\(\int \text {csch}^4(c+d x) (a+b \tanh ^3(c+d x))^3 \, dx\) [72]

Optimal result

Integrand size = 23, antiderivative size = 138 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {a^3 \coth (c+d x)}{d}-\frac {a^3 \coth ^3(c+d x)}{3 d}+\frac {3 a^2 b \log (\tanh (c+d x))}{d}-\frac {3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac {a b^2 \tanh ^3(c+d x)}{d}-\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^6(c+d x)}{6 d}-\frac {b^3 \tanh ^8(c+d x)}{8 d} \]

[Out]

a^3*coth(d*x+c)/d-1/3*a^3*coth(d*x+c)^3/d+3*a^2*b*ln(tanh(d*x+c))/d-3/2*a^2*b*tanh(d*x+c)^2/d+a*b^2*tanh(d*x+c
)^3/d-3/5*a*b^2*tanh(d*x+c)^5/d+1/6*b^3*tanh(d*x+c)^6/d-1/8*b^3*tanh(d*x+c)^8/d

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3744, 1816} \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=-\frac {a^3 \coth ^3(c+d x)}{3 d}+\frac {a^3 \coth (c+d x)}{d}-\frac {3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac {3 a^2 b \log (\tanh (c+d x))}{d}-\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {a b^2 \tanh ^3(c+d x)}{d}-\frac {b^3 \tanh ^8(c+d x)}{8 d}+\frac {b^3 \tanh ^6(c+d x)}{6 d} \]

[In]

Int[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^3)^3,x]

[Out]

(a^3*Coth[c + d*x])/d - (a^3*Coth[c + d*x]^3)/(3*d) + (3*a^2*b*Log[Tanh[c + d*x]])/d - (3*a^2*b*Tanh[c + d*x]^
2)/(2*d) + (a*b^2*Tanh[c + d*x]^3)/d - (3*a*b^2*Tanh[c + d*x]^5)/(5*d) + (b^3*Tanh[c + d*x]^6)/(6*d) - (b^3*Ta
nh[c + d*x]^8)/(8*d)

Rule 1816

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 3744

Int[sin[(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^(m + 1)/f), Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)
^(m/2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right ) \left (a+b x^3\right )^3}{x^4} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^3}{x^4}-\frac {a^3}{x^2}+\frac {3 a^2 b}{x}-3 a^2 b x+3 a b^2 x^2-3 a b^2 x^4+b^3 x^5-b^3 x^7\right ) \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {a^3 \coth (c+d x)}{d}-\frac {a^3 \coth ^3(c+d x)}{3 d}+\frac {3 a^2 b \log (\tanh (c+d x))}{d}-\frac {3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac {a b^2 \tanh ^3(c+d x)}{d}-\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^6(c+d x)}{6 d}-\frac {b^3 \tanh ^8(c+d x)}{8 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.74 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.54 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {2 a^3 \coth (c+d x)}{3 d}-\frac {a^3 \coth (c+d x) \text {csch}^2(c+d x)}{3 d}-\frac {3 a^2 b \log (\cosh (c+d x))}{d}+\frac {3 a^2 b \log (\sinh (c+d x))}{d}+\frac {3 a^2 b \text {sech}^2(c+d x)}{2 d}-\frac {b^3 \text {sech}^4(c+d x)}{4 d}+\frac {b^3 \text {sech}^6(c+d x)}{3 d}-\frac {b^3 \text {sech}^8(c+d x)}{8 d}+\frac {2 a b^2 \tanh (c+d x)}{5 d}+\frac {a b^2 \text {sech}^2(c+d x) \tanh (c+d x)}{5 d}-\frac {3 a b^2 \text {sech}^4(c+d x) \tanh (c+d x)}{5 d} \]

[In]

Integrate[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^3)^3,x]

[Out]

