3.6.81 \(\int e^{4 \coth ^{-1}(a x)} (c-a^2 c x^2)^5 \, dx\) [581]

Optimal. Leaf size=66 \[ \frac {c^5 (1+a x)^8}{a}-\frac {4 c^5 (1+a x)^9}{3 a}+\frac {3 c^5 (1+a x)^{10}}{5 a}-\frac {c^5 (1+a x)^{11}}{11 a} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 66, 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 {3 c^5 (a x+1)^{10}}{5 a}-\frac {4 c^5 (a x+1)^9}{3 a}+\frac {c^5 (a x+1)^8}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^5,x]

[Out]

(c^5*(1 + a*x)^8)/a - (4*c^5*(1 + a*x)^9)/(3*a) + (3*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^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx\\ &=c^5 \int (1-a x)^3 (1+a x)^7 \, dx\\ &=c^5 \int \left (8 (1+a x)^7-12 (1+a x)^8+6 (1+a x)^9-(1+a x)^{10}\right ) \, dx\\ &=\frac {c^5 (1+a x)^8}{a}-\frac {4 c^5 (1+a x)^9}{3 a}+\frac {3 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 = 0.02, size = 39, normalized size = 0.59 \begin {gather*} -\frac {c^5 (1+a x)^8 \left (-29+67 a x-54 a^2 x^2+15 a^3 x^3\right )}{165 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^5,x]

[Out]

-1/165*(c^5*(1 + a*x)^8*(-29 + 67*a*x - 54*a^2*x^2 + 15*a^3*x^3))/a

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Maple [A]
time = 0.28, size = 75, normalized size = 1.14

method	result	size
default	\(c^{5} \left (-\frac {1}{11} a^{10} x^{11}-\frac {2}{5} a^{9} x^{10}-\frac {1}{3} a^{8} x^{9}+a^{7} x^{8}+2 a^{6} x^{7}-\frac {14}{5} a^{4} x^{5}-2 a^{3} x^{4}+a^{2} x^{3}+2 a \,x^{2}+x \right )\)	\(75\)
gosper	\(-\frac {c^{5} x \left (15 a^{10} x^{10}+66 a^{9} x^{9}+55 a^{8} x^{8}-165 a^{7} x^{7}-330 a^{6} x^{6}+462 a^{4} x^{4}+330 a^{3} x^{3}-165 a^{2} x^{2}-330 a x -165\right )}{165}\)	\(77\)
risch	\(-\frac {1}{11} a^{10} c^{5} x^{11}-\frac {2}{5} a^{9} c^{5} x^{10}-\frac {1}{3} a^{8} c^{5} x^{9}+a^{7} c^{5} x^{8}+2 a^{6} c^{5} x^{7}-\frac {14}{5} a^{4} c^{5} x^{5}-2 c^{5} a^{3} x^{4}+c^{5} a^{2} x^{3}+2 a \,c^{5} x^{2}+c^{5} x\)	\(102\)
norman	\(\frac {-c^{5} x +a^{7} c^{5} x^{8}+c^{5} a^{2} x^{3}-a \,c^{5} x^{2}+\frac {4}{5} a^{4} c^{5} x^{5}-\frac {14}{5} a^{5} c^{5} x^{6}-2 a^{6} c^{5} x^{7}+\frac {4}{3} a^{8} c^{5} x^{9}+\frac {1}{15} a^{9} c^{5} x^{10}-\frac {17}{55} a^{10} c^{5} x^{11}-\frac {1}{11} a^{11} c^{5} x^{12}+3 c^{5} a^{3} x^{4}}{a x -1}\)	\(133\)
meijerg	\(\frac {c^{5} \left (-\frac {x a \left (-2730 x^{11} a^{11}-3276 a^{10} x^{10}-4004 a^{9} x^{9}-5005 a^{8} x^{8}-6435 a^{7} x^{7}-8580 a^{6} x^{6}-12012 a^{5} x^{5}-18018 a^{4} x^{4}-30030 a^{3} x^{3}-60060 a^{2} x^{2}-180180 a x +360360\right )}{30030 \left (-a x +1\right )}-12 \ln \left (-a x +1\right )\right )}{a}-\frac {4 c^{5} \left (-\frac {x a \left (-308 a^{9} x^{9}-385 a^{8} x^{8}-495 a^{7} x^{7}-660 a^{6} x^{6}-924 a^{5} x^{5}-1386 a^{4} x^{4}-2310 a^{3} x^{3}-4620 a^{2} x^{2}-13860 a x +27720\right )}{2772 \left (-a x +1\right )}-10 \ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (-\frac {a x \left (-45 a^{7} x^{7}-60 a^{6} x^{6}-84 a^{5} x^{5}-126 a^{4} x^{4}-210 a^{3} x^{3}-420 a^{2} x^{2}-1260 a x +2520\right )}{315 \left (-a x +1\right )}-8 \ln \left (-a x +1\right )\right )}{a}-\frac {5 c^{5} \left (-\frac {a x \left (-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{15 \left (-a x +1\right )}-4 \ln \left (-a x +1\right )\right )}{a}+\frac {4 c^{5} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}-\frac {2 c^{5} \left (\frac {x a \left (-252 a^{10} x^{10}-308 a^{9} x^{9}-385 a^{8} x^{8}-495 a^{7} x^{7}-660 a^{6} x^{6}-924 a^{5} x^{5}-1386 a^{4} x^{4}-2310 a^{3} x^{3}-4620 a^{2} x^{2}-13860 a x +27720\right )}{-2520 a x +2520}+11 \ln \left (-a x +1\right )\right )}{a}+\frac {10 c^{5} \left (\frac {a x \left (-35 a^{8} x^{8}-45 a^{7} x^{7}-60 a^{6} x^{6}-84 a^{5} x^{5}-126 a^{4} x^{4}-210 a^{3} x^{3}-420 a^{2} x^{2}-1260 a x +2520\right )}{-280 a x +280}+9 \ln \left (-a x +1\right )\right )}{a}-\frac {20 c^{5} \left (\frac {a x \left (-20 a^{6} x^{6}-28 a^{5} x^{5}-42 a^{4} x^{4}-70 a^{3} x^{3}-140 a^{2} x^{2}-420 a x +840\right )}{-120 a x +120}+7 \ln \left (-a x +1\right )\right )}{a}+\frac {20 c^{5} \left (\frac {a x \left (-3 a^{4} x^{4}-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{-12 a x +12}+5 \ln \left (-a x +1\right )\right )}{a}-\frac {10 c^{5} \left (\frac {a x \left (-2 a^{2} x^{2}-6 a x +12\right )}{-4 a x +4}+3 \ln \left (-a x +1\right )\right )}{a}+\frac {2 c^{5} \left (\frac {a x}{-a x +1}+\ln \left (-a x +1\right )\right )}{a}+\frac {c^{5} x}{-a x +1}\)	\(809\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^5*(-1/11*a^10*x^11-2/5*a^9*x^10-1/3*a^8*x^9+a^7*x^8+2*a^6*x^7-14/5*a^4*x^5-2*a^3*x^4+a^2*x^3+2*a*x^2+x)

