3.2.63 \(\int \coth ^3(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx\) [163]

Optimal. Leaf size=72 \[ -\frac {a^3 \coth ^2(c+d x)}{2 d}+\frac {(a+b)^3 \log (\cosh (c+d x))}{d}+\frac {a^2 (a+3 b) \log (\tanh (c+d x))}{d}-\frac {b^3 \tanh ^2(c+d x)}{2 d} \]

[Out]

-1/2*a^3*coth(d*x+c)^2/d+(a+b)^3*ln(cosh(d*x+c))/d+a^2*(a+3*b)*ln(tanh(d*x+c))/d-1/2*b^3*tanh(d*x+c)^2/d

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Rubi [A]
time = 0.08, antiderivative size = 72, 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^3 \coth ^2(c+d x)}{2 d}+\frac {a^2 (a+3 b) \log (\tanh (c+d x))}{d}+\frac {(a+b)^3 \log (\cosh (c+d x))}{d}-\frac {b^3 \tanh ^2(c+d x)}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

-1/2*(a^3*Coth[c + d*x]^2)/d + ((a + b)^3*Log[Cosh[c + d*x]])/d + (a^2*(a + 3*b)*Log[Tanh[c + d*x]])/d - (b^3*
Tanh[c + d*x]^2)/(2*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 ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^3}{x^3 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {(a+b x)^3}{(1-x) x^2} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (-b^3-\frac {(a+b)^3}{-1+x}+\frac {a^3}{x^2}+\frac {a^2 (a+3 b)}{x}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=-\frac {a^3 \coth ^2(c+d x)}{2 d}+\frac {(a+b)^3 \log (\cosh (c+d x))}{d}+\frac {a^2 (a+3 b) \log (\tanh (c+d x))}{d}-\frac {b^3 \tanh ^2(c+d x)}{2 d}\\ \end {align*}

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Mathematica [A]
time = 0.33, size = 63, normalized size = 0.88 \begin {gather*} -\frac {a^3 \coth ^2(c+d x)-2 (a+b)^3 \log (\cosh (c+d x))-2 a^2 (a+3 b) \log (\tanh (c+d x))+b^3 \tanh ^2(c+d x)}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

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

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Maple [A]
time = 1.97, size = 76, normalized size = 1.06

method	result	size
derivativedivides	\(\frac {a^{3} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}\right )+3 a^{2} b \ln \left (\sinh \left (d x +c \right )\right )+3 a \,b^{2} \ln \left (\cosh \left (d x +c \right )\right )+b^{3} \left (\ln \left (\cosh \left (d x +c \right )\right )-\frac {\left (\tanh ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(76\)
default	\(\frac {a^{3} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}\right )+3 a^{2} b \ln \left (\sinh \left (d x +c \right )\right )+3 a \,b^{2} \ln \left (\cosh \left (d x +c \right )\right )+b^{3} \left (\ln \left (\cosh \left (d x +c \right )\right )-\frac {\left (\tanh ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(76\)
risch	\(-a^{3} x -3 a^{2} b x -3 a \,b^{2} x -b^{3} x -\frac {2 a^{3} c}{d}-\frac {6 a^{2} b c}{d}-\frac {6 a \,b^{2} c}{d}-\frac {2 b^{3} c}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (a^{3} {\mathrm e}^{4 d x +4 c}-b^{3} {\mathrm e}^{4 d x +4 c}+2 a^{3} {\mathrm e}^{2 d x +2 c}+2 b^{3} {\mathrm e}^{2 d x +2 c}+a^{3}-b^{3}\right )}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b}{d}+\frac {3 \ln \left (1+{\mathrm e}^{2 d x +2 c}\right ) a \,b^{2}}{d}+\frac {\ln \left (1+{\mathrm e}^{2 d x +2 c}\right ) b^{3}}{d}\)	\(250\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^3*(ln(sinh(d*x+c))-1/2*coth(d*x+c)^2)+3*a^2*b*ln(sinh(d*x+c))+3*a*b^2*ln(cosh(d*x+c))+b^3*(ln(cosh(d*x+
c))-1/2*tanh(d*x+c)^2))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (68) = 136\).
time = 0.49, size = 203, normalized size = 2.82 \begin {gather*} a^{3} {\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 )} + b^{3} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-2 \, d x - 2 \, 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 )} + \frac {3 \, a b^{2} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}{d} + \frac {3 \, a^{2} b \log \left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

