3.2.34 \(\int \coth ^6(c+d x) (a+b \text {sech}^2(c+d x))^3 \, dx\) [134]

Optimal. Leaf size=69 \[ a^3 x-\frac {\left (a^3+b^3\right ) \coth (c+d x)}{d}-\frac {(a-2 b) (a+b)^2 \coth ^3(c+d x)}{3 d}-\frac {(a+b)^3 \coth ^5(c+d x)}{5 d} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4226, 1816, 213} \begin {gather*} -\frac {\left (a^3+b^3\right ) \coth (c+d x)}{d}+a^3 x-\frac {(a+b)^3 \coth ^5(c+d x)}{5 d}-\frac {(a-2 b) (a+b)^2 \coth ^3(c+d x)}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[c + d*x]^6*(a + b*Sech[c + d*x]^2)^3,x]

[Out]

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

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

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 4226

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2
*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] ||
EqQ[n, 2])

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(303\) vs. \(2(69)=138\).
time = 0.72, size = 303, normalized size = 4.39 \begin {gather*} \frac {\text {csch}(c) \text {csch}^5(c+d x) \left (-150 a^3 d x \cosh (d x)+150 a^3 d x \cosh (2 c+d x)+75 a^3 d x \cosh (2 c+3 d x)-75 a^3 d x \cosh (4 c+3 d x)-15 a^3 d x \cosh (4 c+5 d x)+15 a^3 d x \cosh (6 c+5 d x)+280 a^3 \sinh (d x)+180 a^2 b \sinh (d x)+60 a b^2 \sinh (d x)+160 b^3 \sinh (d x)+180 a^3 \sinh (2 c+d x)-180 a b^2 \sinh (2 c+d x)-140 a^3 \sinh (2 c+3 d x)+60 a b^2 \sinh (2 c+3 d x)-80 b^3 \sinh (2 c+3 d x)-90 a^3 \sinh (4 c+3 d x)-90 a^2 b \sinh (4 c+3 d x)+46 a^3 \sinh (4 c+5 d x)+18 a^2 b \sinh (4 c+5 d x)-12 a b^2 \sinh (4 c+5 d x)+16 b^3 \sinh (4 c+5 d x)\right )}{480 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[c + d*x]^6*(a + b*Sech[c + d*x]^2)^3,x]

[Out]

(Csch[c]*Csch[c + d*x]^5*(-150*a^3*d*x*Cosh[d*x] + 150*a^3*d*x*Cosh[2*c + d*x] + 75*a^3*d*x*Cosh[2*c + 3*d*x]
- 75*a^3*d*x*Cosh[4*c + 3*d*x] - 15*a^3*d*x*Cosh[4*c + 5*d*x] + 15*a^3*d*x*Cosh[6*c + 5*d*x] + 280*a^3*Sinh[d*
x] + 180*a^2*b*Sinh[d*x] + 60*a*b^2*Sinh[d*x] + 160*b^3*Sinh[d*x] + 180*a^3*Sinh[2*c + d*x] - 180*a*b^2*Sinh[2
*c + d*x] - 140*a^3*Sinh[2*c + 3*d*x] + 60*a*b^2*Sinh[2*c + 3*d*x] - 80*b^3*Sinh[2*c + 3*d*x] - 90*a^3*Sinh[4*
c + 3*d*x] - 90*a^2*b*Sinh[4*c + 3*d*x] + 46*a^3*Sinh[4*c + 5*d*x] + 18*a^2*b*Sinh[4*c + 5*d*x] - 12*a*b^2*Sin
h[4*c + 5*d*x] + 16*b^3*Sinh[4*c + 5*d*x]))/(480*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(206\) vs. \(2(65)=130\).
time = 2.83, size = 207, normalized size = 3.00

method	result	size
risch	\(a^{3} x -\frac {2 \left (45 a^{3} {\mathrm e}^{8 d x +8 c}+45 a^{2} b \,{\mathrm e}^{8 d x +8 c}-90 a^{3} {\mathrm e}^{6 d x +6 c}+90 a \,b^{2} {\mathrm e}^{6 d x +6 c}+140 a^{3} {\mathrm e}^{4 d x +4 c}+90 a^{2} b \,{\mathrm e}^{4 d x +4 c}+30 a \,b^{2} {\mathrm e}^{4 d x +4 c}+80 b^{3} {\mathrm e}^{4 d x +4 c}-70 a^{3} {\mathrm e}^{2 d x +2 c}+30 a \,b^{2} {\mathrm e}^{2 d x +2 c}-40 b^{3} {\mathrm e}^{2 d x +2 c}+23 a^{3}+9 a^{2} b -6 a \,b^{2}+8 b^{3}\right )}{15 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{5}}\)	\(207\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)^6*(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

