3.1.24 \(\int \text {csch}^4(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx\) [24]

Optimal. Leaf size=98 \[ \frac {a^2 (a-3 b) \coth (c+d x)}{d}-\frac {a^3 \coth ^3(c+d x)}{3 d}-\frac {3 a (a-b) b \tanh (c+d x)}{d}-\frac {(3 a-b) b^2 \tanh ^3(c+d x)}{3 d}-\frac {b^3 \tanh ^5(c+d x)}{5 d} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 98, 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, 459} \begin {gather*} -\frac {a^3 \coth ^3(c+d x)}{3 d}+\frac {a^2 (a-3 b) \coth (c+d x)}{d}-\frac {b^2 (3 a-b) \tanh ^3(c+d x)}{3 d}-\frac {3 a b (a-b) \tanh (c+d x)}{d}-\frac {b^3 \tanh ^5(c+d x)}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

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

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 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}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right ) \left (a+b x^2\right )^3}{x^4} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-3 a (a-b) b+\frac {a^3}{x^4}-\frac {a^2 (a-3 b)}{x^2}-(3 a-b) b^2 x^2-b^3 x^4\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {a^2 (a-3 b) \coth (c+d x)}{d}-\frac {a^3 \coth ^3(c+d x)}{3 d}-\frac {3 a (a-b) b \tanh (c+d x)}{d}-\frac {(3 a-b) b^2 \tanh ^3(c+d x)}{3 d}-\frac {b^3 \tanh ^5(c+d x)}{5 d}\\ \end {align*}

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Mathematica [A]
time = 0.85, size = 87, normalized size = 0.89 \begin {gather*} \frac {-5 a^2 \coth (c+d x) \left (-2 a+9 b+a \text {csch}^2(c+d x)\right )+b \left (-45 a^2+30 a b+2 b^2+b (15 a+b) \text {sech}^2(c+d x)-3 b^2 \text {sech}^4(c+d x)\right ) \tanh (c+d x)}{15 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

(-5*a^2*Coth[c + d*x]*(-2*a + 9*b + a*Csch[c + d*x]^2) + b*(-45*a^2 + 30*a*b + 2*b^2 + b*(15*a + b)*Sech[c + d
*x]^2 - 3*b^2*Sech[c + d*x]^4)*Tanh[c + d*x])/(15*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(402\) vs. \(2(92)=184\).
time = 2.62, size = 403, normalized size = 4.11

method	result	size
risch	\(-\frac {4 \left (-15 a \,b^{2}+45 a^{2} b \,{\mathrm e}^{12 d x +12 c}+45 a \,b^{2} {\mathrm e}^{12 d x +12 c}+90 a^{2} b \,{\mathrm e}^{10 d x +10 c}-30 a \,b^{2} {\mathrm e}^{10 d x +10 c}-45 a^{2} b \,{\mathrm e}^{4 d x +4 c}+90 a^{2} b \,{\mathrm e}^{2 d x +2 c}+45 a^{2} b -5 a^{3}-b^{3}-105 a \,b^{2} {\mathrm e}^{8 d x +8 c}+60 a \,b^{2} {\mathrm e}^{6 d x +6 c}+75 a \,b^{2} {\mathrm e}^{4 d x +4 c}-45 a^{2} b \,{\mathrm e}^{8 d x +8 c}-30 a \,b^{2} {\mathrm e}^{2 d x +2 c}-180 a^{2} b \,{\mathrm e}^{6 d x +6 c}-10 a^{3} {\mathrm e}^{2 d x +2 c}+15 a^{3} {\mathrm e}^{12 d x +12 c}+15 b^{3} {\mathrm e}^{12 d x +12 c}+70 a^{3} {\mathrm e}^{10 d x +10 c}-2 b^{3} {\mathrm e}^{2 d x +2 c}+65 b^{3} {\mathrm e}^{8 d x +8 c}+25 a^{3} {\mathrm e}^{4 d x +4 c}+17 b^{3} {\mathrm e}^{4 d x +4 c}-50 b^{3} {\mathrm e}^{10 d x +10 c}+125 a^{3} {\mathrm e}^{8 d x +8 c}+100 a^{3} {\mathrm e}^{6 d x +6 c}-44 b^{3} {\mathrm e}^{6 d x +6 c}\right )}{15 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{5} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\)	\(403\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^4*(a+b*tanh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