a^3*(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))) + b^3*(x + c/d + log(e^(-2*d*x - 2*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))) + 3*a*b^2*log(e^(d*x + c) + e^(-d*x - c))/d + 3*a^2*b*log(e^(d*x + c) - e^(-d*x
- c))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1686 vs. \(2 (68) = 136\).
time = 0.38, size = 1686, normalized size = 23.42 \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)^3*(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

-((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^8 + 8*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)*si
nh(d*x + c)^7 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*sinh(d*x + c)^8 + 2*(a^3 - b^3)*cosh(d*x + c)^6 + 2*(14*(a
^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^2 + a^3 - b^3)*sinh(d*x + c)^6 + 4*(14*(a^3 + 3*a^2*b + 3*a*b^
2 + b^3)*d*x*cosh(d*x + c)^3 + 3*(a^3 - b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(2*a^3 + 2*b^3 - (a^3 + 3*a^2*
b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^4 + 2*(35*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^4 + 2*a^3 +
2*b^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x + 15*(a^3 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(a^3 + 3*
a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^5 + 5*(a^3 - b^3)*cosh(d*x + c)^3 + (2*a^3 + 2*b^3 - (a^3 + 3*a^2*b +
3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x + 2*(a^3 - b^3)*cosh
(d*x + c)^2 + 2*(14*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^6 + 15*(a^3 - b^3)*cosh(d*x + c)^4 + a^3
- b^3 + 6*(2*a^3 + 2*b^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((3*a*b^2
+ b^3)*cosh(d*x + c)^8 + 56*(3*a*b^2 + b^3)*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*(3*a*b^2 + b^3)*cosh(d*x + c)
^2*sinh(d*x + c)^6 + 8*(3*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a*b^2 + b^3)*sinh(d*x + c)^8 - 2*(3*
a*b^2 + b^3)*cosh(d*x + c)^4 + 2*(35*(3*a*b^2 + b^3)*cosh(d*x + c)^4 - 3*a*b^2 - b^3)*sinh(d*x + c)^4 + 8*(7*(
3*a*b^2 + b^3)*cosh(d*x + c)^5 - (3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*a*b^2 + b^3 + 4*(7*(3*a*b^
2 + b^3)*cosh(d*x + c)^6 - 3*(3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a*b^2 + b^3)*cosh(d*x +
c)^7 - (3*a*b^2 + b^3)*cosh(d*x + c)^3)*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) -
((a^3 + 3*a^2*b)*cosh(d*x + c)^8 + 56*(a^3 + 3*a^2*b)*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*(a^3 + 3*a^2*b)*cos
h(d*x + c)^2*sinh(d*x + c)^6 + 8*(a^3 + 3*a^2*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^3 + 3*a^2*b)*sinh(d*x + c)
^8 - 2*(a^3 + 3*a^2*b)*cosh(d*x + c)^4 + 2*(35*(a^3 + 3*a^2*b)*cosh(d*x + c)^4 - a^3 - 3*a^2*b)*sinh(d*x + c)^
4 + 8*(7*(a^3 + 3*a^2*b)*cosh(d*x + c)^5 - (a^3 + 3*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 + 3*a^2*b + 4*
(7*(a^3 + 3*a^2*b)*cosh(d*x + c)^6 - 3*(a^3 + 3*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^3 + 3*a^2*b)*c
osh(d*x + c)^7 - (a^3 + 3*a^2*b)*cosh(d*x + c)^3)*sinh(d*x + c))*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x
+ c))) + 4*(2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^7 + 3*(a^3 - b^3)*cosh(d*x + c)^5 + 2*(2*a^3
+ 2*b^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 + (a^3 - b^3)*cosh(d*x + c))*sinh(d*x + c))/(d*
cosh(d*x + c)^8 + 56*d*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*d*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*d*cosh(d*x +
c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8 - 2*d*cosh(d*x + c)^4 + 2*(35*d*cosh(d*x + c)^4 - d)*sinh(d*x + c)^4 +
8*(7*d*cosh(d*x + c)^5 - d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*d*cosh(d*x + c)^6 - 3*d*cosh(d*x + c)^2)*sin
h(d*x + c)^2 + 8*(d*cosh(d*x + c)^7 - d*cosh(d*x + c)^3)*sinh(d*x + c) + d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \coth ^{3}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)**3*(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Integral((a + b*tanh(c + d*x)**2)**3*coth(c + d*x)**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (68) = 136\).
time = 0.58, size = 274, normalized size = 3.81 \begin {gather*} \frac {2 \, {\left (3 \, a b^{2} + b^{3}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2\right ) + 2 \, {\left (a^{3} + 3 \, a^{2} b\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2\right ) - \frac {a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 3 \, a^{2} b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 3 \, a b^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + b^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 8 \, a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} - 8 \, b^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 12 \, a^{3} - 12 \, a^{2} b - 12 \, a b^{2} + 12 \, b^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} - 4}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/4*(2*(3*a*b^2 + b^3)*log(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c) + 2) + 2*(a^3 + 3*a^2*b)*log(e^(2*d*x + 2*c) + e
^(-2*d*x - 2*c) - 2) - (a^3*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^2 + 3*a^2*b*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*
c))^2 + 3*a*b^2*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^2 + b^3*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^2 + 8*a^3*(e
^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) - 8*b^3*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) + 12*a^3 - 12*a^2*b - 12*a*b^2
+ 12*b^3)/((e^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^2 - 4))/d

