3.1.70 \(\int \text {csch}^2(c+d x) (a+b \tanh ^3(c+d x))^3 \, dx\) [70]

Optimal. Leaf size=71 \[ -\frac {a^3 \coth (c+d x)}{d}+\frac {3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^8(c+d x)}{8 d} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 71, 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, 276} \begin {gather*} -\frac {a^3 \coth (c+d x)}{d}+\frac {3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^8(c+d x)}{8 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

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

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Mathematica [A]
time = 0.53, size = 113, normalized size = 1.59 \begin {gather*} \frac {-40 a^3 \coth (c+d x)+b \left (-20 b^2 \text {sech}^6(c+d x)+5 b^2 \text {sech}^8(c+d x)+24 a b \tanh (c+d x)+6 b \text {sech}^4(c+d x) (5 b+4 a \tanh (c+d x))-4 \text {sech}^2(c+d x) \left (15 a^2+5 b^2+12 a b \tanh (c+d x)\right )\right )}{40 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-40*a^3*Coth[c + d*x] + b*(-20*b^2*Sech[c + d*x]^6 + 5*b^2*Sech[c + d*x]^8 + 24*a*b*Tanh[c + d*x] + 6*b*Sech[
c + d*x]^4*(5*b + 4*a*Tanh[c + d*x]) - 4*Sech[c + d*x]^2*(15*a^2 + 5*b^2 + 12*a*b*Tanh[c + d*x])))/(40*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(507\) vs. \(2(65)=130\).
time = 3.45, size = 508, normalized size = 7.15

method	result	size
risch	\(-\frac {2 \left (-3 a \,b^{2}+30 a \,b^{2} {\mathrm e}^{14 d x +14 c}+135 a^{2} b \,{\mathrm e}^{12 d x +12 c}+30 a \,b^{2} {\mathrm e}^{12 d x +12 c}+75 a^{2} b \,{\mathrm e}^{10 d x +10 c}+30 a \,b^{2} {\mathrm e}^{10 d x +10 c}+15 a^{2} b \,{\mathrm e}^{16 d x +16 c}+15 a \,b^{2} {\mathrm e}^{16 d x +16 c}+75 a^{2} b \,{\mathrm e}^{14 d x +14 c}-75 a^{2} b \,{\mathrm e}^{4 d x +4 c}-15 a^{2} b \,{\mathrm e}^{2 d x +2 c}+5 a^{3}-12 a \,b^{2} {\mathrm e}^{8 d x +8 c}-54 a \,b^{2} {\mathrm e}^{6 d x +6 c}-30 a \,b^{2} {\mathrm e}^{4 d x +4 c}-75 a^{2} b \,{\mathrm e}^{8 d x +8 c}-6 a \,b^{2} {\mathrm e}^{2 d x +2 c}-135 a^{2} b \,{\mathrm e}^{6 d x +6 c}+5 b^{3} {\mathrm e}^{16 d x +16 c}-5 b^{3} {\mathrm e}^{14 d x +14 c}+40 a^{3} {\mathrm e}^{2 d x +2 c}+140 a^{3} {\mathrm e}^{12 d x +12 c}+35 b^{3} {\mathrm e}^{12 d x +12 c}+280 a^{3} {\mathrm e}^{10 d x +10 c}+40 a^{3} {\mathrm e}^{14 d x +14 c}-5 b^{3} {\mathrm e}^{2 d x +2 c}+5 a^{3} {\mathrm e}^{16 d x +16 c}+35 b^{3} {\mathrm e}^{8 d x +8 c}+140 a^{3} {\mathrm e}^{4 d x +4 c}+5 b^{3} {\mathrm e}^{4 d x +4 c}-35 b^{3} {\mathrm e}^{10 d x +10 c}+350 a^{3} {\mathrm e}^{8 d x +8 c}+280 a^{3} {\mathrm e}^{6 d x +6 c}-35 b^{3} {\mathrm e}^{6 d x +6 c}\right )}{5 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{8} \left ({\mathrm e}^{2 d x +2 c}-1\right )}\)	\(508\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(-3*a*b^2+30*a*b^2*exp(14*d*x+14*c)+135*a^2*b*exp(12*d*x+12*c)+30*a*b^2*exp(12*d*x+12*c)+75*a^2*b*exp(10*
d*x+10*c)+30*a*b^2*exp(10*d*x+10*c)+15*a^2*b*exp(16*d*x+16*c)+15*a*b^2*exp(16*d*x+16*c)+75*a^2*b*exp(14*d*x+14
*c)-75*a^2*b*exp(4*d*x+4*c)-15*a^2*b*exp(2*d*x+2*c)+5*a^3-12*a*b^2*exp(8*d*x+8*c)-54*a*b^2*exp(6*d*x+6*c)-30*a
*b^2*exp(4*d*x+4*c)-75*a^2*b*exp(8*d*x+8*c)-6*a*b^2*exp(2*d*x+2*c)-135*a^2*b*exp(6*d*x+6*c)+5*b^3*exp(16*d*x+1
6*c)-5*b^3*exp(14*d*x+14*c)+40*a^3*exp(2*d*x+2*c)+140*a^3*exp(12*d*x+12*c)+35*b^3*exp(12*d*x+12*c)+280*a^3*exp
(10*d*x+10*c)+40*a^3*exp(14*d*x+14*c)-5*b^3*exp(2*d*x+2*c)+5*a^3*exp(16*d*x+16*c)+35*b^3*exp(8*d*x+8*c)+140*a^
3*exp(4*d*x+4*c)+5*b^3*exp(4*d*x+4*c)-35*b^3*exp(10*d*x+10*c)+350*a^3*exp(8*d*x+8*c)+280*a^3*exp(6*d*x+6*c)-35
*b^3*exp(6*d*x+6*c))/d/(1+exp(2*d*x+2*c))^8/(exp(2*d*x+2*c)-1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (65) = 130\).
time = 0.28, size = 679, normalized size = 9.56 \begin {gather*} -2 \, b^{3} {\left (\frac {e^{\left (-2 \, d x - 2 \, 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 {7 \, 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 {7 \, 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 {e^{\left (-14 \, d x - 14 \, 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 )} + \frac {6}{5} \, a b^{2} {\left (\frac {10 \, 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 {5 \, e^{\left (-8 \, d x - 8 \, 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 {2 \, a^{3}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\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(csch(d*x+c)^2*(a+b*tanh(d*x+c)^3)^3,x, algorithm="maxima")

