3.2.12 \(\int (a+b \text {sech}^2(c+d x))^2 \tanh ^4(c+d x) \, dx\) [112]

Optimal. Leaf size=77 \[ a^2 x-\frac {a^2 \tanh (c+d x)}{d}-\frac {a^2 \tanh ^3(c+d x)}{3 d}+\frac {b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^7(c+d x)}{7 d} \]

[Out]

a^2*x-a^2*tanh(d*x+c)/d-1/3*a^2*tanh(d*x+c)^3/d+1/5*b*(2*a+b)*tanh(d*x+c)^5/d-1/7*b^2*tanh(d*x+c)^7/d

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Rubi [A]
time = 0.07, antiderivative size = 77, 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, 212} \begin {gather*} -\frac {a^2 \tanh ^3(c+d x)}{3 d}-\frac {a^2 \tanh (c+d x)}{d}+a^2 x+\frac {b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^7(c+d x)}{7 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*x - (a^2*Tanh[c + d*x])/d - (a^2*Tanh[c + d*x]^3)/(3*d) + (b*(2*a + b)*Tanh[c + d*x]^5)/(5*d) - (b^2*Tanh[
c + d*x]^7)/(7*d)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[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 \left (a+b \text {sech}^2(c+d x)\right )^2 \tanh ^4(c+d x) \, dx &=\frac {\text {Subst}\left (\int \frac {x^4 \left (a+b \left (1-x^2\right )\right )^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-a^2-a^2 x^2+b (2 a+b) x^4-b^2 x^6+\frac {a^2}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a^2 \tanh (c+d x)}{d}-\frac {a^2 \tanh ^3(c+d x)}{3 d}+\frac {b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^7(c+d x)}{7 d}+\frac {a^2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^2 x-\frac {a^2 \tanh (c+d x)}{d}-\frac {a^2 \tanh ^3(c+d x)}{3 d}+\frac {b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^7(c+d x)}{7 d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(395\) vs. \(2(77)=154\).
time = 0.79, size = 395, normalized size = 5.13 \begin {gather*} \frac {\text {sech}(c) \text {sech}^7(c+d x) \left (3675 a^2 d x \cosh (d x)+3675 a^2 d x \cosh (2 c+d x)+2205 a^2 d x \cosh (2 c+3 d x)+2205 a^2 d x \cosh (4 c+3 d x)+735 a^2 d x \cosh (4 c+5 d x)+735 a^2 d x \cosh (6 c+5 d x)+105 a^2 d x \cosh (6 c+7 d x)+105 a^2 d x \cosh (8 c+7 d x)-5320 a^2 \sinh (d x)+1680 a b \sinh (d x)+840 b^2 \sinh (d x)+4480 a^2 \sinh (2 c+d x)-1260 a b \sinh (2 c+d x)+420 b^2 \sinh (2 c+d x)-3780 a^2 \sinh (2 c+3 d x)+924 a b \sinh (2 c+3 d x)-168 b^2 \sinh (2 c+3 d x)+2100 a^2 \sinh (4 c+3 d x)-840 a b \sinh (4 c+3 d x)-420 b^2 \sinh (4 c+3 d x)-1540 a^2 \sinh (4 c+5 d x)+168 a b \sinh (4 c+5 d x)+84 b^2 \sinh (4 c+5 d x)+420 a^2 \sinh (6 c+5 d x)-420 a b \sinh (6 c+5 d x)-280 a^2 \sinh (6 c+7 d x)+84 a b \sinh (6 c+7 d x)+12 b^2 \sinh (6 c+7 d x)\right )}{13440 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sech[c]*Sech[c + d*x]^7*(3675*a^2*d*x*Cosh[d*x] + 3675*a^2*d*x*Cosh[2*c + d*x] + 2205*a^2*d*x*Cosh[2*c + 3*d*
x] + 2205*a^2*d*x*Cosh[4*c + 3*d*x] + 735*a^2*d*x*Cosh[4*c + 5*d*x] + 735*a^2*d*x*Cosh[6*c + 5*d*x] + 105*a^2*
d*x*Cosh[6*c + 7*d*x] + 105*a^2*d*x*Cosh[8*c + 7*d*x] - 5320*a^2*Sinh[d*x] + 1680*a*b*Sinh[d*x] + 840*b^2*Sinh
[d*x] + 4480*a^2*Sinh[2*c + d*x] - 1260*a*b*Sinh[2*c + d*x] + 420*b^2*Sinh[2*c + d*x] - 3780*a^2*Sinh[2*c + 3*
d*x] + 924*a*b*Sinh[2*c + 3*d*x] - 168*b^2*Sinh[2*c + 3*d*x] + 2100*a^2*Sinh[4*c + 3*d*x] - 840*a*b*Sinh[4*c +
3*d*x] - 420*b^2*Sinh[4*c + 3*d*x] - 1540*a^2*Sinh[4*c + 5*d*x] + 168*a*b*Sinh[4*c + 5*d*x] + 84*b^2*Sinh[4*c
+ 5*d*x] + 420*a^2*Sinh[6*c + 5*d*x] - 420*a*b*Sinh[6*c + 5*d*x] - 280*a^2*Sinh[6*c + 7*d*x] + 84*a*b*Sinh[6*
c + 7*d*x] + 12*b^2*Sinh[6*c + 7*d*x]))/(13440*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(271\) vs. \(2(71)=142\).
time = 2.44, size = 272, normalized size = 3.53