a^3*x-2/15*(45*a^3*exp(8*d*x+8*c)+45*a^2*b*exp(8*d*x+8*c)-90*a^3*exp(6*d*x+6*c)+90*a*b^2*exp(6*d*x+6*c)+140*a^
3*exp(4*d*x+4*c)+90*a^2*b*exp(4*d*x+4*c)+30*a*b^2*exp(4*d*x+4*c)+80*b^3*exp(4*d*x+4*c)-70*a^3*exp(2*d*x+2*c)+3
0*a*b^2*exp(2*d*x+2*c)-40*b^3*exp(2*d*x+2*c)+23*a^3+9*a^2*b-6*a*b^2+8*b^3)/d/(exp(2*d*x+2*c)-1)^5

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 826 vs. \(2 (65) = 130\).
time = 0.28, size = 826, normalized size = 11.97 \begin {gather*} \frac {1}{15} \, a^{3} {\left (15 \, x + \frac {15 \, c}{d} - \frac {2 \, {\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} - 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} - 45 \, e^{\left (-8 \, d x - 8 \, c\right )} - 23\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 )}}\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 {16}{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 {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 {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 {6}{5} \, a^{2} b {\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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^6*(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

1/15*a^3*(15*x + 15*c/d - 2*(70*e^(-2*d*x - 2*c) - 140*e^(-4*d*x - 4*c) + 90*e^(-6*d*x - 6*c) - 45*e^(-8*d*x -
8*c) - 23)/(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))) + 4/5*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))) - 16/15*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)) - 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)) - 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))) + 6/5*a^2*b*(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)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (65) = 130\).
time = 0.38, size = 521, normalized size = 7.55 \begin {gather*} -\frac {{\left (23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - {\left (15 \, a^{3} d x + 23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \sinh \left (d x + c\right )^{5} - 5 \, {\left (5 \, a^{3} - 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 5 \, {\left (15 \, a^{3} d x + 23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3} - 2 \, {\left (15 \, a^{3} d x + 23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (2 \, {\left (23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (5 \, a^{3} - 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, {\left (5 \, a^{3} + 9 \, a^{2} b + 12 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right ) - 5 \, {\left (30 \, a^{3} d x + {\left (15 \, a^{3} d x + 23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 46 \, a^{3} + 18 \, a^{2} b - 12 \, a b^{2} + 16 \, b^{3} - 3 \, {\left (15 \, a^{3} d x + 23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{15 \, {\left (d \sinh \left (d x + c\right )^{5} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (d \cosh \left (d x + c\right )^{4} - 3 \, d \cosh \left (d x + c\right )^{2} + 2 \, d\right )} \sinh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^6*(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

-1/15*((23*a^3 + 9*a^2*b - 6*a*b^2 + 8*b^3)*cosh(d*x + c)^5 + 5*(23*a^3 + 9*a^2*b - 6*a*b^2 + 8*b^3)*cosh(d*x
+ c)*sinh(d*x + c)^4 - (15*a^3*d*x + 23*a^3 + 9*a^2*b - 6*a*b^2 + 8*b^3)*sinh(d*x + c)^5 - 5*(5*a^3 - 9*a^2*b
- 6*a*b^2 + 8*b^3)*cosh(d*x + c)^3 + 5*(15*a^3*d*x + 23*a^3 + 9*a^2*b - 6*a*b^2 + 8*b^3 - 2*(15*a^3*d*x + 23*a
^3 + 9*a^2*b - 6*a*b^2 + 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 5*(2*(23*a^3 + 9*a^2*b - 6*a*b^2 + 8*b^3)*c
osh(d*x + c)^3 - 3*(5*a^3 - 9*a^2*b - 6*a*b^2 + 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 10*(5*a^3 + 9*a^2*b +
12*a*b^2 + 8*b^3)*cosh(d*x + c) - 5*(30*a^3*d*x + (15*a^3*d*x + 23*a^3 + 9*a^2*b - 6*a*b^2 + 8*b^3)*cosh(d*x +
c)^4 + 46*a^3 + 18*a^2*b - 12*a*b^2 + 16*b^3 - 3*(15*a^3*d*x + 23*a^3 + 9*a^2*b - 6*a*b^2 + 8*b^3)*cosh(d*x +
c)^2)*sinh(d*x + c))/(d*sinh(d*x + c)^5 + 5*(2*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^3 + 5*(d*cosh(d*x + c)^4
- 3*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c))

