3.2.55 \(\int \coth ^7(c+d x) (a+b \tanh ^2(c+d x))^2 \, dx\) [155]

Optimal. Leaf size=92 \[ -\frac {(a+b)^2 \coth ^2(c+d x)}{2 d}-\frac {a (a+2 b) \coth ^4(c+d x)}{4 d}-\frac {a^2 \coth ^6(c+d x)}{6 d}+\frac {(a+b)^2 \log (\cosh (c+d x))}{d}+\frac {(a+b)^2 \log (\tanh (c+d x))}{d} \]

[Out]

-1/2*(a+b)^2*coth(d*x+c)^2/d-1/4*a*(a+2*b)*coth(d*x+c)^4/d-1/6*a^2*coth(d*x+c)^6/d+(a+b)^2*ln(cosh(d*x+c))/d+(
a+b)^2*ln(tanh(d*x+c))/d

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Rubi [A]
time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3751, 457, 90} \begin {gather*} -\frac {a^2 \coth ^6(c+d x)}{6 d}-\frac {a (a+2 b) \coth ^4(c+d x)}{4 d}-\frac {(a+b)^2 \coth ^2(c+d x)}{2 d}+\frac {(a+b)^2 \log (\tanh (c+d x))}{d}+\frac {(a+b)^2 \log (\cosh (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[c + d*x]^7*(a + b*Tanh[c + d*x]^2)^2,x]

[Out]

-1/2*((a + b)^2*Coth[c + d*x]^2)/d - (a*(a + 2*b)*Coth[c + d*x]^4)/(4*d) - (a^2*Coth[c + d*x]^6)/(6*d) + ((a +
b)^2*Log[Cosh[c + d*x]])/d + ((a + b)^2*Log[Tanh[c + d*x]])/d

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 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps

\begin {align*} \int \coth ^7(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^2}{x^7 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {(a+b x)^2}{(1-x) x^4} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {(a+b)^2}{-1+x}+\frac {a^2}{x^4}+\frac {a (a+2 b)}{x^3}+\frac {(a+b)^2}{x^2}+\frac {(a+b)^2}{x}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=-\frac {(a+b)^2 \coth ^2(c+d x)}{2 d}-\frac {a (a+2 b) \coth ^4(c+d x)}{4 d}-\frac {a^2 \coth ^6(c+d x)}{6 d}+\frac {(a+b)^2 \log (\cosh (c+d x))}{d}+\frac {(a+b)^2 \log (\tanh (c+d x))}{d}\\ \end {align*}

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Mathematica [A]
time = 0.35, size = 74, normalized size = 0.80 \begin {gather*} -\frac {6 (a+b)^2 \coth ^2(c+d x)+3 a (a+2 b) \coth ^4(c+d x)+2 a^2 \coth ^6(c+d x)-12 (a+b)^2 (\log (\cosh (c+d x))+\log (\tanh (c+d x)))}{12 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[c + d*x]^7*(a + b*Tanh[c + d*x]^2)^2,x]

[Out]

-1/12*(6*(a + b)^2*Coth[c + d*x]^2 + 3*a*(a + 2*b)*Coth[c + d*x]^4 + 2*a^2*Coth[c + d*x]^6 - 12*(a + b)^2*(Log
[Cosh[c + d*x]] + Log[Tanh[c + d*x]]))/d

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Maple [A]
time = 1.81, size = 102, normalized size = 1.11