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Mupad [B]
time = 2.48, size = 327, normalized size = 4.54 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )\,\left (d\,a^3+3\,d\,a^2\,b+3\,d\,a\,b^2+d\,b^3\right )}{2\,d^2}-\frac {\frac {4\,\left (a^3+b^3\right )}{d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3-b^3\right )}{d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-1}-\frac {\frac {4\,\left (a^3+b^3\right )}{d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3-b^3\right )}{d}}{{\mathrm {e}}^{8\,c+8\,d\,x}-2\,{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (a^3\,\sqrt {-d^2}-b^3\,\sqrt {-d^2}-3\,a\,b^2\,\sqrt {-d^2}+3\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^6+6\,a^5\,b+3\,a^4\,b^2-20\,a^3\,b^3+3\,a^2\,b^4+6\,a\,b^5+b^6}}\right )\,\sqrt {a^6+6\,a^5\,b+3\,a^4\,b^2-20\,a^3\,b^3+3\,a^2\,b^4+6\,a\,b^5+b^6}}{\sqrt {-d^2}}-x\,{\left (a+b\right )}^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(c + d*x)^3*(a + b*tanh(c + d*x)^2)^3,x)

[Out]

(log(exp(4*c + 4*d*x) - 1)*(a^3*d + b^3*d + 3*a*b^2*d + 3*a^2*b*d))/(2*d^2) - ((4*(a^3 + b^3))/d + (2*exp(2*c
+ 2*d*x)*(a^3 - b^3))/d)/(exp(4*c + 4*d*x) - 1) - ((4*(a^3 + b^3))/d + (4*exp(2*c + 2*d*x)*(a^3 - b^3))/d)/(ex
p(8*c + 8*d*x) - 2*exp(4*c + 4*d*x) + 1) - (atan((exp(2*c)*exp(2*d*x)*(a^3*(-d^2)^(1/2) - b^3*(-d^2)^(1/2) - 3
*a*b^2*(-d^2)^(1/2) + 3*a^2*b*(-d^2)^(1/2)))/(d*(6*a*b^5 + 6*a^5*b + a^6 + b^6 + 3*a^2*b^4 - 20*a^3*b^3 + 3*a^
4*b^2)^(1/2)))*(6*a*b^5 + 6*a^5*b + a^6 + b^6 + 3*a^2*b^4 - 20*a^3*b^3 + 3*a^4*b^2)^(1/2))/(-d^2)^(1/2) - x*(a
+ b)^3

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