(2*a^3*Coth[c + d*x])/(3*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (3*a^2*b*Log[Cosh[c + d*x]])/d + (3*
a^2*b*Log[Sinh[c + d*x]])/d + (3*a^2*b*Sech[c + d*x]^2)/(2*d) - (b^3*Sech[c + d*x]^4)/(4*d) + (b^3*Sech[c + d*
x]^6)/(3*d) - (b^3*Sech[c + d*x]^8)/(8*d) + (2*a*b^2*Tanh[c + d*x])/(5*d) + (a*b^2*Sech[c + d*x]^2*Tanh[c + d*
x])/(5*d) - (3*a*b^2*Sech[c + d*x]^4*Tanh[c + d*x])/(5*d)

Maple [A] (verified)

Time = 62.88 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.13

method	result	size
derivativedivides	\(\frac {a^{3} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (-\frac {\sinh \left (d x +c \right )}{4 \cosh \left (d x +c \right )^{5}}+\frac {\left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{4}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{4}}{4 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )^{2}}{6 \cosh \left (d x +c \right )^{8}}-\frac {1}{24 \cosh \left (d x +c \right )^{8}}\right )}{d}\)	\(156\)
default	\(\frac {a^{3} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (-\frac {\sinh \left (d x +c \right )}{4 \cosh \left (d x +c \right )^{5}}+\frac {\left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{4}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{4}}{4 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )^{2}}{6 \cosh \left (d x +c \right )^{8}}-\frac {1}{24 \cosh \left (d x +c \right )^{8}}\right )}{d}\)	\(156\)
risch	\(-\frac {2 \left (-6 a \,b^{2}+90 a \,b^{2} {\mathrm e}^{18 d x +18 c}+230 a^{3} {\mathrm e}^{16 d x +16 c}-130 b^{3} {\mathrm e}^{16 d x +16 c}-490 b^{3} {\mathrm e}^{12 d x +12 c}-10 a^{3}-30 \,{\mathrm e}^{4 d x +4 c} b^{3}+760 a^{3} {\mathrm e}^{14 d x +14 c}+310 b^{3} {\mathrm e}^{14 d x +14 c}-30 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+45 a^{2} b \,{\mathrm e}^{2 d x +2 c}+48 a \,b^{2} {\mathrm e}^{4 d x +4 c}+135 a^{2} b \,{\mathrm e}^{4 d x +4 c}+1400 a^{3} {\mathrm e}^{12 d x +12 c}-360 a^{2} b \,{\mathrm e}^{8 d x +8 c}-108 a \,b^{2} {\mathrm e}^{8 d x +8 c}-30 a \,b^{2} {\mathrm e}^{16 d x +16 c}+360 a^{2} b \,{\mathrm e}^{14 d x +14 c}-240 a \,b^{2} {\mathrm e}^{14 d x +14 c}-270 a^{2} b \,{\mathrm e}^{10 d x +10 c}-45 a^{2} b \,{\mathrm e}^{20 d x +20 c}+490 b^{3} {\mathrm e}^{10 d x +10 c}+1540 a^{3} {\mathrm e}^{10 d x +10 c}+280 a^{3} {\mathrm e}^{6 d x +6 c}+130 \,{\mathrm e}^{6 d x +6 c} b^{3}-40 a^{3} {\mathrm e}^{4 d x +4 c}+96 a \,b^{2} {\mathrm e}^{12 d x +12 c}+270 a^{2} b \,{\mathrm e}^{12 d x +12 c}-50 a^{3} {\mathrm e}^{2 d x +2 c}-310 b^{3} {\mathrm e}^{8 d x +8 c}+980 a^{3} {\mathrm e}^{8 d x +8 c}+180 a \,b^{2} {\mathrm e}^{10 d x +10 c}+30 a^{3} {\mathrm e}^{18 d x +18 c}+30 b^{3} {\mathrm e}^{18 d x +18 c}-135 a^{2} b \,{\mathrm e}^{18 d x +18 c}\right )}{15 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{8} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}-\frac {3 b \ln \left ({\mathrm e}^{2 d x +2 c}+1\right ) a^{2}}{d}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d}\)	\(565\)