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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)**6*(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. 213 vs. \(2 (65) = 130\).
time = 0.48, size = 213, normalized size = 3.09 \begin {gather*} \frac {15 \, {\left (d x + c\right )} a^{3} - \frac {2 \, {\left (45 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 45 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 90 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 90 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 140 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 90 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 30 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 80 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 70 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 30 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 40 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}}}{15 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^6*(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/15*(15*(d*x + c)*a^3 - 2*(45*a^3*e^(8*d*x + 8*c) + 45*a^2*b*e^(8*d*x + 8*c) - 90*a^3*e^(6*d*x + 6*c) + 90*a*
b^2*e^(6*d*x + 6*c) + 140*a^3*e^(4*d*x + 4*c) + 90*a^2*b*e^(4*d*x + 4*c) + 30*a*b^2*e^(4*d*x + 4*c) + 80*b^3*e
^(4*d*x + 4*c) - 70*a^3*e^(2*d*x + 2*c) + 30*a*b^2*e^(2*d*x + 2*c) - 40*b^3*e^(2*d*x + 2*c) + 23*a^3 + 9*a^2*b
- 6*a*b^2 + 8*b^3)/(e^(2*d*x + 2*c) - 1)^5)/d

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Mupad [B]
time = 1.60, size = 547, normalized size = 7.93 \begin {gather*} a^3\,x-\frac {\frac {6\,\left (a^3+b\,a^2\right )}{5\,d}+\frac {24\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2\,b+a\,b^2\right )}{5\,d}+\frac {24\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a^2\,b+a\,b^2\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (a^3+b\,a^2\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (5\,a^3+9\,a^2\,b+12\,a\,b^2+8\,b^3\right )}{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 {6\,\left (a^2\,b+a\,b^2\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3+b\,a^2\right )}{5\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\frac {6\,\left (a^2\,b+a\,b^2\right )}{5\,d}+\frac {18\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^2\,b+a\,b^2\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a^3+b\,a^2\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (5\,a^3+9\,a^2\,b+12\,a\,b^2+8\,b^3\right )}{5\,d}}{6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}-\frac {\frac {2\,\left (5\,a^3+9\,a^2\,b+12\,a\,b^2+8\,b^3\right )}{15\,d}+\frac {12\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2\,b+a\,b^2\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^3+b\,a^2\right )}{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 {6\,\left (a^3+b\,a^2\right )}{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(coth(c + d*x)^6*(a + b/cosh(c + d*x)^2)^3,x)

[Out]

a^3*x - ((6*(a^2*b + a^3))/(5*d) + (24*exp(2*c + 2*d*x)*(a*b^2 + a^2*b))/(5*d) + (24*exp(6*c + 6*d*x)*(a*b^2 +
a^2*b))/(5*d) + (6*exp(8*c + 8*d*x)*(a^2*b + a^3))/(5*d) + (4*exp(4*c + 4*d*x)*(12*a*b^2 + 9*a^2*b + 5*a^3 +
8*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) - ((6*(a*b^2 + a^2*b))/(5*d) + (6*exp(2*c + 2*d*x)*(a^2*b + a^3))/(5*d))/(exp(4*c + 4*d*x) - 2
*exp(2*c + 2*d*x) + 1) - ((6*(a*b^2 + a^2*b))/(5*d) + (18*exp(4*c + 4*d*x)*(a*b^2 + a^2*b))/(5*d) + (6*exp(6*c
+ 6*d*x)*(a^2*b + a^3))/(5*d) + (2*exp(2*c + 2*d*x)*(12*a*b^2 + 9*a^2*b + 5*a^3 + 8*b^3))/(5*d))/(6*exp(4*c +
4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1) - ((2*(12*a*b^2 + 9*a^2*b + 5*a^3 +
8*b^3))/(15*d) + (12*exp(2*c + 2*d*x)*(a*b^2 + a^2*b))/(5*d) + (6*exp(4*c + 4*d*x)*(a^2*b + a^3))/(5*d))/(3*ex
p(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) - (6*(a^2*b + a^3))/(5*d*(exp(2*c + 2*d*x) - 1))

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