-4/15*(-15*a*b^2+45*a^2*b*exp(12*d*x+12*c)+45*a*b^2*exp(12*d*x+12*c)+90*a^2*b*exp(10*d*x+10*c)-30*a*b^2*exp(10
*d*x+10*c)-45*a^2*b*exp(4*d*x+4*c)+90*a^2*b*exp(2*d*x+2*c)+45*a^2*b-5*a^3-b^3-105*a*b^2*exp(8*d*x+8*c)+60*a*b^
2*exp(6*d*x+6*c)+75*a*b^2*exp(4*d*x+4*c)-45*a^2*b*exp(8*d*x+8*c)-30*a*b^2*exp(2*d*x+2*c)-180*a^2*b*exp(6*d*x+6
*c)-10*a^3*exp(2*d*x+2*c)+15*a^3*exp(12*d*x+12*c)+15*b^3*exp(12*d*x+12*c)+70*a^3*exp(10*d*x+10*c)-2*b^3*exp(2*
d*x+2*c)+65*b^3*exp(8*d*x+8*c)+25*a^3*exp(4*d*x+4*c)+17*b^3*exp(4*d*x+4*c)-50*b^3*exp(10*d*x+10*c)+125*a^3*exp
(8*d*x+8*c)+100*a^3*exp(6*d*x+6*c)-44*b^3*exp(6*d*x+6*c))/d/(1+exp(2*d*x+2*c))^5/(exp(2*d*x+2*c)-1)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (92) = 184\).
time = 0.30, size = 493, normalized size = 5.03 \begin {gather*} \frac {4}{15} \, b^{3} {\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 )} + 4 \, a b^{2} {\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} \, 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 {12 \, a^{2} b}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} - 1\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)^2)^3,x, algorithm="maxima")

[Out]

4/15*b^3*(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*a*b^2*(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*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)
)) + 12*a^2*b/(d*(e^(-4*d*x - 4*c) - 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 925 vs. \(2 (92) = 184\).
time = 0.35, size = 925, normalized size = 9.44 \begin {gather*} -\frac {8 \, {\left ({\left (5 \, a^{3} + 45 \, a^{2} b + 15 \, a b^{2} + 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} + 12 \, {\left (5 \, a^{3} + 15 \, a b^{2} + 4 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + {\left (5 \, a^{3} + 45 \, a^{2} b + 15 \, a b^{2} + 7 \, b^{3}\right )} \sinh \left (d x + c\right )^{6} + 2 \, {\left (15 \, a^{3} + 45 \, a^{2} b - 15 \, a b^{2} - 13 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + {\left (30 \, a^{3} + 90 \, a^{2} b - 30 \, a b^{2} - 26 \, b^{3} + 15 \, {\left (5 \, a^{3} + 45 \, a^{2} b + 15 \, a b^{2} + 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (5 \, {\left (5 \, a^{3} + 15 \, a b^{2} + 4 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 4 \, {\left (5 \, a^{3} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 50 \, a^{3} - 90 \, a^{2} b + 30 \, a b^{2} - 22 \, b^{3} + {\left (75 \, a^{3} - 45 \, a^{2} b - 15 \, a b^{2} + 41 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + {\left (15 \, {\left (5 \, a^{3} + 45 \, a^{2} b + 15 \, a b^{2} + 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 75 \, a^{3} - 45 \, a^{2} b - 15 \, a b^{2} + 41 \, b^{3} + 12 \, {\left (15 \, a^{3} + 45 \, a^{2} b - 15 \, a b^{2} - 13 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (3 \, {\left (5 \, a^{3} + 15 \, a b^{2} + 4 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 8 \, {\left (5 \, a^{3} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (25 \, a^{3} - 45 \, a b^{2} + 12 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}}{15 \, {\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} + 2 \, d \cosh \left (d x + c\right )^{8} + {\left (45 \, 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} + 2 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{7} - 3 \, d \cosh \left (d x + c\right )^{6} + {\left (210 \, d \cosh \left (d x + c\right )^{4} + 56 \, d \cosh \left (d x + c\right )^{2} - 3 \, d\right )} \sinh \left (d x + c\right )^{6} + 2 \, {\left (126 \, d \cosh \left (d x + c\right )^{5} + 56 \, d \cosh \left (d x + c\right )^{3} - 3 \, 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} + 140 \, d \cosh \left (d x + c\right )^{4} - 45 \, 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} + 28 \, d \cosh \left (d x + c\right )^{5} - 5 \, d \cosh \left (d x + c\right )^{3} - 4 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 2 \, d \cosh \left (d x + c\right )^{2} + {\left (45 \, d \cosh \left (d x + c\right )^{8} + 56 \, d \cosh \left (d x + c\right )^{6} - 45 \, d \cosh \left (d x + c\right )^{4} - 48 \, d \cosh \left (d x + c\right )^{2} + 2 \, d\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (5 \, d \cosh \left (d x + c\right )^{9} + 8 \, d \cosh \left (d x + c\right )^{7} - 3 \, d \cosh \left (d x + c\right )^{5} - 8 \, d \cosh \left (d x + c\right )^{3} - 2 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 6 \, d\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)^2)^3,x, algorithm="fricas")