[In]

int(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^3*(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)+3*a^2*b*(1/2/cosh(d*x+c)^2+ln(tanh(d*x+c)))+3*a*b^2*(-1/4*sinh(d*
x+c)/cosh(d*x+c)^5+1/4*(8/15+1/5*sech(d*x+c)^4+4/15*sech(d*x+c)^2)*tanh(d*x+c))+b^3*(-1/4*sinh(d*x+c)^4/cosh(d
*x+c)^8-1/6*sinh(d*x+c)^2/cosh(d*x+c)^8-1/24/cosh(d*x+c)^8))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 9459 vs. \(2 (128) = 256\).

Time = 0.35 (sec) , antiderivative size = 9459, normalized size of antiderivative = 68.54 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \]

[In]

integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{3} \operatorname {csch}^{4}{\left (c + d x \right )}\, dx \]

[In]

integrate(csch(d*x+c)**4*(a+b*tanh(d*x+c)**3)**3,x)

[Out]

Integral((a + b*tanh(c + d*x)**3)**3*csch(c + d*x)**4, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (128) = 256\).

Time = 0.29 (sec) , antiderivative size = 997, normalized size of antiderivative = 7.22 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \]

[In]

integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^3,x, algorithm="maxima")

[Out]

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 + 2*e^(-2*d*x - 2*c)/
(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1))) + 4/5*a*b^2*(5*e^(-2*d*x - 2*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e
^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) - 5*e^(-4*d*x - 4*c)/(d*
(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)
) + 15*e^(-6*d*x - 6*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c
) + e^(-10*d*x - 10*c) + 1)) + 1/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*
d*x - 8*c) + e^(-10*d*x - 10*c) + 1))) + 4/3*a^3*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*
c) + e^(-6*d*x - 6*c) - 1)) - 1/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1))) - 4/3*b^
3*(3*e^(-4*d*x - 4*c)/(d*(8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c)
+ 56*e^(-10*d*x - 10*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)) - 4*e^(-6*d
*x - 6*c)/(d*(8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10
*d*x - 10*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)) + 10*e^(-8*d*x - 8*c)/(
d*(8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 10*c
) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)) - 4*e^(-10*d*x - 10*c)/(d*(8*e^(-2
*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 10*c) + 28*e^(
-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)) + 3*e^(-12*d*x - 12*c)/(d*(8*e^(-2*d*x - 2*c
) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 10*c) + 28*e^(-12*d*x -
12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (128) = 256\).

Time = 0.58 (sec) , antiderivative size = 437, normalized size of antiderivative = 3.17 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=-\frac {2520 \, a^{2} b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 2520 \, a^{2} b \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {140 \, {\left (33 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 99 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 24 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 99 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 8 \, a^{3} - 33 \, a^{2} b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}} - \frac {6849 \, a^{2} b e^{\left (16 \, d x + 16 \, c\right )} + 59832 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 222012 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} - 10080 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} - 3360 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 459144 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 26880 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 4480 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 580230 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 23520 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 11200 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 459144 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 10752 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 4480 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 222012 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 8736 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 3360 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 59832 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 5376 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6849 \, a^{2} b - 672 \, a b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{840 \, d} \]

[In]

integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^3,x, algorithm="giac")

[Out]