method	result	size
derivativedivides	\(\frac {a^{2} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (\coth ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\coth ^{6}\left (d x +c \right )\right )}{6}\right )+2 a b \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (\coth ^{4}\left (d x +c \right )\right )}{4}\right )+b^{2} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(102\)
default	\(\frac {a^{2} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (\coth ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\coth ^{6}\left (d x +c \right )\right )}{6}\right )+2 a b \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (\coth ^{4}\left (d x +c \right )\right )}{4}\right )+b^{2} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(102\)
risch	\(-a^{2} x -2 a b x -b^{2} x -\frac {2 a^{2} c}{d}-\frac {4 a b c}{d}-\frac {2 b^{2} c}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (9 a^{2} {\mathrm e}^{8 d x +8 c}+12 a b \,{\mathrm e}^{8 d x +8 c}+3 b^{2} {\mathrm e}^{8 d x +8 c}-18 a^{2} {\mathrm e}^{6 d x +6 c}-36 a b \,{\mathrm e}^{6 d x +6 c}-12 b^{2} {\mathrm e}^{6 d x +6 c}+34 a^{2} {\mathrm e}^{4 d x +4 c}+48 a b \,{\mathrm e}^{4 d x +4 c}+18 b^{2} {\mathrm e}^{4 d x +4 c}-18 a^{2} {\mathrm e}^{2 d x +2 c}-36 a b \,{\mathrm e}^{2 d x +2 c}-12 b^{2} {\mathrm e}^{2 d x +2 c}+9 a^{2}+12 a b +3 b^{2}\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{6}}+\frac {a^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d}+\frac {2 a \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b}{d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b^{2}}{d}\)	\(308\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)^7*(a+b*tanh(d*x+c)^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^2*(ln(sinh(d*x+c))-1/2*coth(d*x+c)^2-1/4*coth(d*x+c)^4-1/6*coth(d*x+c)^6)+2*a*b*(ln(sinh(d*x+c))-1/2*co
th(d*x+c)^2-1/4*coth(d*x+c)^4)+b^2*(ln(sinh(d*x+c))-1/2*coth(d*x+c)^2))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 390 vs. \(2 (86) = 172\).
time = 0.28, size = 390, normalized size = 4.24 \begin {gather*} \frac {1}{3} \, a^{2} {\left (3 \, x + \frac {3 \, c}{d} + \frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 18 \, e^{\left (-4 \, d x - 4 \, c\right )} + 34 \, e^{\left (-6 \, d x - 6 \, c\right )} - 18 \, e^{\left (-8 \, d x - 8 \, c\right )} + 9 \, e^{\left (-10 \, d x - 10 \, c\right )}\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} + 2 \, a b {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {4 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + b^{2} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^7*(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/3*a^2*(3*x + 3*c/d + 3*log(e^(-d*x - c) + 1)/d + 3*log(e^(-d*x - c) - 1)/d + 2*(9*e^(-2*d*x - 2*c) - 18*e^(-
4*d*x - 4*c) + 34*e^(-6*d*x - 6*c) - 18*e^(-8*d*x - 8*c) + 9*e^(-10*d*x - 10*c))/(d*(6*e^(-2*d*x - 2*c) - 15*e
^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1)))
+ 2*a*b*(x + c/d + log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d + 4*(e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c)
+ e^(-6*d*x - 6*c))/(d*(4*e^(-2*d*x - 2*c) - 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) - 1))
) + b^2*(x + c/d + log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d + 2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x - 2*
c) - e^(-4*d*x - 4*c) - 1)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3454 vs. \(2 (86) = 172\).
time = 0.39, size = 3454, normalized size = 37.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^7*(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

