3.2.31 \(\int \coth ^3(c+d x) (a+b \text {sech}^2(c+d x))^3 \, dx\) [131]

Optimal. Leaf size=81 \[ -\frac {(a+b)^3 \text {csch}^2(c+d x)}{2 d}+\frac {b^2 (3 a+2 b) \log (\cosh (c+d x))}{d}+\frac {(a-2 b) (a+b)^2 \log (\sinh (c+d x))}{d}-\frac {b^3 \text {sech}^2(c+d x)}{2 d} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 81, 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 = {4223, 457, 90} \begin {gather*} \frac {b^2 (3 a+2 b) \log (\cosh (c+d x))}{d}-\frac {(a+b)^3 \text {csch}^2(c+d x)}{2 d}+\frac {(a-2 b) (a+b)^2 \log (\sinh (c+d x))}{d}-\frac {b^3 \text {sech}^2(c+d x)}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((a + b)^3*Csch[c + d*x]^2)/d + (b^2*(3*a + 2*b)*Log[Cosh[c + d*x]])/d + ((a - 2*b)*(a + b)^2*Log[Sinh[c
+ d*x]])/d - (b^3*Sech[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 4223

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Module[{ff =
FreeFactors[Cos[e + f*x], x]}, Dist[-(f*ff^(m + n*p - 1))^(-1), Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*
(ff*x)^n)^p/x^(m + n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]
time = 0.85, size = 110, normalized size = 1.36 \begin {gather*} -\frac {4 \cosh ^6(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \left ((a+b)^3 \text {csch}^2(c+d x)-2 b^2 (3 a+2 b) \log (\cosh (c+d x))-2 (a-2 b) (a+b)^2 \log (\sinh (c+d x))+b^3 \text {sech}^2(c+d x)\right )}{d (a+2 b+a \cosh (2 c+2 d x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Cosh[c + d*x]^6*(a + b*Sech[c + d*x]^2)^3*((a + b)^3*Csch[c + d*x]^2 - 2*b^2*(3*a + 2*b)*Log[Cosh[c + d*x]
] - 2*(a - 2*b)*(a + b)^2*Log[Sinh[c + d*x]] + b^3*Sech[c + d*x]^2))/(d*(a + 2*b + a*Cosh[2*c + 2*d*x])^3)

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Maple [A]
time = 2.14, size = 110, normalized size = 1.36

method	result	size
derivativedivides	\(\frac {a^{3} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}\right )-\frac {3 a^{2} b}{2 \sinh \left (d x +c \right )^{2}}+3 a \,b^{2} \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2}}-\ln \left (\tanh \left (d x +c \right )\right )\right )+b^{3} \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right )^{2}}-\frac {1}{\cosh \left (d x +c \right )^{2}}-2 \ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\)	\(110\)
default	\(\frac {a^{3} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}\right )-\frac {3 a^{2} b}{2 \sinh \left (d x +c \right )^{2}}+3 a \,b^{2} \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2}}-\ln \left (\tanh \left (d x +c \right )\right )\right )+b^{3} \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right )^{2}}-\frac {1}{\cosh \left (d x +c \right )^{2}}-2 \ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\)	\(110\)
risch	\(-a^{3} x -\frac {2 a^{3} c}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (a^{3} {\mathrm e}^{4 d x +4 c}+3 a^{2} b \,{\mathrm e}^{4 d x +4 c}+3 a \,b^{2} {\mathrm e}^{4 d x +4 c}+2 b^{3} {\mathrm e}^{4 d x +4 c}+2 a^{3} {\mathrm e}^{2 d x +2 c}+6 a^{2} b \,{\mathrm e}^{2 d x +2 c}+6 a \,b^{2} {\mathrm e}^{2 d x +2 c}+a^{3}+3 a^{2} b +3 a \,b^{2}+2 b^{3}\right )}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{3}}{d}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a \,b^{2}}{d}-\frac {2 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b^{3}}{d}+\frac {3 b^{2} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right ) a}{d}+\frac {2 b^{3} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d}\)	\(280\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (77) = 154\).
time = 0.49, size = 314, normalized size = 3.88 \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 )} - 2 \, b^{3} {\left (\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 {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} - \frac {2 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} - 3 \, a b^{2} {\left (\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 {\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 {6 \, a^{2} b}{d {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^3*(a+b*sech(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))) - 2*b^3*(log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d - log(e^(-2*d*x - 2*c) + 1
)/d - 2*(e^(-2*d*x - 2*c) + e^(-6*d*x - 6*c))/(d*(2*e^(-4*d*x - 4*c) - e^(-8*d*x - 8*c) - 1))) - 3*a*b^2*(log(
e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/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))) - 6*a^2*b/(d*(e^(d*x + c) - e^(-d*x - c))^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1701 vs. \(2 (77) = 154\).
time = 0.44, size = 1701, normalized size = 21.00 \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*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