-1/840*(2520*a^2*b*log(e^(2*d*x + 2*c) + 1) - 2520*a^2*b*log(abs(e^(2*d*x + 2*c) - 1)) + 140*(33*a^2*b*e^(6*d*
x + 6*c) - 99*a^2*b*e^(4*d*x + 4*c) + 24*a^3*e^(2*d*x + 2*c) + 99*a^2*b*e^(2*d*x + 2*c) - 8*a^3 - 33*a^2*b)/(e
^(2*d*x + 2*c) - 1)^3 - (6849*a^2*b*e^(16*d*x + 16*c) + 59832*a^2*b*e^(14*d*x + 14*c) + 222012*a^2*b*e^(12*d*x
+ 12*c) - 10080*a*b^2*e^(12*d*x + 12*c) - 3360*b^3*e^(12*d*x + 12*c) + 459144*a^2*b*e^(10*d*x + 10*c) - 26880
*a*b^2*e^(10*d*x + 10*c) + 4480*b^3*e^(10*d*x + 10*c) + 580230*a^2*b*e^(8*d*x + 8*c) - 23520*a*b^2*e^(8*d*x +
8*c) - 11200*b^3*e^(8*d*x + 8*c) + 459144*a^2*b*e^(6*d*x + 6*c) - 10752*a*b^2*e^(6*d*x + 6*c) + 4480*b^3*e^(6*
d*x + 6*c) + 222012*a^2*b*e^(4*d*x + 4*c) - 8736*a*b^2*e^(4*d*x + 4*c) - 3360*b^3*e^(4*d*x + 4*c) + 59832*a^2*
b*e^(2*d*x + 2*c) - 5376*a*b^2*e^(2*d*x + 2*c) + 6849*a^2*b - 672*a*b^2)/(e^(2*d*x + 2*c) + 1)^8)/d

Mupad [B] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 646, normalized size of antiderivative = 4.68 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {96\,\left (10\,b^3+a\,b^2\right )}{5\,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 )}-\frac {640\,b^3}{3\,d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,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 )}-\frac {4\,\left (25\,b^3+12\,a\,b^2\right )}{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 )}+\frac {128\,b^3}{d\,\left (7\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}+35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}+7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}+1\right )}-\frac {2\,\left (3\,a^2\,b+6\,a\,b^2+2\,b^3\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {6\,\mathrm {atan}\left (\frac {a^2\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-d^2}}{d\,\sqrt {a^4\,b^2}}\right )\,\sqrt {a^4\,b^2}}{\sqrt {-d^2}}-\frac {32\,b^3}{d\,\left (8\,{\mathrm {e}}^{2\,c+2\,d\,x}+28\,{\mathrm {e}}^{4\,c+4\,d\,x}+56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}+56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}+8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1\right )}-\frac {4\,a^3}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,a^3}{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 {8\,\left (11\,b^3+15\,a\,b^2\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 {6\,a^2\,b}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]

[In]

int((a + b*tanh(c + d*x)^3)^3/sinh(c + d*x)^4,x)

[Out]

(96*(a*b^2 + 10*b^3))/(5*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
) + exp(10*c + 10*d*x) + 1)) - (640*b^3)/(3*d*(6*exp(2*c + 2*d*x) + 15*exp(4*c + 4*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)) - (4*(12*a*b^2 + 25*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)) + (128*b^3)/(d*(7*exp(2*c + 2*d*
x) + 21*exp(4*c + 4*d*x) + 35*exp(6*c + 6*d*x) + 35*exp(8*c + 8*d*x) + 21*exp(10*c + 10*d*x) + 7*exp(12*c + 12
*d*x) + exp(14*c + 14*d*x) + 1)) - (2*(6*a*b^2 + 3*a^2*b + 2*b^3))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) +
1)) - (6*atan((a^2*b*exp(2*c)*exp(2*d*x)*(-d^2)^(1/2))/(d*(a^4*b^2)^(1/2)))*(a^4*b^2)^(1/2))/(-d^2)^(1/2) - (
32*b^3)/(d*(8*exp(2*c + 2*d*x) + 28*exp(4*c + 4*d*x) + 56*exp(6*c + 6*d*x) + 70*exp(8*c + 8*d*x) + 56*exp(10*c
+ 10*d*x) + 28*exp(12*c + 12*d*x) + 8*exp(14*c + 14*d*x) + exp(16*c + 16*d*x) + 1)) - (4*a^3)/(d*(exp(4*c + 4
*d*x) - 2*exp(2*c + 2*d*x) + 1)) - (8*a^3)/(3*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) -
1)) + (8*(15*a*b^2 + 11*b^3))/(3*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) + (6*a^2*
b)/(d*(exp(2*c + 2*d*x) + 1))