[Out]

-2*b^3*(e^(-2*d*x - 2*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)) + 7*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)) + 7*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)) + e^(-14*d*x - 14*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))) + 6/5*a*b^2*(10*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)) + 5
*e^(-8*d*x - 8*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))) + 2*a^3/(d*(e^(-2*d*x - 2*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. 1192 vs. \(2 (65) = 130\).
time = 0.33, size = 1192, normalized size = 16.79 \begin {gather*} -\frac {2 \, {\left ({\left (10 \, a^{3} + 15 \, a^{2} b + 12 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{8} + 8 \, {\left (15 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + {\left (10 \, a^{3} + 15 \, a^{2} b + 12 \, a b^{2} + 5 \, b^{3}\right )} \sinh \left (d x + c\right )^{8} + 2 \, {\left (40 \, a^{3} + 30 \, a^{2} b + 12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} + 2 \, {\left (40 \, a^{3} + 30 \, a^{2} b + 12 \, a b^{2} - 5 \, b^{3} + 14 \, {\left (10 \, a^{3} + 15 \, a^{2} b + 12 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{6} + 4 \, {\left (14 \, {\left (15 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 27 \, {\left (5 \, a^{2} b + 2 \, a b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 20 \, {\left (14 \, a^{3} + 3 \, a^{2} b + 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 10 \, {\left (7 \, {\left (10 \, a^{3} + 15 \, a^{2} b + 12 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 28 \, a^{3} + 6 \, a^{2} b + 4 \, b^{3} + 3 \, {\left (40 \, a^{3} + 30 \, a^{2} b + 12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, {\left (15 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 45 \, {\left (5 \, a^{2} b + 2 \, a b^{2}\right )} \cosh \left (d x + c\right )^{3} + 15 \, {\left (7 \, a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 350 \, a^{3} - 75 \, a^{2} b - 12 \, a b^{2} + 35 \, b^{3} + 2 \, {\left (280 \, a^{3} - 30 \, a^{2} b - 12 \, a b^{2} - 35 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (14 \, {\left (10 \, a^{3} + 15 \, a^{2} b + 12 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} + 15 \, {\left (40 \, a^{3} + 30 \, a^{2} b + 12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 280 \, a^{3} - 30 \, a^{2} b - 12 \, a b^{2} - 35 \, b^{3} + 60 \, {\left (14 \, a^{3} + 3 \, a^{2} b + 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (2 \, {\left (15 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 27 \, {\left (5 \, a^{2} b + 2 \, a b^{2}\right )} \cosh \left (d x + c\right )^{5} + 30 \, {\left (7 \, a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{3} + 21 \, {\left (5 \, a^{2} b + 2 \, a b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}}{5 \, {\left (d \cosh \left (d x + c\right )^{10} + 10 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{9} + d \sinh \left (d x + c\right )^{10} + 6 \, d \cosh \left (d x + c\right )^{8} + 3 \, {\left (15 \, d \cosh \left (d x + c\right )^{2} + 2 \, d\right )} \sinh \left (d x + c\right )^{8} + 8 \, {\left (15 \, d \cosh \left (d x + c\right )^{3} + 8 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{7} + 13 \, d \cosh \left (d x + c\right )^{6} + {\left (210 \, d \cosh \left (d x + c\right )^{4} + 168 \, d \cosh \left (d x + c\right )^{2} + 13 \, d\right )} \sinh \left (d x + c\right )^{6} + 2 \, {\left (126 \, d \cosh \left (d x + c\right )^{5} + 224 \, d \cosh \left (d x + c\right )^{3} + 81 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 8 \, d \cosh \left (d x + c\right )^{4} + {\left (210 \, d \cosh \left (d x + c\right )^{6} + 420 \, d \cosh \left (d x + c\right )^{4} + 195 \, d \cosh \left (d x + c\right )^{2} + 8 \, d\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (30 \, d \cosh \left (d x + c\right )^{7} + 112 \, d \cosh \left (d x + c\right )^{5} + 135 \, d \cosh \left (d x + c\right )^{3} + 48 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 14 \, d \cosh \left (d x + c\right )^{2} + {\left (45 \, d \cosh \left (d x + c\right )^{8} + 168 \, d \cosh \left (d x + c\right )^{6} + 195 \, d \cosh \left (d x + c\right )^{4} + 48 \, d \cosh \left (d x + c\right )^{2} - 14 \, d\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (5 \, d \cosh \left (d x + c\right )^{9} + 32 \, d \cosh \left (d x + c\right )^{7} + 81 \, d \cosh \left (d x + c\right )^{5} + 96 \, d \cosh \left (d x + c\right )^{3} + 42 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 14 \, d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*((10*a^3 + 15*a^2*b + 12*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 8*(15*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)*s
inh(d*x + c)^7 + (10*a^3 + 15*a^2*b + 12*a*b^2 + 5*b^3)*sinh(d*x + c)^8 + 2*(40*a^3 + 30*a^2*b + 12*a*b^2 - 5*
b^3)*cosh(d*x + c)^6 + 2*(40*a^3 + 30*a^2*b + 12*a*b^2 - 5*b^3 + 14*(10*a^3 + 15*a^2*b + 12*a*b^2 + 5*b^3)*cos
h(d*x + c)^2)*sinh(d*x + c)^6 + 4*(14*(15*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 27*(5*a^2*b + 2*a*b^2)*c
osh(d*x + c))*sinh(d*x + c)^5 + 20*(14*a^3 + 3*a^2*b + 2*b^3)*cosh(d*x + c)^4 + 10*(7*(10*a^3 + 15*a^2*b + 12*
a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 28*a^3 + 6*a^2*b + 4*b^3 + 3*(40*a^3 + 30*a^2*b + 12*a*b^2 - 5*b^3)*cosh(d*x
+ c)^2)*sinh(d*x + c)^4 + 8*(7*(15*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 45*(5*a^2*b + 2*a*b^2)*cosh(d*x
+ c)^3 + 15*(7*a^2*b + 2*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 350*a^3 - 75*a^2*b - 12*a*b^2 + 35*b^3
+ 2*(280*a^3 - 30*a^2*b - 12*a*b^2 - 35*b^3)*cosh(d*x + c)^2 + 2*(14*(10*a^3 + 15*a^2*b + 12*a*b^2 + 5*b^3)*c
osh(d*x + c)^6 + 15*(40*a^3 + 30*a^2*b + 12*a*b^2 - 5*b^3)*cosh(d*x + c)^4 + 280*a^3 - 30*a^2*b - 12*a*b^2 - 3
5*b^3 + 60*(14*a^3 + 3*a^2*b + 2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(2*(15*a^2*b + 18*a*b^2 + 5*b^3)*co
sh(d*x + c)^7 + 27*(5*a^2*b + 2*a*b^2)*cosh(d*x + c)^5 + 30*(7*a^2*b + 2*a*b^2 + b^3)*cosh(d*x + c)^3 + 21*(5*
a^2*b + 2*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^10 + 10*d*cosh(d*x + c)*sinh(d*x + c)^9 + d*si
nh(d*x + c)^10 + 6*d*cosh(d*x + c)^8 + 3*(15*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c)^8 + 8*(15*d*cosh(d*x + c)^
3 + 8*d*cosh(d*x + c))*sinh(d*x + c)^7 + 13*d*cosh(d*x + c)^6 + (210*d*cosh(d*x + c)^4 + 168*d*cosh(d*x + c)^2
+ 13*d)*sinh(d*x + c)^6 + 2*(126*d*cosh(d*x + c)^5 + 224*d*cosh(d*x + c)^3 + 81*d*cosh(d*x + c))*sinh(d*x + c
)^5 + 8*d*cosh(d*x + c)^4 + (210*d*cosh(d*x + c)^6 + 420*d*cosh(d*x + c)^4 + 195*d*cosh(d*x + c)^2 + 8*d)*sinh
(d*x + c)^4 + 4*(30*d*cosh(d*x + c)^7 + 112*d*cosh(d*x + c)^5 + 135*d*cosh(d*x + c)^3 + 48*d*cosh(d*x + c))*si
nh(d*x + c)^3 - 14*d*cosh(d*x + c)^2 + (45*d*cosh(d*x + c)^8 + 168*d*cosh(d*x + c)^6 + 195*d*cosh(d*x + c)^4 +
48*d*cosh(d*x + c)^2 - 14*d)*sinh(d*x + c)^2 + 2*(5*d*cosh(d*x + c)^9 + 32*d*cosh(d*x + c)^7 + 81*d*cosh(d*x
+ c)^5 + 96*d*cosh(d*x + c)^3 + 42*d*cosh(d*x + c))*sinh(d*x + c) - 14*d)