method	result	size
risch	\(a^{2} x +\frac {4 a^{2} {\mathrm e}^{12 d x +12 c}-4 a b \,{\mathrm e}^{12 d x +12 c}+20 a^{2} {\mathrm e}^{10 d x +10 c}-8 a b \,{\mathrm e}^{10 d x +10 c}-4 b^{2} {\mathrm e}^{10 d x +10 c}+\frac {128 a^{2} {\mathrm e}^{8 d x +8 c}}{3}-12 a b \,{\mathrm e}^{8 d x +8 c}+4 b^{2} {\mathrm e}^{8 d x +8 c}+\frac {152 a^{2} {\mathrm e}^{6 d x +6 c}}{3}-16 a b \,{\mathrm e}^{6 d x +6 c}-8 b^{2} {\mathrm e}^{6 d x +6 c}+36 a^{2} {\mathrm e}^{4 d x +4 c}-\frac {44 a b \,{\mathrm e}^{4 d x +4 c}}{5}+\frac {8 b^{2} {\mathrm e}^{4 d x +4 c}}{5}+\frac {44 a^{2} {\mathrm e}^{2 d x +2 c}}{3}-\frac {8 a b \,{\mathrm e}^{2 d x +2 c}}{5}-\frac {4 b^{2} {\mathrm e}^{2 d x +2 c}}{5}+\frac {8 a^{2}}{3}-\frac {4 a b}{5}-\frac {4 b^{2}}{35}}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{7}}\)	\(272\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*x+4/105*(105*a^2*exp(12*d*x+12*c)-105*a*b*exp(12*d*x+12*c)+525*a^2*exp(10*d*x+10*c)-210*a*b*exp(10*d*x+10*
c)-105*b^2*exp(10*d*x+10*c)+1120*a^2*exp(8*d*x+8*c)-315*a*b*exp(8*d*x+8*c)+105*b^2*exp(8*d*x+8*c)+1330*a^2*exp
(6*d*x+6*c)-420*a*b*exp(6*d*x+6*c)-210*b^2*exp(6*d*x+6*c)+945*a^2*exp(4*d*x+4*c)-231*a*b*exp(4*d*x+4*c)+42*b^2
*exp(4*d*x+4*c)+385*a^2*exp(2*d*x+2*c)-42*a*b*exp(2*d*x+2*c)-21*b^2*exp(2*d*x+2*c)+70*a^2-21*a*b-3*b^2)/d/(1+e
xp(2*d*x+2*c))^7

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (71) = 142\).
time = 0.28, size = 649, normalized size = 8.43 \begin {gather*} \frac {2 \, a b \tanh \left (d x + c\right )^{5}}{5 \, d} + \frac {1}{3} \, a^{2} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\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 )}}\right )} + \frac {4}{35} \, b^{2} {\left (\frac {7 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}} - \frac {14 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}} + \frac {70 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}} - \frac {35 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}} + \frac {35 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}} + \frac {1}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)^2)^2*tanh(d*x+c)^4,x, algorithm="maxima")

