3.2.26 \(\int (a+b \text {sech}^2(c+d x))^3 \tanh ^2(c+d x) \, dx\) [126]

Optimal. Leaf size=92 \[ a^3 x-\frac {a^3 \tanh (c+d x)}{d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^2 (3 a+2 b) \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^7(c+d x)}{7 d} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 92, 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^3 \tanh (c+d x)}{d}+a^3 x+\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^2 (3 a+2 b) \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^7(c+d x)}{7 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*x - (a^3*Tanh[c + d*x])/d + (b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^3)/(3*d) - (b^2*(3*a + 2*b)*Tanh[c + d*
x]^5)/(5*d) + (b^3*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 )^3 \tanh ^2(c+d x) \, dx &=\frac {\text {Subst}\left (\int \frac {x^2 \left (a+b \left (1-x^2\right )\right )^3}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-a^3+b \left (3 a^2+3 a b+b^2\right ) x^2-b^2 (3 a+2 b) x^4+b^3 x^6+\frac {a^3}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a^3 \tanh (c+d x)}{d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^2 (3 a+2 b) \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^7(c+d x)}{7 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 {a^3 \tanh (c+d x)}{d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^2 (3 a+2 b) \tanh ^5(c+d x)}{5 d}+\frac {b^3 \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. \(479\) vs. \(2(92)=184\).
time = 1.21, size = 479, normalized size = 5.21 \begin {gather*} \frac {\text {sech}(c) \text {sech}^7(c+d x) \left (3675 a^3 d x \cosh (d x)+3675 a^3 d x \cosh (2 c+d x)+2205 a^3 d x \cosh (2 c+3 d x)+2205 a^3 d x \cosh (4 c+3 d x)+735 a^3 d x \cosh (4 c+5 d x)+735 a^3 d x \cosh (6 c+5 d x)+105 a^3 d x \cosh (6 c+7 d x)+105 a^3 d x \cosh (8 c+7 d x)-4200 a^3 \sinh (d x)+3360 a^2 b \sinh (d x)+840 a b^2 \sinh (d x)-560 b^3 \sinh (d x)+3150 a^3 \sinh (2 c+d x)-3990 a^2 b \sinh (2 c+d x)-2100 a b^2 \sinh (2 c+d x)-1120 b^3 \sinh (2 c+d x)-3150 a^3 \sinh (2 c+3 d x)+1890 a^2 b \sinh (2 c+3 d x)+504 a b^2 \sinh (2 c+3 d x)+336 b^3 \sinh (2 c+3 d x)+1260 a^3 \sinh (4 c+3 d x)-2520 a^2 b \sinh (4 c+3 d x)-1260 a b^2 \sinh (4 c+3 d x)-1260 a^3 \sinh (4 c+5 d x)+840 a^2 b \sinh (4 c+5 d x)+588 a b^2 \sinh (4 c+5 d x)+112 b^3 \sinh (4 c+5 d x)+210 a^3 \sinh (6 c+5 d x)-630 a^2 b \sinh (6 c+5 d x)-210 a^3 \sinh (6 c+7 d x)+210 a^2 b \sinh (6 c+7 d x)+84 a b^2 \sinh (6 c+7 d x)+16 b^3 \sinh (6 c+7 d x)\right )}{13440 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sech[c]*Sech[c + d*x]^7*(3675*a^3*d*x*Cosh[d*x] + 3675*a^3*d*x*Cosh[2*c + d*x] + 2205*a^3*d*x*Cosh[2*c + 3*d*
x] + 2205*a^3*d*x*Cosh[4*c + 3*d*x] + 735*a^3*d*x*Cosh[4*c + 5*d*x] + 735*a^3*d*x*Cosh[6*c + 5*d*x] + 105*a^3*
d*x*Cosh[6*c + 7*d*x] + 105*a^3*d*x*Cosh[8*c + 7*d*x] - 4200*a^3*Sinh[d*x] + 3360*a^2*b*Sinh[d*x] + 840*a*b^2*
Sinh[d*x] - 560*b^3*Sinh[d*x] + 3150*a^3*Sinh[2*c + d*x] - 3990*a^2*b*Sinh[2*c + d*x] - 2100*a*b^2*Sinh[2*c +
d*x] - 1120*b^3*Sinh[2*c + d*x] - 3150*a^3*Sinh[2*c + 3*d*x] + 1890*a^2*b*Sinh[2*c + 3*d*x] + 504*a*b^2*Sinh[2
*c + 3*d*x] + 336*b^3*Sinh[2*c + 3*d*x] + 1260*a^3*Sinh[4*c + 3*d*x] - 2520*a^2*b*Sinh[4*c + 3*d*x] - 1260*a*b
^2*Sinh[4*c + 3*d*x] - 1260*a^3*Sinh[4*c + 5*d*x] + 840*a^2*b*Sinh[4*c + 5*d*x] + 588*a*b^2*Sinh[4*c + 5*d*x]
+ 112*b^3*Sinh[4*c + 5*d*x] + 210*a^3*Sinh[6*c + 5*d*x] - 630*a^2*b*Sinh[6*c + 5*d*x] - 210*a^3*Sinh[6*c + 7*d
*x] + 210*a^2*b*Sinh[6*c + 7*d*x] + 84*a*b^2*Sinh[6*c + 7*d*x] + 16*b^3*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. \(352\) vs. \(2(86)=172\).