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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}^{2}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (65) = 130\).
time = 0.69, size = 311, normalized size = 4.38 \begin {gather*} -\frac {2 \, {\left (\frac {5 \, a^{3}}{e^{\left (2 \, d x + 2 \, c\right )} - 1} + \frac {15 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 15 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 5 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 90 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 45 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 225 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 75 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 35 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 300 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 105 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 225 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 93 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 35 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 90 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 39 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}\right )}}{5 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(5*a^3/(e^(2*d*x + 2*c) - 1) + (15*a^2*b*e^(14*d*x + 14*c) + 15*a*b^2*e^(14*d*x + 14*c) + 5*b^3*e^(14*d*x
+ 14*c) + 90*a^2*b*e^(12*d*x + 12*c) + 45*a*b^2*e^(12*d*x + 12*c) + 225*a^2*b*e^(10*d*x + 10*c) + 75*a*b^2*e^
(10*d*x + 10*c) + 35*b^3*e^(10*d*x + 10*c) + 300*a^2*b*e^(8*d*x + 8*c) + 105*a*b^2*e^(8*d*x + 8*c) + 225*a^2*b
*e^(6*d*x + 6*c) + 93*a*b^2*e^(6*d*x + 6*c) + 35*b^3*e^(6*d*x + 6*c) + 90*a^2*b*e^(4*d*x + 4*c) + 39*a*b^2*e^(
4*d*x + 4*c) + 15*a^2*b*e^(2*d*x + 2*c) + 9*a*b^2*e^(2*d*x + 2*c) + 5*b^3*e^(2*d*x + 2*c) + 3*a*b^2)/(e^(2*d*x
+ 2*c) + 1)^8)/d