-(a^3*d*x*cosh(d*x + c)^8 + 8*a^3*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + a^3*d*x*sinh(d*x + c)^8 + 2*(a^3 + 3*a^2
*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^6 + 2*(14*a^3*d*x*cosh(d*x + c)^2 + a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*sinh(
d*x + c)^6 + 4*(14*a^3*d*x*cosh(d*x + c)^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c))*sinh(d*x + c)^
5 + a^3*d*x - 2*(a^3*d*x - 2*a^3 - 6*a^2*b - 6*a*b^2)*cosh(d*x + c)^4 + 2*(35*a^3*d*x*cosh(d*x + c)^4 - a^3*d*
x + 2*a^3 + 6*a^2*b + 6*a*b^2 + 15*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*a
^3*d*x*cosh(d*x + c)^5 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^3 - (a^3*d*x - 2*a^3 - 6*a^2*b - 6*
a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^2 + 2*(14*a^3*d*x*co
sh(d*x + c)^6 + 15*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^4 + a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3 - 6*(a^
3*d*x - 2*a^3 - 6*a^2*b - 6*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((3*a*b^2 + 2*b^3)*cosh(d*x + c)^8 + 56*
(3*a*b^2 + 2*b^3)*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*(3*a*b^2 + 2*b^3)*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*(
3*a*b^2 + 2*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a*b^2 + 2*b^3)*sinh(d*x + c)^8 - 2*(3*a*b^2 + 2*b^3)*cosh(
d*x + c)^4 + 2*(35*(3*a*b^2 + 2*b^3)*cosh(d*x + c)^4 - 3*a*b^2 - 2*b^3)*sinh(d*x + c)^4 + 8*(7*(3*a*b^2 + 2*b^
3)*cosh(d*x + c)^5 - (3*a*b^2 + 2*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*a*b^2 + 2*b^3 + 4*(7*(3*a*b^2 + 2*b^
3)*cosh(d*x + c)^6 - 3*(3*a*b^2 + 2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a*b^2 + 2*b^3)*cosh(d*x + c)
^7 - (3*a*b^2 + 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*b^2 - 2*b^3)*cosh(d*x + c)^8 + 56*(a^3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*(a^
3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*(a^3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c)*sinh(d*x + c)^7
+ (a^3 - 3*a*b^2 - 2*b^3)*sinh(d*x + c)^8 - 2*(a^3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c)^4 + 2*(35*(a^3 - 3*a*b^2
- 2*b^3)*cosh(d*x + c)^4 - a^3 + 3*a*b^2 + 2*b^3)*sinh(d*x + c)^4 + 8*(7*(a^3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c)
^5 - (a^3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 - 3*a*b^2 - 2*b^3 + 4*(7*(a^3 - 3*a*b^2 - 2*
b^3)*cosh(d*x + c)^6 - 3*(a^3 - 3*a*b^2 - 2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^3 - 3*a*b^2 - 2*b^3)
*cosh(d*x + c)^7 - (a^3 - 3*a*b^2 - 2*b^3)*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*d*x*cosh(d*x + c)^7 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^5 - 2*(a^
3*d*x - 2*a^3 - 6*a^2*b - 6*a*b^2)*cosh(d*x + c)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + 2*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)*sinh(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(-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)**3*(a+b*sech(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (77) = 154\).
time = 0.46, size = 247, normalized size = 3.05 \begin {gather*} \frac {2 \, {\left (3 \, a b^{2} + 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 b^{2} - 2 \, b^{3}\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} + 8 \, a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 24 \, a^{2} b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 24 \, a b^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 16 \, b^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 12 \, a^{3} + 48 \, a^{2} b + 48 \, a b^{2}}{{\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*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/4*(2*(3*a*b^2 + 2*b^3)*log(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c) + 2) + 2*(a^3 - 3*a*b^2 - 2*b^3)*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 + 8*a^3*(e^(2*d*x + 2*c) + e^(-2*
d*x - 2*c)) + 24*a^2*b*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) + 24*a*b^2*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) +
16*b^3*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) + 12*a^3 + 48*a^2*b + 48*a*b^2)/((e^(2*d*x + 2*c) + e^(-2*d*x - 2*
c))^2 - 4))/d

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Mupad [B]
time = 1.72, size = 324, normalized size = 4.00 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (4\,b^3\,\sqrt {-d^2}-a^3\,\sqrt {-d^2}+6\,a\,b^2\,\sqrt {-d^2}\right )}{d\,\sqrt {a^6-12\,a^4\,b^2-8\,a^3\,b^3+36\,a^2\,b^4+48\,a\,b^5+16\,b^6}}\right )\,\sqrt {a^6-12\,a^4\,b^2-8\,a^3\,b^3+36\,a^2\,b^4+48\,a\,b^5+16\,b^6}}{\sqrt {-d^2}}-\frac {\frac {4\,\left (a^3+3\,a^2\,b+3\,a\,b^2\right )}{d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+2\,b^3\right )}{d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-1}-\frac {\frac {4\,\left (a^3+3\,a^2\,b+3\,a\,b^2\right )}{d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+2\,b^3\right )}{d}}{{\mathrm {e}}^{8\,c+8\,d\,x}-2\,{\mathrm {e}}^{4\,c+4\,d\,x}+1}-a^3\,x+\frac {a^3\,\ln \left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )}{2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((exp(2*c)*exp(2*d*x)*(4*b^3*(-d^2)^(1/2) - a^3*(-d^2)^(1/2) + 6*a*b^2*(-d^2)^(1/2)))/(d*(48*a*b^5 + a^6
+ 16*b^6 + 36*a^2*b^4 - 8*a^3*b^3 - 12*a^4*b^2)^(1/2)))*(48*a*b^5 + a^6 + 16*b^6 + 36*a^2*b^4 - 8*a^3*b^3 - 12
*a^4*b^2)^(1/2))/(-d^2)^(1/2) - ((4*(3*a*b^2 + 3*a^2*b + a^3))/d + (2*exp(2*c + 2*d*x)*(3*a*b^2 + 3*a^2*b + a^
3 + 2*b^3))/d)/(exp(4*c + 4*d*x) - 1) - ((4*(3*a*b^2 + 3*a^2*b + a^3))/d + (4*exp(2*c + 2*d*x)*(3*a*b^2 + 3*a^
2*b + a^3 + 2*b^3))/d)/(exp(8*c + 8*d*x) - 2*exp(4*c + 4*d*x) + 1) - a^3*x + (a^3*log(exp(4*c + 4*d*x) - 1))/(
2*d)

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