time = 2.40, size = 353, normalized size = 3.84

method	result	size
risch	\(a^{3} x +\frac {-\frac {4 a \,b^{2}}{5}-6 a^{2} b \,{\mathrm e}^{12 d x +12 c}-24 a^{2} b \,{\mathrm e}^{10 d x +10 c}-12 a \,b^{2} {\mathrm e}^{10 d x +10 c}-18 a^{2} b \,{\mathrm e}^{4 d x +4 c}-8 a^{2} b \,{\mathrm e}^{2 d x +2 c}-2 a^{2} b +2 a^{3}-\frac {16 b^{3}}{105}-20 a \,b^{2} {\mathrm e}^{8 d x +8 c}-8 a \,b^{2} {\mathrm e}^{6 d x +6 c}-\frac {24 a \,b^{2} {\mathrm e}^{4 d x +4 c}}{5}-38 a^{2} b \,{\mathrm e}^{8 d x +8 c}-\frac {28 a \,b^{2} {\mathrm e}^{2 d x +2 c}}{5}-32 a^{2} b \,{\mathrm e}^{6 d x +6 c}+12 a^{3} {\mathrm e}^{2 d x +2 c}+2 a^{3} {\mathrm e}^{12 d x +12 c}+12 a^{3} {\mathrm e}^{10 d x +10 c}-\frac {16 b^{3} {\mathrm e}^{2 d x +2 c}}{15}-\frac {32 b^{3} {\mathrm e}^{8 d x +8 c}}{3}+30 a^{3} {\mathrm e}^{4 d x +4 c}-\frac {16 b^{3} {\mathrm e}^{4 d x +4 c}}{5}+30 a^{3} {\mathrm e}^{8 d x +8 c}+40 a^{3} {\mathrm e}^{6 d x +6 c}+\frac {16 b^{3} {\mathrm e}^{6 d x +6 c}}{3}}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{7}}\)	\(353\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*x+2/105*(-42*a*b^2-315*a^2*b*exp(12*d*x+12*c)-1260*a^2*b*exp(10*d*x+10*c)-630*a*b^2*exp(10*d*x+10*c)-945*a
^2*b*exp(4*d*x+4*c)-420*a^2*b*exp(2*d*x+2*c)-105*a^2*b+105*a^3-8*b^3-1050*a*b^2*exp(8*d*x+8*c)-420*a*b^2*exp(6
*d*x+6*c)-252*a*b^2*exp(4*d*x+4*c)-1995*a^2*b*exp(8*d*x+8*c)-294*a*b^2*exp(2*d*x+2*c)-1680*a^2*b*exp(6*d*x+6*c
)+630*a^3*exp(2*d*x+2*c)+105*a^3*exp(12*d*x+12*c)+630*a^3*exp(10*d*x+10*c)-56*b^3*exp(2*d*x+2*c)-560*b^3*exp(8
*d*x+8*c)+1575*a^3*exp(4*d*x+4*c)-168*b^3*exp(4*d*x+4*c)+1575*a^3*exp(8*d*x+8*c)+2100*a^3*exp(6*d*x+6*c)+280*b
^3*exp(6*d*x+6*c))/d/(1+exp(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. 788 vs. \(2 (86) = 172\).
