+\frac {1}{3} a^{8} c^{5} x^{9}-\frac {1}{5} a^{9} c^{5} x^{10}-\frac {1}{11} a^{10} c^{5} x^{11}+2 c^{5} a^{3} x^{4}\)	\(114\)
meijerg	\(\frac {c^{5} \left (-\frac {x a \left (2520 a^{10} x^{10}+2772 a^{9} x^{9}+3080 a^{8} x^{8}+3465 a^{7} x^{7}+3960 a^{6} x^{6}+4620 a^{5} x^{5}+5544 a^{4} x^{4}+6930 a^{3} x^{3}+9240 a^{2} x^{2}+13860 a x +27720\right )}{27720}-\ln \left (-a x +1\right )\right )}{a}-\frac {5 c^{5} \left (-\frac {a x \left (280 a^{8} x^{8}+315 a^{7} x^{7}+360 a^{6} x^{6}+420 a^{5} x^{5}+504 a^{4} x^{4}+630 a^{3} x^{3}+840 a^{2} x^{2}+1260 a x +2520\right )}{2520}-\ln \left (-a x +1\right )\right )}{a}+\frac {10 c^{5} \left (-\frac {a x \left (120 a^{6} x^{6}+140 a^{5} x^{5}+168 a^{4} x^{4}+210 a^{3} x^{3}+280 a^{2} x^{2}+420 a x +840\right )}{840}-\ln \left (-a x +1\right )\right )}{a}-\frac {10 c^{5} \left (-\frac {a x \left (12 a^{4} x^{4}+15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}-\ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (-\frac {a x \left (4 a^{2} x^{2}+6 a x +12\right )}{12}-\ln \left (-a x +1\right )\right )}{a}-\frac {c^{5} \left (-a x -\ln \left (-a x +1\right )\right )}{a}-\frac {c^{5} \left (\frac {x a \left (2772 a^{9} x^{9}+3080 a^{8} x^{8}+3465 a^{7} x^{7}+3960 a^{6} x^{6}+4620 a^{5} x^{5}+5544 a^{4} x^{4}+6930 a^{3} x^{3}+9240 a^{2} x^{2}+13860 a x +27720\right )}{27720}+\ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (\frac {a x \left (315 a^{7} x^{7}+360 a^{6} x^{6}+420 a^{5} x^{5}+504 a^{4} x^{4}+630 a^{3} x^{3}+840 a^{2} x^{2}+1260 a x +2520\right )}{2520}+\ln \left (-a x +1\right )\right )}{a}-\frac {10 c^{5} \left (\frac {a x \left (70 a^{5} x^{5}+84 a^{4} x^{4}+105 a^{3} x^{3}+140 a^{2} x^{2}+210 a x +420\right )}{420}+\ln \left (-a x +1\right )\right )}{a}+\frac {10 c^{5} \left (\frac {a x \left (15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}+\ln \left (-a x +1\right )\right )}{a}-\frac {5 c^{5} \left (\frac {a x \left (3 a x +6\right )}{6}+\ln \left (-a x +1\right )\right )}{a}+\frac {c^{5} \ln \left (-a x +1\right )}{a}\)	\(667\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^5,x,method=_RETURNVERBOSE)

[Out]

c^5*(-1/11*a^10*x^11-1/5*a^9*x^10+1/3*a^8*x^9+a^7*x^8-2/7*a^6*x^7-2*a^5*x^6-2/5*a^4*x^5+2*a^3*x^4+a^2*x^3-a*x^
2-x)

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Maxima [A]
time = 0.26, size = 113, normalized size = 1.35 \begin {gather*} -\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {1}{5} \, a^{9} c^{5} x^{10} + \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac {2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac {2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \end {gather*}

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)^5,x, algorithm="maxima")

[Out]

-1/11*a^10*c^5*x^11 - 1/5*a^9*c^5*x^10 + 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 - 2/7*a^6*c^5*x^7 - 2*a^5*c^5*x^6 - 2/5
*a^4*c^5*x^5 + 2*a^3*c^5*x^4 + a^2*c^5*x^3 - a*c^5*x^2 - c^5*x

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Fricas [A]
time = 0.34, size = 113, normalized size = 1.35 \begin {gather*} -\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {1}{5} \, a^{9} c^{5} x^{10} + \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac {2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac {2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \end {gather*}

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)^5,x, algorithm="fricas")

[Out]

-1/11*a^10*c^5*x^11 - 1/5*a^9*c^5*x^10 + 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 - 2/7*a^6*c^5*x^7 - 2*a^5*c^5*x^6 - 2/5
*a^4*c^5*x^5 + 2*a^3*c^5*x^4 + a^2*c^5*x^3 - a*c^5*x^2 - c^5*x

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Sympy [A]
time = 0.04, size = 119, normalized size = 1.42 \begin {gather*} - \frac {a^{10} c^{5} x^{11}}{11} - \frac {a^{9} c^{5} x^{10}}{5} + \frac {a^{8} c^{5} x^{9}}{3} + a^{7} c^{5} x^{8} - \frac {2 a^{6} c^{5} x^{7}}{7} - 2 a^{5} c^{5} x^{6} - \frac {2 a^{4} c^{5} x^{5}}{5} + 2 a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \end {gather*}

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)**5,x)

[Out]

-a**10*c**5*x**11/11 - a**9*c**5*x**10/5 + a**8*c**5*x**9/3 + a**7*c**5*x**8 - 2*a**6*c**5*x**7/7 - 2*a**5*c**
5*x**6 - 2*a**4*c**5*x**5/5 + 2*a**3*c**5*x**4 + a**2*c**5*x**3 - a*c**5*x**2 - c**5*x

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Giac [A]
time = 0.42, size = 113, normalized size = 1.35 \begin {gather*} -\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {1}{5} \, a^{9} c^{5} x^{10} + \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac {2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac {2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \end {gather*}

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)^5,x, algorithm="giac")

[Out]

-1/11*a^10*c^5*x^11 - 1/5*a^9*c^5*x^10 + 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 - 2/7*a^6*c^5*x^7 - 2*a^5*c^5*x^6 - 2/5
*a^4*c^5*x^5 + 2*a^3*c^5*x^4 + a^2*c^5*x^3 - a*c^5*x^2 - c^5*x

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Mupad [B]
time = 1.25, size = 113, normalized size = 1.35 \begin {gather*} -\frac {a^{10}\,c^5\,x^{11}}{11}-\frac {a^9\,c^5\,x^{10}}{5}+\frac {a^8\,c^5\,x^9}{3}+a^7\,c^5\,x^8-\frac {2\,a^6\,c^5\,x^7}{7}-2\,a^5\,c^5\,x^6-\frac {2\,a^4\,c^5\,x^5}{5}+2\,a^3\,c^5\,x^4+a^2\,c^5\,x^3-a\,c^5\,x^2-c^5\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - a^2*c*x^2)^5*(a*x + 1))/(a*x - 1),x)

[Out]

a^2*c^5*x^3 - a*c^5*x^2 - c^5*x + 2*a^3*c^5*x^4 - (2*a^4*c^5*x^5)/5 - 2*a^5*c^5*x^6 - (2*a^6*c^5*x^7)/7 + a^7*
c^5*x^8 + (a^8*c^5*x^9)/3 - (a^9*c^5*x^10)/5 - (a^10*c^5*x^11)/11

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