3.2.29 \(\int \coth (c+d x) (a+b \text {sech}^2(c+d x))^3 \, dx\) [129]

Optimal. Leaf size=84 \[ -\frac {b \left (3 a^2+3 a b+b^2\right ) \log (\cosh (c+d x))}{d}+\frac {(a+b)^3 \log (\sinh (c+d x))}{d}+\frac {b^2 (3 a+b) \text {sech}^2(c+d x)}{2 d}+\frac {b^3 \text {sech}^4(c+d x)}{4 d} \]

[Out]

-b*(3*a^2+3*a*b+b^2)*ln(cosh(d*x+c))/d+(a+b)^3*ln(sinh(d*x+c))/d+1/2*b^2*(3*a+b)*sech(d*x+c)^2/d+1/4*b^3*sech(
d*x+c)^4/d

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Rubi [A]
time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4223, 457, 90} \begin {gather*} -\frac {b \left (3 a^2+3 a b+b^2\right ) \log (\cosh (c+d x))}{d}+\frac {b^2 (3 a+b) \text {sech}^2(c+d x)}{2 d}+\frac {(a+b)^3 \log (\sinh (c+d x))}{d}+\frac {b^3 \text {sech}^4(c+d x)}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(3*a^2 + 3*a*b + b^2)*Log[Cosh[c + d*x]])/d) + ((a + b)^3*Log[Sinh[c + d*x]])/d + (b^2*(3*a + b)*Sech[c +
d*x]^2)/(2*d) + (b^3*Sech[c + d*x]^4)/(4*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 (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^5 \left (1-x^2\right )} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {(b+a x)^3}{(1-x) x^3} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\text {Subst}\left (\int \left (-\frac {(a+b)^3}{-1+x}+\frac {b^3}{x^3}+\frac {b^2 (3 a+b)}{x^2}+\frac {b \left (3 a^2+3 a b+b^2\right )}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {b \left (3 a^2+3 a b+b^2\right ) \log (\cosh (c+d x))}{d}+\frac {(a+b)^3 \log (\sinh (c+d x))}{d}+\frac {b^2 (3 a+b) \text {sech}^2(c+d x)}{2 d}+\frac {b^3 \text {sech}^4(c+d x)}{4 d}\\ \end {align*}

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Mathematica [A]
time = 0.44, size = 114, normalized size = 1.36 \begin {gather*} -\frac {2 \cosh ^6(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \left (4 b \left (3 a^2+3 a b+b^2\right ) \log (\cosh (c+d x))-4 (a+b)^3 \log (\sinh (c+d x))-2 b^2 (3 a+b) \text {sech}^2(c+d x)-b^3 \text {sech}^4(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]*(a + b*Sech[c + d*x]^2)^3,x]

[Out]

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

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Maple [A]
time = 1.83, size = 86, normalized size = 1.02