[Out]

2/5*a*b*tanh(d*x + c)^5/d + 1/3*a^2*(3*x + 3*c/d - 4*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + 2)/(d*(3*e^(-2
*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1))) + 4/35*b^2*(7*e^(-2*d*x - 2*c)/(d*(7*e^(-2*d*x - 2*
c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x -
12*c) + e^(-14*d*x - 14*c) + 1)) - 14*e^(-4*d*x - 4*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6
*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) +
70*e^(-6*d*x - 6*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) +
21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) - 35*e^(-8*d*x - 8*c)/(d*(7*e^(-2*d*x
- 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d
*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 35*e^(-10*d*x - 10*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 3
5*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) +
1)) + 1/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*
d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 721 vs. \(2 (71) = 142\).
time = 0.40, size = 721, normalized size = 9.36 \begin {gather*} \frac {{\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )^{7} + 7 \, {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - 2 \, {\left (70 \, a^{2} - 21 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{7} + 7 \, {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} - 14 \, {\left (3 \, {\left (70 \, a^{2} - 21 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 40 \, a^{2} + 9 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{5} + 35 \, {\left ({\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 21 \, {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 14 \, {\left (5 \, {\left (70 \, a^{2} - 21 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 10 \, {\left (40 \, a^{2} + 9 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 60 \, a^{2} - 3 \, a b + 21 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 7 \, {\left (3 \, {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 10 \, {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 9 \, {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \, {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right ) - 14 \, {\left ({\left (70 \, a^{2} - 21 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{6} + 5 \, {\left (40 \, a^{2} + 9 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 9 \, {\left (20 \, a^{2} - a b + 7 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 30 \, a^{2} - 15 \, a b - 45 \, b^{2}\right )} \sinh \left (d x + c\right )}{105 \, {\left (d \cosh \left (d x + c\right )^{7} + 7 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + 7 \, d \cosh \left (d x + c\right )^{5} + 35 \, {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 21 \, d \cosh \left (d x + c\right )^{3} + 7 \, {\left (3 \, d \cosh \left (d x + c\right )^{5} + 10 \, d \cosh \left (d x + c\right )^{3} + 9 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \, d \cosh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)^2)^2*tanh(d*x+c)^4,x, algorithm="fricas")

[Out]

1/105*((105*a^2*d*x + 140*a^2 - 42*a*b - 6*b^2)*cosh(d*x + c)^7 + 7*(105*a^2*d*x + 140*a^2 - 42*a*b - 6*b^2)*c
osh(d*x + c)*sinh(d*x + c)^6 - 2*(70*a^2 - 21*a*b - 3*b^2)*sinh(d*x + c)^7 + 7*(105*a^2*d*x + 140*a^2 - 42*a*b
- 6*b^2)*cosh(d*x + c)^5 - 14*(3*(70*a^2 - 21*a*b - 3*b^2)*cosh(d*x + c)^2 + 40*a^2 + 9*a*b - 3*b^2)*sinh(d*x
+ c)^5 + 35*((105*a^2*d*x + 140*a^2 - 42*a*b - 6*b^2)*cosh(d*x + c)^3 + (105*a^2*d*x + 140*a^2 - 42*a*b - 6*b
^2)*cosh(d*x + c))*sinh(d*x + c)^4 + 21*(105*a^2*d*x + 140*a^2 - 42*a*b - 6*b^2)*cosh(d*x + c)^3 - 14*(5*(70*a
^2 - 21*a*b - 3*b^2)*cosh(d*x + c)^4 + 10*(40*a^2 + 9*a*b - 3*b^2)*cosh(d*x + c)^2 + 60*a^2 - 3*a*b + 21*b^2)*
sinh(d*x + c)^3 + 7*(3*(105*a^2*d*x + 140*a^2 - 42*a*b - 6*b^2)*cosh(d*x + c)^5 + 10*(105*a^2*d*x + 140*a^2 -
42*a*b - 6*b^2)*cosh(d*x + c)^3 + 9*(105*a^2*d*x + 140*a^2 - 42*a*b - 6*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 +
35*(105*a^2*d*x + 140*a^2 - 42*a*b - 6*b^2)*cosh(d*x + c) - 14*((70*a^2 - 21*a*b - 3*b^2)*cosh(d*x + c)^6 + 5*
(40*a^2 + 9*a*b - 3*b^2)*cosh(d*x + c)^4 + 9*(20*a^2 - a*b + 7*b^2)*cosh(d*x + c)^2 + 30*a^2 - 15*a*b - 45*b^2
)*sinh(d*x + c))/(d*cosh(d*x + c)^7 + 7*d*cosh(d*x + c)*sinh(d*x + c)^6 + 7*d*cosh(d*x + c)^5 + 35*(d*cosh(d*x
+ c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^4 + 21*d*cosh(d*x + c)^3 + 7*(3*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c
)^3 + 9*d*cosh(d*x + c))*sinh(d*x + c)^2 + 35*d*cosh(d*x + c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \tanh ^{4}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)**2)**2*tanh(d*x+c)**4,x)