[Out]

-8/15*((5*a^3 + 45*a^2*b + 15*a*b^2 + 7*b^3)*cosh(d*x + c)^6 + 12*(5*a^3 + 15*a*b^2 + 4*b^3)*cosh(d*x + c)*sin
h(d*x + c)^5 + (5*a^3 + 45*a^2*b + 15*a*b^2 + 7*b^3)*sinh(d*x + c)^6 + 2*(15*a^3 + 45*a^2*b - 15*a*b^2 - 13*b^
3)*cosh(d*x + c)^4 + (30*a^3 + 90*a^2*b - 30*a*b^2 - 26*b^3 + 15*(5*a^3 + 45*a^2*b + 15*a*b^2 + 7*b^3)*cosh(d*
x + c)^2)*sinh(d*x + c)^4 + 8*(5*(5*a^3 + 15*a*b^2 + 4*b^3)*cosh(d*x + c)^3 + 4*(5*a^3 - 3*b^3)*cosh(d*x + c))
*sinh(d*x + c)^3 + 50*a^3 - 90*a^2*b + 30*a*b^2 - 22*b^3 + (75*a^3 - 45*a^2*b - 15*a*b^2 + 41*b^3)*cosh(d*x +
c)^2 + (15*(5*a^3 + 45*a^2*b + 15*a*b^2 + 7*b^3)*cosh(d*x + c)^4 + 75*a^3 - 45*a^2*b - 15*a*b^2 + 41*b^3 + 12*
(15*a^3 + 45*a^2*b - 15*a*b^2 - 13*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(3*(5*a^3 + 15*a*b^2 + 4*b^3)*cos
h(d*x + c)^5 + 8*(5*a^3 - 3*b^3)*cosh(d*x + c)^3 + (25*a^3 - 45*a*b^2 + 12*b^3)*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*sinh(d*x + c)^10 + 2*d*cosh(d*x + c)^8 + (45*d*co
sh(d*x + c)^2 + 2*d)*sinh(d*x + c)^8 + 8*(15*d*cosh(d*x + c)^3 + 2*d*cosh(d*x + c))*sinh(d*x + c)^7 - 3*d*cosh
(d*x + c)^6 + (210*d*cosh(d*x + c)^4 + 56*d*cosh(d*x + c)^2 - 3*d)*sinh(d*x + c)^6 + 2*(126*d*cosh(d*x + c)^5
+ 56*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))*sinh(d*x + c)^5 - 8*d*cosh(d*x + c)^4 + (210*d*cosh(d*x + c)^6 + 1
40*d*cosh(d*x + c)^4 - 45*d*cosh(d*x + c)^2 - 8*d)*sinh(d*x + c)^4 + 4*(30*d*cosh(d*x + c)^7 + 28*d*cosh(d*x +
c)^5 - 5*d*cosh(d*x + c)^3 - 4*d*cosh(d*x + c))*sinh(d*x + c)^3 + 2*d*cosh(d*x + c)^2 + (45*d*cosh(d*x + c)^8
+ 56*d*cosh(d*x + c)^6 - 45*d*cosh(d*x + c)^4 - 48*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c)^2 + 2*(5*d*cosh(d*x
+ c)^9 + 8*d*cosh(d*x + c)^7 - 3*d*cosh(d*x + c)^5 - 8*d*cosh(d*x + c)^3 - 2*d*cosh(d*x + c))*sinh(d*x + c) +
6*d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tanh ^{2}{\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)**2)**3,x)

[Out]