method	result	size
derivativedivides	\(\frac {a^{3} \ln \left (\sinh \left (d x +c \right )\right )+3 a^{2} b \ln \left (\tanh \left (d x +c \right )\right )+3 a \,b^{2} \left (\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )+b^{3} \left (\frac {1}{4 \cosh \left (d x +c \right )^{4}}+\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\)	\(86\)
default	\(\frac {a^{3} \ln \left (\sinh \left (d x +c \right )\right )+3 a^{2} b \ln \left (\tanh \left (d x +c \right )\right )+3 a \,b^{2} \left (\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )+b^{3} \left (\frac {1}{4 \cosh \left (d x +c \right )^{4}}+\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\)	\(86\)
risch	\(-a^{3} x -\frac {2 a^{3} c}{d}+\frac {2 b^{2} {\mathrm e}^{2 d x +2 c} \left (3 a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+6 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+3 a +b \right )}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{3}}{d}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{2} b}{d}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a \,b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b^{3}}{d}-\frac {3 b \ln \left (1+{\mathrm e}^{2 d x +2 c}\right ) a^{2}}{d}-\frac {3 b^{2} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right ) a}{d}-\frac {b^{3} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d}\)	\(241\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^3*ln(sinh(d*x+c))+3*a^2*b*ln(tanh(d*x+c))+3*a*b^2*(1/2/cosh(d*x+c)^2+ln(tanh(d*x+c)))+b^3*(1/4/cosh(d*x
+c)^4+1/2/cosh(d*x+c)^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. 300 vs. \(2 (80) = 160\).
time = 0.49, size = 300, normalized size = 3.57 \begin {gather*} 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 )} + 4 \, 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 )} + 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 )} + 3 \, a^{2} b {\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}\right )} + \frac {a^{3} \log \left (\sinh \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)*(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2376 vs. \(2 (80) = 160\).
time = 0.38, size = 2376, normalized size = 28.29 \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)*(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*(2*a^3*d*x -
3*a*b^2 - b^3)*cosh(d*x + c)^6 + 2*(14*a^3*d*x*cosh(d*x + c)^2 + 2*a^3*d*x - 3*a*b^2 - b^3)*sinh(d*x + c)^6 +
4*(14*a^3*d*x*cosh(d*x + c)^3 + 3*(2*a^3*d*x - 3*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + a^3*d*x + 2*(3
*a^3*d*x - 6*a*b^2 - 4*b^3)*cosh(d*x + c)^4 + 2*(35*a^3*d*x*cosh(d*x + c)^4 + 3*a^3*d*x - 6*a*b^2 - 4*b^3 + 15
*(2*a^3*d*x - 3*a*b^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*(2*a^3*d*x -
3*a*b^2 - b^3)*cosh(d*x + c)^3 + (3*a^3*d*x - 6*a*b^2 - 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(2*a^3*d*x -
3*a*b^2 - b^3)*cosh(d*x + c)^2 + 2*(14*a^3*d*x*cosh(d*x + c)^6 + 2*a^3*d*x + 15*(2*a^3*d*x - 3*a*b^2 - b^3)*c
osh(d*x + c)^4 - 3*a*b^2 - b^3 + 6*(3*a^3*d*x - 6*a*b^2 - 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((3*a^2*b
+ 3*a*b^2 + b^3)*cosh(d*x + c)^8 + 8*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^2*b + 3*a*
b^2 + b^3)*sinh(d*x + c)^8 + 4*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 + 4*(3*a^2*b + 3*a*b^2 + b^3 + 7*(3*a
^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + 3*(3
*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + 2*(35*(
3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + 9*a^2*b + 9*a*b^2 + 3*b^3 + 30*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x +
c)^2)*sinh(d*x + c)^4 + 8*(7*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 10*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*
x + c)^3 + 3*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*a^2*b + 3*a*b^2 + b^3 + 4*(3*a^2*b +
3*a*b^2 + b^3)*cosh(d*x + c)^2 + 4*(7*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 + 15*(3*a^2*b + 3*a*b^2 + b^3
)*cosh(d*x + c)^4 + 3*a^2*b + 3*a*b^2 + b^3 + 9*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8
*((3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^7 + 3*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 3*(3*a^2*b + 3*a*b
^2 + b^3)*cosh(d*x + c)^3 + (3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(
d*x + c) - sinh(d*x + c))) - ((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^8 + 8*(a^3 + 3*a^2*b + 3*a*b^2 + b
^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^8 + 4*(a^3 + 3*a^2*b + 3*a*b
^2 + b^3)*cosh(d*x + c)^6 + 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3 + 7*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)
^2)*sinh(d*x + c)^6 + 8*(7*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)
*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + 2*(35*(a^3 + 3*a^2*b + 3
*a*b^2 + b^3)*cosh(d*x + c)^4 + 3*a^3 + 9*a^2*b + 9*a*b^2 + 3*b^3 + 30*(a^3 + 3*a^2*b + 3*a*b^2 + 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)*cosh(d*x + c)^5 + 10*(a^3 + 3*a^2*b + 3*a*b^2
+ b^3)*cosh(d*x + c)^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 + 3*a^2*b + 3
*a*b^2 + b^3 + 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2 + 4*(7*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d
*x + c)^6 + 15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 + 9*(a^3 + 3*a^
2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^7 + 3
*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + (a^3 +
3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*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*(2*a^3*d*x - 3*a*b^2 - b^3)*cosh(d*x + c)^5 + 2*(3*a^3*d*x - 6*a*b^2 - 4*b^3)
*cosh(d*x + c)^3 + (2*a^3*d*x - 3*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^8 + 8*d*cosh(d*x
+ c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8 + 4*d*cosh(d*x + c)^6 + 4*(7*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^6
+ 8*(7*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^5 + 6*d*cosh(d*x + c)^4 + 2*(35*d*cosh(d*x + c)^4
+ 30*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^4 + 8*(7*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c)^3 + 3*d*cosh(d*x +
c))*sinh(d*x + c)^3 + 4*d*cosh(d*x + c)^2 + 4*(7*d*cosh(d*x + c)^6 + 15*d*cosh(d*x + c)^4 + 9*d*cosh(d*x + c)
^2 + d)*sinh(d*x + c)^2 + 8*(d*cosh(d*x + c)^7 + 3*d*cosh(d*x + c)^5 + 3*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*
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)*(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. 283 vs. \(2 (80) = 160\).
time = 0.42, size = 283, normalized size = 3.37 \begin {gather*} -\frac {2 \, {\left (3 \, a^{2} b + 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 + 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 ) - \frac {9 \, a^{2} b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 9 \, a b^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 3 \, b^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 36 \, a^{2} b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 60 \, a b^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 20 \, b^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 36 \, a^{2} b + 84 \, a b^{2} + 44 \, b^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2\right )}^{2}}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*(3*a^2*b + 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 + 3*a*b^2 + b
^3)*log(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c) - 2) - (9*a^2*b*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^2 + 9*a*b^2*(e
^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^2 + 3*b^3*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^2 + 36*a^2*b*(e^(2*d*x + 2*c
) + e^(-2*d*x - 2*c)) + 60*a*b^2*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) + 20*b^3*(e^(2*d*x + 2*c) + e^(-2*d*x -
2*c)) + 36*a^2*b + 84*a*b^2 + 44*b^3)/(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c) + 2)^2)/d

