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

Optimal. Leaf size=138 \[ \frac {a^3 \coth (c+d x)}{d}-\frac {a^3 \coth ^3(c+d x)}{3 d}+\frac {3 a^2 b \log (\tanh (c+d x))}{d}-\frac {3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac {a b^2 \tanh ^3(c+d x)}{d}-\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^6(c+d x)}{6 d}-\frac {b^3 \tanh ^8(c+d x)}{8 d} \]

[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

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Rubi [A]
time = 0.08, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3744, 1816} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.14, size = 213, normalized size = 1.54 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(564\) vs. \(2(128)=256\).
time = 3.69, size = 565, normalized size = 4.09

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(-6*a*b^2-240*a*b^2*exp(14*d*x+14*c)+270*a^2*b*exp(12*d*x+12*c)+96*a*b^2*exp(12*d*x+12*c)-270*a^2*b*exp(
10*d*x+10*c)+180*a*b^2*exp(10*d*x+10*c)+90*a*b^2*exp(18*d*x+18*c)-30*a*b^2*exp(16*d*x+16*c)+360*a^2*b*exp(14*d
*x+14*c)+135*a^2*b*exp(4*d*x+4*c)+45*a^2*b*exp(2*d*x+2*c)-10*a^3-108*a*b^2*exp(8*d*x+8*c)+48*a*b^2*exp(4*d*x+4
*c)-360*a^2*b*exp(8*d*x+8*c)-30*a*b^2*exp(2*d*x+2*c)-135*a^2*b*exp(18*d*x+18*c)+30*b^3*exp(18*d*x+18*c)-130*b^
3*exp(16*d*x+16*c)+310*b^3*exp(14*d*x+14*c)-50*a^3*exp(2*d*x+2*c)-45*a^2*b*exp(20*d*x+20*c)+1400*a^3*exp(12*d*
x+12*c)-490*b^3*exp(12*d*x+12*c)+1540*a^3*exp(10*d*x+10*c)+760*a^3*exp(14*d*x+14*c)+230*a^3*exp(16*d*x+16*c)-3
10*b^3*exp(8*d*x+8*c)-40*a^3*exp(4*d*x+4*c)-30*b^3*exp(4*d*x+4*c)+30*a^3*exp(18*d*x+18*c)+490*b^3*exp(10*d*x+1
0*c)+980*a^3*exp(8*d*x+8*c)+280*a^3*exp(6*d*x+6*c)+130*b^3*exp(6*d*x+6*c))/d/(1+exp(2*d*x+2*c))^8/(exp(2*d*x+2
*c)-1)^3+3*a^2*b/d*ln(exp(2*d*x+2*c)-1)-3*b/d*ln(1+exp(2*d*x+2*c))*a^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (128) = 256\).
time = 0.50, size = 997, normalized size = 7.22 \begin {gather*} 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} + \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 {4}{5} \, a b^{2} {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} - \frac {5 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {15 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + \frac {4}{3} \, a^{3} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} - \frac {4}{3} \, b^{3} {\left (\frac {3 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}} - \frac {4 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}} + \frac {10 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}} - \frac {4 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}} + \frac {3 \, e^{\left (-12 \, d x - 12 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 9459 vs. \(2 (128) = 256\).
time = 0.48, size = 9459, normalized size = 68.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(90*a^2*b*cosh(d*x + c)^20 + 1800*a^2*b*cosh(d*x + c)*sinh(d*x + c)^19 + 90*a^2*b*sinh(d*x + c)^20 - 30*(
2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^18 + 30*(570*a^2*b*cosh(d*x + c)^2 - 2*a^3 + 9*a^2*b - 6*a*b^
2 - 2*b^3)*sinh(d*x + c)^18 + 540*(190*a^2*b*cosh(d*x + c)^3 - (2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x +
c))*sinh(d*x + c)^17 - 20*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^16 + 10*(43605*a^2*b*cosh(d*x + c)^4 - 46*
a^3 + 6*a*b^2 + 26*b^3 - 459*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^16 + 160*(8721
*a^2*b*cosh(d*x + c)^5 - 153*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^3 - 2*(23*a^3 - 3*a*b^2 - 13*b^
3)*cosh(d*x + c))*sinh(d*x + c)^15 - 20*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^14 + 20*(174420*
a^2*b*cosh(d*x + c)^6 - 4590*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^4 - 76*a^3 - 36*a^2*b + 24*a*b^
2 - 31*b^3 - 120*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^14 + 40*(174420*a^2*b*cosh(d*x + c
)^7 - 6426*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^5 - 280*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)
^3 - 7*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c))*sinh(d*x + c)^13 - 4*(700*a^3 + 135*a^2*b + 48*a
*b^2 - 245*b^3)*cosh(d*x + c)^12 + 4*(2834325*a^2*b*cosh(d*x + c)^8 - 139230*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^