[Out]

Integral((a + b*sech(c + d*x)**2)**2*tanh(c + d*x)**4, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (71) = 142\).
time = 0.45, size = 278, normalized size = 3.61 \begin {gather*} \frac {105 \, {\left (d x + c\right )} a^{2} + \frac {4 \, {\left (105 \, a^{2} e^{\left (12 \, d x + 12 \, c\right )} - 105 \, a b e^{\left (12 \, d x + 12 \, c\right )} + 525 \, a^{2} e^{\left (10 \, d x + 10 \, c\right )} - 210 \, a b e^{\left (10 \, d x + 10 \, c\right )} - 105 \, b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1120 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} - 315 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 105 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 1330 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} - 420 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 210 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 945 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 231 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 42 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 385 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 42 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 21 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 70 \, a^{2} - 21 \, a b - 3 \, b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}}}{105 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)^2)^2*tanh(d*x+c)^4,x, algorithm="giac")

[Out]

1/105*(105*(d*x + c)*a^2 + 4*(105*a^2*e^(12*d*x + 12*c) - 105*a*b*e^(12*d*x + 12*c) + 525*a^2*e^(10*d*x + 10*c
) - 210*a*b*e^(10*d*x + 10*c) - 105*b^2*e^(10*d*x + 10*c) + 1120*a^2*e^(8*d*x + 8*c) - 315*a*b*e^(8*d*x + 8*c)
+ 105*b^2*e^(8*d*x + 8*c) + 1330*a^2*e^(6*d*x + 6*c) - 420*a*b*e^(6*d*x + 6*c) - 210*b^2*e^(6*d*x + 6*c) + 94
5*a^2*e^(4*d*x + 4*c) - 231*a*b*e^(4*d*x + 4*c) + 42*b^2*e^(4*d*x + 4*c) + 385*a^2*e^(2*d*x + 2*c) - 42*a*b*e^
(2*d*x + 2*c) - 21*b^2*e^(2*d*x + 2*c) + 70*a^2 - 21*a*b - 3*b^2)/(e^(2*d*x + 2*c) + 1)^7)/d