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Mupad [B]
time = 1.64, size = 360, normalized size = 4.29 \begin {gather*} \frac {2\,\left (b^3+3\,a\,b^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-a^3\,x-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (a^3\,\sqrt {-d^2}+2\,b^3\,\sqrt {-d^2}+6\,a\,b^2\,\sqrt {-d^2}+6\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^6+12\,a^5\,b+48\,a^4\,b^2+76\,a^3\,b^3+60\,a^2\,b^4+24\,a\,b^5+4\,b^6}}\right )\,\sqrt {a^6+12\,a^5\,b+48\,a^4\,b^2+76\,a^3\,b^3+60\,a^2\,b^4+24\,a\,b^5+4\,b^6}}{\sqrt {-d^2}}+\frac {a^3\,\ln \left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )}{2\,d}-\frac {8\,b^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 {2\,\left (3\,a\,b^2-b^3\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {4\,b^3}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(3*a*b^2 + b^3))/(d*(exp(2*c + 2*d*x) + 1)) - a^3*x - (atan((exp(2*c)*exp(2*d*x)*(a^3*(-d^2)^(1/2) + 2*b^3*
(-d^2)^(1/2) + 6*a*b^2*(-d^2)^(1/2) + 6*a^2*b*(-d^2)^(1/2)))/(d*(24*a*b^5 + 12*a^5*b + a^6 + 4*b^6 + 60*a^2*b^
4 + 76*a^3*b^3 + 48*a^4*b^2)^(1/2)))*(24*a*b^5 + 12*a^5*b + a^6 + 4*b^6 + 60*a^2*b^4 + 76*a^3*b^3 + 48*a^4*b^2
)^(1/2))/(-d^2)^(1/2) + (a^3*log(exp(4*c + 4*d*x) - 1))/(2*d) - (8*b^3)/(d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4
*d*x) + exp(6*c + 6*d*x) + 1)) - (2*(3*a*b^2 - b^3))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) + (4*b^3)
/(d*(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1))

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