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Mupad [B]
time = 1.38, size = 1515, normalized size = 21.34 \begin {gather*} \frac {\frac {-15\,a^2\,b+3\,a\,b^2+7\,b^3}{28\,d}-\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a^2\,b+3\,a\,b^2+b^3\right )}{4\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\frac {15\,a^2\,b+9\,a\,b^2-35\,b^3}{140\,d}+\frac {{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (3\,a^2\,b+3\,a\,b^2+b^3\right )}{4\,d}+\frac {3\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (9\,a^2\,b-3\,a\,b^2+7\,b^3\right )}{28\,d}-\frac {3\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (-15\,a^2\,b+3\,a\,b^2+7\,b^3\right )}{28\,d}}{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}-\frac {\frac {{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (3\,a^2\,b+3\,a\,b^2+b^3\right )}{4\,d}-\frac {9\,a^2\,b+3\,a\,b^2+7\,b^3}{28\,d}+\frac {5\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (9\,a^2\,b-3\,a\,b^2+7\,b^3\right )}{14\,d}-\frac {5\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (-15\,a^2\,b+3\,a\,b^2+7\,b^3\right )}{28\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (-15\,a^2\,b+9\,a\,b^2+35\,b^3\right )}{28\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (15\,a^2\,b+9\,a\,b^2-35\,b^3\right )}{14\,d}}{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}-\frac {\frac {9\,a^2\,b-3\,a\,b^2+7\,b^3}{28\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (3\,a^2\,b+3\,a\,b^2+b^3\right )}{4\,d}-\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (-15\,a^2\,b+3\,a\,b^2+7\,b^3\right )}{14\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}-\frac {\frac {-15\,a^2\,b+9\,a\,b^2+35\,b^3}{140\,d}+\frac {{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (3\,a^2\,b+3\,a\,b^2+b^3\right )}{4\,d}+\frac {3\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (9\,a^2\,b-3\,a\,b^2+7\,b^3\right )}{14\,d}-\frac {{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (-15\,a^2\,b+3\,a\,b^2+7\,b^3\right )}{7\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (15\,a^2\,b+9\,a\,b^2-35\,b^3\right )}{35\,d}}{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}-\frac {\frac {{\mathrm {e}}^{12\,c+12\,d\,x}\,\left (3\,a^2\,b+3\,a\,b^2+b^3\right )}{4\,d}-\frac {15\,a^2\,b+3\,a\,b^2-7\,b^3}{28\,d}-\frac {3\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (9\,a^2\,b+3\,a\,b^2+7\,b^3\right )}{14\,d}+\frac {15\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (9\,a^2\,b-3\,a\,b^2+7\,b^3\right )}{28\,d}-\frac {3\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (-15\,a^2\,b+3\,a\,b^2+7\,b^3\right )}{14\,d}+\frac {3\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (-15\,a^2\,b+9\,a\,b^2+35\,b^3\right )}{28\,d}+\frac {{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (15\,a^2\,b+9\,a\,b^2-35\,b^3\right )}{7\,d}}{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}+\frac {\frac {3\,a^2\,b-3\,a\,b^2+b^3}{4\,d}-\frac {{\mathrm {e}}^{14\,c+14\,d\,x}\,\left (3\,a^2\,b+3\,a\,b^2+b^3\right )}{4\,d}+\frac {3\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (9\,a^2\,b+3\,a\,b^2+7\,b^3\right )}{4\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (15\,a^2\,b+3\,a\,b^2-7\,b^3\right )}{4\,d}-\frac {3\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (9\,a^2\,b-3\,a\,b^2+7\,b^3\right )}{4\,d}+\frac {{\mathrm {e}}^{12\,c+12\,d\,x}\,\left (-15\,a^2\,b+3\,a\,b^2+7\,b^3\right )}{4\,d}-\frac {{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (-15\,a^2\,b+9\,a\,b^2+35\,b^3\right )}{4\,d}-\frac {{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (15\,a^2\,b+9\,a\,b^2-35\,b^3\right )}{4\,d}}{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}-\frac {2\,a^3}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {3\,a^2\,b+3\,a\,b^2+b^3}{4\,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)^2,x)

[Out]