3)*cosh(d*x + c)^6 - 9100*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^4 - 700*a^3 - 135*a^2*b - 48*a*b^2 + 245*b
^3 - 455*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^12 + 16*(944775*a^2*b*cosh(d*x
+ c)^9 - 59670*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^7 - 5460*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*
x + c)^5 - 455*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^3 - 3*(700*a^3 + 135*a^2*b + 48*a*b^2 - 2
45*b^3)*cosh(d*x + c))*sinh(d*x + c)^11 - 20*(154*a^3 - 27*a^2*b + 18*a*b^2 + 49*b^3)*cosh(d*x + c)^10 + 4*(41
57010*a^2*b*cosh(d*x + c)^10 - 328185*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^8 - 40040*(23*a^3 - 3*
a*b^2 - 13*b^3)*cosh(d*x + c)^6 - 5005*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^4 - 770*a^3 + 135
*a^2*b - 90*a*b^2 - 245*b^3 - 66*(700*a^3 + 135*a^2*b + 48*a*b^2 - 245*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10
+ 40*(377910*a^2*b*cosh(d*x + c)^11 - 36465*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^9 - 5720*(23*a^3
- 3*a*b^2 - 13*b^3)*cosh(d*x + c)^7 - 1001*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^5 - 22*(700*
a^3 + 135*a^2*b + 48*a*b^2 - 245*b^3)*cosh(d*x + c)^3 - 5*(154*a^3 - 27*a^2*b + 18*a*b^2 + 49*b^3)*cosh(d*x +
c))*sinh(d*x + c)^9 - 4*(490*a^3 - 180*a^2*b - 54*a*b^2 - 155*b^3)*cosh(d*x + c)^8 + 4*(2834325*a^2*b*cosh(d*x
+ c)^12 - 328185*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^10 - 64350*(23*a^3 - 3*a*b^2 - 13*b^3)*cos
h(d*x + c)^8 - 15015*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^6 - 495*(700*a^3 + 135*a^2*b + 48*a
*b^2 - 245*b^3)*cosh(d*x + c)^4 - 490*a^3 + 180*a^2*b + 54*a*b^2 + 155*b^3 - 225*(154*a^3 - 27*a^2*b + 18*a*b^
2 + 49*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 32*(218025*a^2*b*cosh(d*x + c)^13 - 29835*(2*a^3 - 9*a^2*b + 6*
a*b^2 + 2*b^3)*cosh(d*x + c)^11 - 7150*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^9 - 2145*(76*a^3 + 36*a^2*b -
24*a*b^2 + 31*b^3)*cosh(d*x + c)^7 - 99*(700*a^3 + 135*a^2*b + 48*a*b^2 - 245*b^3)*cosh(d*x + c)^5 - 75*(154*
a^3 - 27*a^2*b + 18*a*b^2 + 49*b^3)*cosh(d*x + c)^3 - (490*a^3 - 180*a^2*b - 54*a*b^2 - 155*b^3)*cosh(d*x + c)
)*sinh(d*x + c)^7 - 20*(28*a^3 + 13*b^3)*cosh(d*x + c)^6 + 4*(872100*a^2*b*cosh(d*x + c)^14 - 139230*(2*a^3 -
9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^12 - 40040*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^10 - 15015*(76*a
^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^8 - 924*(700*a^3 + 135*a^2*b + 48*a*b^2 - 245*b^3)*cosh(d*x +
c)^6 - 1050*(154*a^3 - 27*a^2*b + 18*a*b^2 + 49*b^3)*cosh(d*x + c)^4 - 140*a^3 - 65*b^3 - 28*(490*a^3 - 180*a
^2*b - 54*a*b^2 - 155*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(174420*a^2*b*cosh(d*x + c)^15 - 32130*(2*a^3
- 9*a^2*b + 6*a*b^2 + 2*b^3)*cosh(d*x + c)^13 - 10920*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^11 - 5005*(76*
a^3 + 36*a^2*b - 24*a*b^2 + 31*b^3)*cosh(d*x + c)^9 - 396*(700*a^3 + 135*a^2*b + 48*a*b^2 - 245*b^3)*cosh(d*x
+ c)^7 - 630*(154*a^3 - 27*a^2*b + 18*a*b^2 + 49*b^3)*cosh(d*x + c)^5 - 28*(490*a^3 - 180*a^2*b - 54*a*b^2 - 1
55*b^3)*cosh(d*x + c)^3 - 15*(28*a^3 + 13*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(40*a^3 - 135*a^2*b - 48*a*b
^2 + 30*b^3)*cosh(d*x + c)^4 + 2*(218025*a^2*b*cosh(d*x + c)^16 - 45900*(2*a^3 - 9*a^2*b + 6*a*b^2 + 2*b^3)*co
sh(d*x + c)^14 - 18200*(23*a^3 - 3*a*b^2 - 13*b^3)*cosh(d*x + c)^12 - 10010*(76*a^3 + 36*a^2*b - 24*a*b^2 + 31
*b^3)*cosh(d*x + c)^10 - 990*(700*a^3 + 135*a^2*b + 48*a*b^2 - 245*b^3)*cosh(d*x + c)^8 - 2100*(154*a^3 - 27*a
^2*b + 18*a*b^2 + 49*b^3)*cosh(d*x + c)^6 - 140*(490*a^3 - 180*a^2*b - 54*a*b^2 - 155*b^3)*cosh(d*x + c)^4 + 4
0*a^3 - 135*a^2*b - 48*a*b^2 + 30*b^3 - 150*(28*a^3 + 13*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(12825*a^2*
b*cosh(d*x + c)^17 - 3060*(2*a^3 - 9*a^2*b + 6*...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{3} \operatorname {csch}^{4}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (128) = 256\).
time = 0.67, size = 437, normalized size = 3.17 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Mupad [B]
time = 0.54, size = 646, normalized size = 4.68 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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