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Maxima [A]
time = 0.26, size = 101, normalized size = 1.53 \begin {gather*} -\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {2}{5} \, a^{9} c^{5} x^{10} - \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac {14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, 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)^2*(a*x+1)^2*(-a^2*c*x^2+c)^5,x, algorithm="maxima")

[Out]

-1/11*a^10*c^5*x^11 - 2/5*a^9*c^5*x^10 - 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 + 2*a^6*c^5*x^7 - 14/5*a^4*c^5*x^5 - 2*
a^3*c^5*x^4 + a^2*c^5*x^3 + 2*a*c^5*x^2 + c^5*x

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Fricas [A]
time = 0.35, size = 101, normalized size = 1.53 \begin {gather*} -\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {2}{5} \, a^{9} c^{5} x^{10} - \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac {14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, 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)^2*(a*x+1)^2*(-a^2*c*x^2+c)^5,x, algorithm="fricas")

[Out]

-1/11*a^10*c^5*x^11 - 2/5*a^9*c^5*x^10 - 1/3*a^8*c^5*x^9 + a^7*c^5*x^8 + 2*a^6*c^5*x^7 - 14/5*a^4*c^5*x^5 - 2*
a^3*c^5*x^4 + a^2*c^5*x^3 + 2*a*c^5*x^2 + c^5*x

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (54) = 108\).
time = 0.04, size = 109, normalized size = 1.65 \begin {gather*} - \frac {a^{10} c^{5} x^{11}}{11} - \frac {2 a^{9} c^{5} x^{10}}{5} - \frac {a^{8} c^{5} x^{9}}{3} + a^{7} c^{5} x^{8} + 2 a^{6} c^{5} x^{7} - \frac {14 a^{4} c^{5} x^{5}}{5} - 2 a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 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)**2*(a*x+1)**2*(-a**2*c*x**2+c)**5,x)

[Out]

-a**10*c**5*x**11/11 - 2*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 - 14*a**4*c*
*5*x**5/5 - 2*a**3*c**5*x**4 + a**2*c**5*x**3 + 2*a*c**5*x**2 + c**5*x

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Giac [A]
time = 0.41, size = 102, normalized size = 1.55 \begin {gather*} -\frac {{\left (15 \, c^{5} + \frac {231 \, c^{5}}{a x - 1} + \frac {1540 \, c^{5}}{{\left (a x - 1\right )}^{2}} + \frac {5775 \, c^{5}}{{\left (a x - 1\right )}^{3}} + \frac {13200 \, c^{5}}{{\left (a x - 1\right )}^{4}} + \frac {18480 \, c^{5}}{{\left (a x - 1\right )}^{5}} + \frac {14784 \, c^{5}}{{\left (a x - 1\right )}^{6}} + \frac {5280 \, c^{5}}{{\left (a x - 1\right )}^{7}}\right )} {\left (a x - 1\right )}^{11}}{165 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^5,x, algorithm="giac")

[Out]

-1/165*(15*c^5 + 231*c^5/(a*x - 1) + 1540*c^5/(a*x - 1)^2 + 5775*c^5/(a*x - 1)^3 + 13200*c^5/(a*x - 1)^4 + 184
80*c^5/(a*x - 1)^5 + 14784*c^5/(a*x - 1)^6 + 5280*c^5/(a*x - 1)^7)*(a*x - 1)^11/a

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Mupad [B]
time = 1.24, size = 101, normalized size = 1.53 \begin {gather*} -\frac {a^{10}\,c^5\,x^{11}}{11}-\frac {2\,a^9\,c^5\,x^{10}}{5}-\frac {a^8\,c^5\,x^9}{3}+a^7\,c^5\,x^8+2\,a^6\,c^5\,x^7-\frac {14\,a^4\,c^5\,x^5}{5}-2\,a^3\,c^5\,x^4+a^2\,c^5\,x^3+2\,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)^2)/(a*x - 1)^2,x)

[Out]

c^5*x + 2*a*c^5*x^2 + a^2*c^5*x^3 - 2*a^3*c^5*x^4 - (14*a^4*c^5*x^5)/5 + 2*a^6*c^5*x^7 + a^7*c^5*x^8 - (a^8*c^
5*x^9)/3 - (2*a^9*c^5*x^10)/5 - (a^10*c^5*x^11)/11

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