((3*a*b^2 - 15*a^2*b + 7*b^3)/(28*d) - (exp(2*c + 2*d*x)*(3*a*b^2 + 3*a^2*b + b^3))/(4*d))/(2*exp(2*c + 2*d*x)
+ exp(4*c + 4*d*x) + 1) - ((9*a*b^2 + 15*a^2*b - 35*b^3)/(140*d) + (exp(6*c + 6*d*x)*(3*a*b^2 + 3*a^2*b + b^3
))/(4*d) + (3*exp(2*c + 2*d*x)*(9*a^2*b - 3*a*b^2 + 7*b^3))/(28*d) - (3*exp(4*c + 4*d*x)*(3*a*b^2 - 15*a^2*b +
7*b^3))/(28*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) - ((exp
(10*c + 10*d*x)*(3*a*b^2 + 3*a^2*b + b^3))/(4*d) - (3*a*b^2 + 9*a^2*b + 7*b^3)/(28*d) + (5*exp(6*c + 6*d*x)*(9
*a^2*b - 3*a*b^2 + 7*b^3))/(14*d) - (5*exp(8*c + 8*d*x)*(3*a*b^2 - 15*a^2*b + 7*b^3))/(28*d) + (exp(2*c + 2*d*
x)*(9*a*b^2 - 15*a^2*b + 35*b^3))/(28*d) + (exp(4*c + 4*d*x)*(9*a*b^2 + 15*a^2*b - 35*b^3))/(14*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) - ((9*a^2*b - 3*a*b^2 + 7*b^3)/(28*d) + (exp(4*c + 4*d*x)*(3*a*b^2 + 3*a^2*b + b^3))/(4*d) - (e
xp(2*c + 2*d*x)*(3*a*b^2 - 15*a^2*b + 7*b^3))/(14*d))/(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d
*x) + 1) - ((9*a*b^2 - 15*a^2*b + 35*b^3)/(140*d) + (exp(8*c + 8*d*x)*(3*a*b^2 + 3*a^2*b + b^3))/(4*d) + (3*ex
p(4*c + 4*d*x)*(9*a^2*b - 3*a*b^2 + 7*b^3))/(14*d) - (exp(6*c + 6*d*x)*(3*a*b^2 - 15*a^2*b + 7*b^3))/(7*d) + (
exp(2*c + 2*d*x)*(9*a*b^2 + 15*a^2*b - 35*b^3))/(35*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) - ((exp(12*c + 12*d*x)*(3*a*b^2 + 3*a^2*b + b^3))/(4*
d) - (3*a*b^2 + 15*a^2*b - 7*b^3)/(28*d) - (3*exp(2*c + 2*d*x)*(3*a*b^2 + 9*a^2*b + 7*b^3))/(14*d) + (15*exp(8
*c + 8*d*x)*(9*a^2*b - 3*a*b^2 + 7*b^3))/(28*d) - (3*exp(10*c + 10*d*x)*(3*a*b^2 - 15*a^2*b + 7*b^3))/(14*d) +
(3*exp(4*c + 4*d*x)*(9*a*b^2 - 15*a^2*b + 35*b^3))/(28*d) + (exp(6*c + 6*d*x)*(9*a*b^2 + 15*a^2*b - 35*b^3))/
(7*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 + 1
0*d*x) + 7*exp(12*c + 12*d*x) + exp(14*c + 14*d*x) + 1) + ((3*a^2*b - 3*a*b^2 + b^3)/(4*d) - (exp(14*c + 14*d*
x)*(3*a*b^2 + 3*a^2*b + b^3))/(4*d) + (3*exp(4*c + 4*d*x)*(3*a*b^2 + 9*a^2*b + 7*b^3))/(4*d) + (exp(2*c + 2*d*
x)*(3*a*b^2 + 15*a^2*b - 7*b^3))/(4*d) - (3*exp(10*c + 10*d*x)*(9*a^2*b - 3*a*b^2 + 7*b^3))/(4*d) + (exp(12*c
+ 12*d*x)*(3*a*b^2 - 15*a^2*b + 7*b^3))/(4*d) - (exp(6*c + 6*d*x)*(9*a*b^2 - 15*a^2*b + 35*b^3))/(4*d) - (exp(
8*c + 8*d*x)*(9*a*b^2 + 15*a^2*b - 35*b^3))/(4*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) - (2*a^3)/(d*(exp(2*c + 2*d*x) - 1)) - (3*a*b^2 + 3*a^2*b + b^3)/(4*d*(exp(2*c + 2*d*x) + 1))

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