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Mupad [B]
time = 0.18, size = 1022, normalized size = 13.27 \begin {gather*} \frac {\frac {4\,\left (7\,a^2+a\,b+8\,b^2\right )}{105\,d}-\frac {4\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (a\,b-a^2\right )}{7\,d}-\frac {16\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (-2\,a^2+a\,b+3\,b^2\right )}{35\,d}+\frac {16\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (2\,a^2+a\,b-b^2\right )}{21\,d}+\frac {8\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (7\,a^2+a\,b+8\,b^2\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 {4\,\left (-2\,a^2+a\,b+3\,b^2\right )}{35\,d}+\frac {4\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a\,b-a^2\right )}{7\,d}-\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (2\,a^2+a\,b-b^2\right )}{7\,d}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (7\,a^2+a\,b+8\,b^2\right )}{35\,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}+a^2\,x+\frac {\frac {4\,\left (7\,a^2+a\,b+8\,b^2\right )}{105\,d}-\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a\,b-a^2\right )}{7\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a^2+a\,b-b^2\right )}{21\,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 {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a^2+a\,b-b^2\right )}{7\,d}-\frac {4\,{\mathrm {e}}^{12\,c+12\,d\,x}\,\left (a\,b-a^2\right )}{7\,d}-\frac {4\,\left (a\,b-a^2\right )}{7\,d}-\frac {16\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (-2\,a^2+a\,b+3\,b^2\right )}{7\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (7\,a^2+a\,b+8\,b^2\right )}{7\,d}+\frac {8\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (2\,a^2+a\,b-b^2\right )}{7\,d}+\frac {4\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (7\,a^2+a\,b+8\,b^2\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 {4\,\left (2\,a^2+a\,b-b^2\right )}{21\,d}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b-a^2\right )}{7\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}+\frac {\frac {4\,\left (2\,a^2+a\,b-b^2\right )}{21\,d}-\frac {4\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (a\,b-a^2\right )}{7\,d}-\frac {8\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (-2\,a^2+a\,b+3\,b^2\right )}{7\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (7\,a^2+a\,b+8\,b^2\right )}{21\,d}+\frac {20\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (2\,a^2+a\,b-b^2\right )}{21\,d}+\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (7\,a^2+a\,b+8\,b^2\right )}{21\,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 {4\,\left (a\,b-a^2\right )}{7\,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(tanh(c + d*x)^4*(a + b/cosh(c + d*x)^2)^2,x)

[Out]

((4*(a*b + 7*a^2 + 8*b^2))/(105*d) - (4*exp(8*c + 8*d*x)*(a*b - a^2))/(7*d) - (16*exp(2*c + 2*d*x)*(a*b - 2*a^
2 + 3*b^2))/(35*d) + (16*exp(6*c + 6*d*x)*(a*b + 2*a^2 - b^2))/(21*d) + (8*exp(4*c + 4*d*x)*(a*b + 7*a^2 + 8*b
^2))/(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) - ((4*(a*b - 2*a^2 + 3*b^2))/(35*d) + (4*exp(6*c + 6*d*x)*(a*b - a^2))/(7*d) - (4*exp(4*c + 4*d*
x)*(a*b + 2*a^2 - b^2))/(7*d) - (4*exp(2*c + 2*d*x)*(a*b + 7*a^2 + 8*b^2))/(35*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) + a^2*x + ((4*(a*b + 7*a^2 + 8*b^2))/(105*d) - (4*e
xp(4*c + 4*d*x)*(a*b - a^2))/(7*d) + (8*exp(2*c + 2*d*x)*(a*b + 2*a^2 - b^2))/(21*d))/(3*exp(2*c + 2*d*x) + 3*
exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1) + ((8*exp(2*c + 2*d*x)*(a*b + 2*a^2 - b^2))/(7*d) - (4*exp(12*c + 12*
d*x)*(a*b - a^2))/(7*d) - (4*(a*b - a^2))/(7*d) - (16*exp(6*c + 6*d*x)*(a*b - 2*a^2 + 3*b^2))/(7*d) + (4*exp(4
*c + 4*d*x)*(a*b + 7*a^2 + 8*b^2))/(7*d) + (8*exp(10*c + 10*d*x)*(a*b + 2*a^2 - b^2))/(7*d) + (4*exp(8*c + 8*d
*x)*(a*b + 7*a^2 + 8*b^2))/(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 + 10*d*x) + 7*exp(12*c + 12*d*x) + exp(14*c + 14*d*x) + 1) + ((4*(a*b + 2*a^2 - b^2))/
(21*d) - (4*exp(2*c + 2*d*x)*(a*b - a^2))/(7*d))/(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1) + ((4*(a*b + 2*a^
2 - b^2))/(21*d) - (4*exp(10*c + 10*d*x)*(a*b - a^2))/(7*d) - (8*exp(4*c + 4*d*x)*(a*b - 2*a^2 + 3*b^2))/(7*d)
+ (4*exp(2*c + 2*d*x)*(a*b + 7*a^2 + 8*b^2))/(21*d) + (20*exp(8*c + 8*d*x)*(a*b + 2*a^2 - b^2))/(21*d) + (8*e
xp(6*c + 6*d*x)*(a*b + 7*a^2 + 8*b^2))/(21*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*(a*b - a^2))/(7*d*(exp(2*c + 2*d*
x) + 1))

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