-1/3*(3*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^12 + 36*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)*sinh(d*x + c)^11 +
3*(a^2 + 2*a*b + b^2)*d*x*sinh(d*x + c)^12 - 6*(3*(a^2 + 2*a*b + b^2)*d*x - 3*a^2 - 4*a*b - b^2)*cosh(d*x + c
)^10 + 6*(33*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^2 - 3*(a^2 + 2*a*b + b^2)*d*x + 3*a^2 + 4*a*b + b^2)*sinh(d
*x + c)^10 + 60*(11*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^3 - (3*(a^2 + 2*a*b + b^2)*d*x - 3*a^2 - 4*a*b - b^2
)*cosh(d*x + c))*sinh(d*x + c)^9 + 3*(15*(a^2 + 2*a*b + b^2)*d*x - 12*a^2 - 24*a*b - 8*b^2)*cosh(d*x + c)^8 +
3*(495*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^4 + 15*(a^2 + 2*a*b + b^2)*d*x - 90*(3*(a^2 + 2*a*b + b^2)*d*x -
3*a^2 - 4*a*b - b^2)*cosh(d*x + c)^2 - 12*a^2 - 24*a*b - 8*b^2)*sinh(d*x + c)^8 + 24*(99*(a^2 + 2*a*b + b^2)*d
*x*cosh(d*x + c)^5 - 30*(3*(a^2 + 2*a*b + b^2)*d*x - 3*a^2 - 4*a*b - b^2)*cosh(d*x + c)^3 + (15*(a^2 + 2*a*b +
b^2)*d*x - 12*a^2 - 24*a*b - 8*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 4*(15*(a^2 + 2*a*b + b^2)*d*x - 17*a^2 -
24*a*b - 9*b^2)*cosh(d*x + c)^6 + 4*(693*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^6 - 315*(3*(a^2 + 2*a*b + b^2)
*d*x - 3*a^2 - 4*a*b - b^2)*cosh(d*x + c)^4 - 15*(a^2 + 2*a*b + b^2)*d*x + 21*(15*(a^2 + 2*a*b + b^2)*d*x - 12
*a^2 - 24*a*b - 8*b^2)*cosh(d*x + c)^2 + 17*a^2 + 24*a*b + 9*b^2)*sinh(d*x + c)^6 + 24*(99*(a^2 + 2*a*b + b^2)
*d*x*cosh(d*x + c)^7 - 63*(3*(a^2 + 2*a*b + b^2)*d*x - 3*a^2 - 4*a*b - b^2)*cosh(d*x + c)^5 + 7*(15*(a^2 + 2*a
*b + b^2)*d*x - 12*a^2 - 24*a*b - 8*b^2)*cosh(d*x + c)^3 - (15*(a^2 + 2*a*b + b^2)*d*x - 17*a^2 - 24*a*b - 9*b
^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*(15*(a^2 + 2*a*b + b^2)*d*x - 12*a^2 - 24*a*b - 8*b^2)*cosh(d*x + c)^4
+ 3*(495*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^8 - 420*(3*(a^2 + 2*a*b + b^2)*d*x - 3*a^2 - 4*a*b - b^2)*cosh(
d*x + c)^6 + 70*(15*(a^2 + 2*a*b + b^2)*d*x - 12*a^2 - 24*a*b - 8*b^2)*cosh(d*x + c)^4 + 15*(a^2 + 2*a*b + b^2
)*d*x - 20*(15*(a^2 + 2*a*b + b^2)*d*x - 17*a^2 - 24*a*b - 9*b^2)*cosh(d*x + c)^2 - 12*a^2 - 24*a*b - 8*b^2)*s
inh(d*x + c)^4 + 4*(165*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^9 - 180*(3*(a^2 + 2*a*b + b^2)*d*x - 3*a^2 - 4*a
*b - b^2)*cosh(d*x + c)^7 + 42*(15*(a^2 + 2*a*b + b^2)*d*x - 12*a^2 - 24*a*b - 8*b^2)*cosh(d*x + c)^5 - 20*(15
*(a^2 + 2*a*b + b^2)*d*x - 17*a^2 - 24*a*b - 9*b^2)*cosh(d*x + c)^3 + 3*(15*(a^2 + 2*a*b + b^2)*d*x - 12*a^2 -
24*a*b - 8*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)*d*x - 6*(3*(a^2 + 2*a*b + b^2)*d*x - 3
*a^2 - 4*a*b - b^2)*cosh(d*x + c)^2 + 6*(33*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^10 - 45*(3*(a^2 + 2*a*b + b^
2)*d*x - 3*a^2 - 4*a*b - b^2)*cosh(d*x + c)^8 + 14*(15*(a^2 + 2*a*b + b^2)*d*x - 12*a^2 - 24*a*b - 8*b^2)*cosh
(d*x + c)^6 - 10*(15*(a^2 + 2*a*b + b^2)*d*x - 17*a^2 - 24*a*b - 9*b^2)*cosh(d*x + c)^4 - 3*(a^2 + 2*a*b + b^2
)*d*x + 3*(15*(a^2 + 2*a*b + b^2)*d*x - 12*a^2 - 24*a*b - 8*b^2)*cosh(d*x + c)^2 + 3*a^2 + 4*a*b + b^2)*sinh(d
*x + c)^2 - 3*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^12 + 12*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^11 +
(a^2 + 2*a*b + b^2)*sinh(d*x + c)^12 - 6*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^10 + 6*(11*(a^2 + 2*a*b + b^2)*cosh
(d*x + c)^2 - a^2 - 2*a*b - b^2)*sinh(d*x + c)^10 + 20*(11*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 - 3*(a^2 + 2*a*
b + b^2)*cosh(d*x + c))*sinh(d*x + c)^9 + 15*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 + 15*(33*(a^2 + 2*a*b + b^2)*
cosh(d*x + c)^4 - 18*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x + c)^8 + 24*(33*(a^2 +
2*a*b + b^2)*cosh(d*x + c)^5 - 30*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + 5*(a^2 + 2*a*b + b^2)*cosh(d*x + c))*s
inh(d*x + c)^7 - 20*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 4*(231*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 - 315*(a^
2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 105*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 - 5*a^2 - 10*a*b - 5*b^2)*sinh(d*x
+ c)^6 + 24*(33*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 - 63*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 + 35*(a^2 + 2*a*b
+ b^2)*cosh(d*x + c)^3 - 5*(a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(a^2 + 2*a*b + b^2)*cosh(d
*x + c)^4 + 15*(33*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 - 84*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 70*(a^2 + 2*
a*b + b^2)*cosh(d*x + c)^4 - 20*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 20*
(11*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^9 - 36*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 + 42*(a^2 + 2*a*b + b^2)*cosh
(d*x + c)^5 - 20*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^3 -
6*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 6*(11*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^10 - 45*(a^2 + 2*a*b + b^2)*co
sh(d*x + c)^8 + 70*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 - 50*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 15*(a^2 + 2*
a*b + b^2)*cosh(d*x + c)^2 - a^2 - 2*a*b - b^2)*sinh(d*x + c)^2 + a^2 + 2*a*b + b^2 + 12*((a^2 + 2*a*b + b^2)*
cosh(d*x + c)^11 - 5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^9 + 10*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 - 10*(a^2 +
2*a*b + b^2)*cosh(d*x + c)^5 + 5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh
(d*x + c))*log(2*sinh(d*x + c)/(cosh(d*x + c) -...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)**7*(a+b*tanh(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (86) = 172\).
time = 0.56, size = 192, normalized size = 2.09 \begin {gather*} -\frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (d x + c\right )} - 3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {2 \, {\left (3 \, {\left (3 \, a^{2} + 4 \, a b + b^{2}\right )} e^{\left (10 \, d x + 10 \, c\right )} - 6 \, {\left (3 \, a^{2} + 6 \, a b + 2 \, b^{2}\right )} e^{\left (8 \, d x + 8 \, c\right )} + 2 \, {\left (17 \, a^{2} + 24 \, a b + 9 \, b^{2}\right )} e^{\left (6 \, d x + 6 \, c\right )} - 6 \, {\left (3 \, a^{2} + 6 \, a b + 2 \, b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, {\left (3 \, a^{2} + 4 \, a b + b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{6}}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^7*(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