Integral((a + b*tanh(c + d*x)**2)**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. 257 vs. \(2 (92) = 184\).
time = 0.54, size = 257, normalized size = 2.62 \begin {gather*} -\frac {2 \, {\left (\frac {5 \, {\left (9 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 18 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a^{3} + 9 \, a^{2} b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}} - \frac {45 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 180 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 90 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 30 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 270 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 210 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 10 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 180 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 150 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 10 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 45 \, a^{2} b - 30 \, a b^{2} - 2 \, b^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}\right )}}{15 \, 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)^2)^3,x, algorithm="giac")

[Out]

-2/15*(5*(9*a^2*b*e^(4*d*x + 4*c) + 6*a^3*e^(2*d*x + 2*c) - 18*a^2*b*e^(2*d*x + 2*c) - 2*a^3 + 9*a^2*b)/(e^(2*
d*x + 2*c) - 1)^3 - (45*a^2*b*e^(8*d*x + 8*c) + 180*a^2*b*e^(6*d*x + 6*c) - 90*a*b^2*e^(6*d*x + 6*c) - 30*b^3*
e^(6*d*x + 6*c) + 270*a^2*b*e^(4*d*x + 4*c) - 210*a*b^2*e^(4*d*x + 4*c) + 10*b^3*e^(4*d*x + 4*c) + 180*a^2*b*e
^(2*d*x + 2*c) - 150*a*b^2*e^(2*d*x + 2*c) - 10*b^3*e^(2*d*x + 2*c) + 45*a^2*b - 30*a*b^2 - 2*b^3)/(e^(2*d*x +
2*c) + 1)^5)/d

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Mupad [B]
time = 0.28, size = 622, normalized size = 6.35 \begin {gather*} \frac {\frac {2\,\left (9\,a^2\,b-12\,a\,b^2+4\,b^3\right )}{15\,d}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (-3\,a^2\,b+3\,a\,b^2+b^3\right )}{5\,d}+\frac {6\,a^2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{5\,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 {2\,\left (-3\,a^2\,b+3\,a\,b^2+b^3\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (-3\,a^2\,b+3\,a\,b^2+b^3\right )}{5\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (9\,a^2\,b-12\,a\,b^2+4\,b^3\right )}{5\,d}-\frac {6\,a^2\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{5\,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 {6\,a^2\,b}{5\,d}-\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (-3\,a^2\,b+3\,a\,b^2+b^3\right )}{5\,d}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (-3\,a^2\,b+3\,a\,b^2+b^3\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (9\,a^2\,b-12\,a\,b^2+4\,b^3\right )}{5\,d}+\frac {6\,a^2\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}}{5\,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 {2\,\left (-3\,a^2\,b+3\,a\,b^2+b^3\right )}{5\,d}-\frac {6\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{5\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\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 {6\,a^2\,b}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {6\,a^2\,b}{5\,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)^2)^3/sinh(c + d*x)^4,x)

[Out]

((2*(9*a^2*b - 12*a*b^2 + 4*b^3))/(15*d) - (4*exp(2*c + 2*d*x)*(3*a*b^2 - 3*a^2*b + b^3))/(5*d) + (6*a^2*b*exp
(4*c + 4*d*x))/(5*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 - 3*a^2*
b + b^3))/(5*d) + (6*exp(4*c + 4*d*x)*(3*a*b^2 - 3*a^2*b + b^3))/(5*d) - (2*exp(2*c + 2*d*x)*(9*a^2*b - 12*a*b
^2 + 4*b^3))/(5*d) - (6*a^2*b*exp(6*c + 6*d*x))/(5*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) + ((6*a^2*b)/(5*d) - (8*exp(6*c + 6*d*x)*(3*a*b^2 - 3*a^2*b + b^3))/(5*d) - (8*
exp(2*c + 2*d*x)*(3*a*b^2 - 3*a^2*b + b^3))/(5*d) + (4*exp(4*c + 4*d*x)*(9*a^2*b - 12*a*b^2 + 4*b^3))/(5*d) +
(6*a^2*b*exp(8*c + 8*d*x))/(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) - ((2*(3*a*b^2 - 3*a^2*b + b^3))/(5*d) - (6*a^2*b*exp(2*c + 2*d*x))/(5*d))/
(2*exp(2*c + 2*d*x) + exp(4*c + 4*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)) - (6*a^2*b)/(d*(exp(2*c + 2*d*x) - 1)
) + (6*a^2*b)/(5*d*(exp(2*c + 2*d*x) + 1))

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