-1/3*(3*(a^2 + 2*a*b + b^2)*(d*x + c) - 3*(a^2 + 2*a*b + b^2)*log(abs(e^(2*d*x + 2*c) - 1)) + 2*(3*(3*a^2 + 4*
a*b + b^2)*e^(10*d*x + 10*c) - 6*(3*a^2 + 6*a*b + 2*b^2)*e^(8*d*x + 8*c) + 2*(17*a^2 + 24*a*b + 9*b^2)*e^(6*d*
x + 6*c) - 6*(3*a^2 + 6*a*b + 2*b^2)*e^(4*d*x + 4*c) + 3*(3*a^2 + 4*a*b + b^2)*e^(2*d*x + 2*c))/(e^(2*d*x + 2*
c) - 1)^6)/d

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Mupad [B]
time = 0.27, size = 362, normalized size = 3.93 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )\,\left (a^2+2\,a\,b+b^2\right )}{d}-\frac {2\,\left (3\,a^2+4\,a\,b+b^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {32\,a^2}{3\,d\,\left (15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}-x\,{\left (a+b\right )}^2-\frac {2\,\left (9\,a^2+8\,a\,b+b^2\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,\left (13\,a^2+6\,b\,a\right )}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {4\,\left (11\,a^2+2\,b\,a\right )}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {32\,a^2}{d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(c + d*x)^7*(a + b*tanh(c + d*x)^2)^2,x)

[Out]

(log(exp(2*c)*exp(2*d*x) - 1)*(2*a*b + a^2 + b^2))/d - (2*(4*a*b + 3*a^2 + b^2))/(d*(exp(2*c + 2*d*x) - 1)) -
(32*a^2)/(3*d*(15*exp(4*c + 4*d*x) - 6*exp(2*c + 2*d*x) - 20*exp(6*c + 6*d*x) + 15*exp(8*c + 8*d*x) - 6*exp(10
*c + 10*d*x) + exp(12*c + 12*d*x) + 1)) - x*(a + b)^2 - (2*(8*a*b + 9*a^2 + b^2))/(d*(exp(4*c + 4*d*x) - 2*exp
(2*c + 2*d*x) + 1)) - (8*(6*a*b + 13*a^2))/(3*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) -
1)) - (4*(2*a*b + 11*a^2))/(d*(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x)
+ 1)) - (32*a^2)/(d*(5*exp(2*c + 2*d*x) - 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) - 5*exp(8*c + 8*d*x) + ex
p(10*c + 10*d*x) - 1))

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