time = 0.28, size = 788, normalized size = 8.57 \begin {gather*} \frac {a^{2} b \tanh \left (d x + c\right )^{3}}{d} + a^{3} {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + \frac {16}{105} \, b^{3} {\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 {21 \, 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 {35 \, 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 {70 \, 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 {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 )} + \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*b*tanh(d*x + c)^3/d + a^3*(x + c/d - 2/(d*(e^(-2*d*x - 2*c) + 1))) + 16/105*b^3*(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)) + 21*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)) - 35*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)) + 70*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)) + 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)))
+ 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)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 881 vs. \(2 (86) = 172\).
time = 0.38, size = 881, normalized size = 9.58 \begin {gather*} \frac {{\left (105 \, a^{3} d x + 105 \, a^{3} - 105 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 7 \, {\left (105 \, a^{3} d x + 105 \, a^{3} - 105 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - {\left (105 \, a^{3} - 105 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \sinh \left (d x + c\right )^{7} + 7 \, {\left (105 \, a^{3} d x + 105 \, a^{3} - 105 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 7 \, {\left (75 \, a^{3} - 15 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3} + 3 \, {\left (105 \, a^{3} - 105 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{5} + 35 \, {\left ({\left (105 \, a^{3} d x + 105 \, a^{3} - 105 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (105 \, a^{3} d x + 105 \, a^{3} - 105 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 21 \, {\left (105 \, a^{3} d x + 105 \, a^{3} - 105 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 7 \, {\left (5 \, {\left (105 \, a^{3} - 105 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 135 \, a^{3} + 45 \, a^{2} b + 54 \, a b^{2} - 24 \, b^{3} + 10 \, {\left (75 \, a^{3} - 15 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 7 \, {\left (3 \, {\left (105 \, a^{3} d x + 105 \, a^{3} - 105 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 10 \, {\left (105 \, a^{3} d x + 105 \, a^{3} - 105 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 9 \, {\left (105 \, a^{3} d x + 105 \, a^{3} - 105 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \, {\left (105 \, a^{3} d x + 105 \, a^{3} - 105 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right ) - 7 \, {\left ({\left (105 \, a^{3} - 105 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} + 5 \, {\left (75 \, a^{3} - 15 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 75 \, a^{3} + 45 \, a^{2} b + 90 \, a b^{2} + 120 \, b^{3} + 9 \, {\left (45 \, a^{3} + 15 \, a^{2} b + 18 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{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)^3*tanh(d*x+c)^2,x, algorithm="fricas")

[Out]

1/105*((105*a^3*d*x + 105*a^3 - 105*a^2*b - 42*a*b^2 - 8*b^3)*cosh(d*x + c)^7 + 7*(105*a^3*d*x + 105*a^3 - 105
*a^2*b - 42*a*b^2 - 8*b^3)*cosh(d*x + c)*sinh(d*x + c)^6 - (105*a^3 - 105*a^2*b - 42*a*b^2 - 8*b^3)*sinh(d*x +
c)^7 + 7*(105*a^3*d*x + 105*a^3 - 105*a^2*b - 42*a*b^2 - 8*b^3)*cosh(d*x + c)^5 - 7*(75*a^3 - 15*a^2*b - 42*a
*b^2 - 8*b^3 + 3*(105*a^3 - 105*a^2*b - 42*a*b^2 - 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 35*((105*a^3*d*x
+ 105*a^3 - 105*a^2*b - 42*a*b^2 - 8*b^3)*cosh(d*x + c)^3 + (105*a^3*d*x + 105*a^3 - 105*a^2*b - 42*a*b^2 - 8*
b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 21*(105*a^3*d*x + 105*a^3 - 105*a^2*b - 42*a*b^2 - 8*b^3)*cosh(d*x + c)^
3 - 7*(5*(105*a^3 - 105*a^2*b - 42*a*b^2 - 8*b^3)*cosh(d*x + c)^4 + 135*a^3 + 45*a^2*b + 54*a*b^2 - 24*b^3 + 1
0*(75*a^3 - 15*a^2*b - 42*a*b^2 - 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 7*(3*(105*a^3*d*x + 105*a^3 - 105*
a^2*b - 42*a*b^2 - 8*b^3)*cosh(d*x + c)^5 + 10*(105*a^3*d*x + 105*a^3 - 105*a^2*b - 42*a*b^2 - 8*b^3)*cosh(d*x
+ c)^3 + 9*(105*a^3*d*x + 105*a^3 - 105*a^2*b - 42*a*b^2 - 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 35*(105*a^
3*d*x + 105*a^3 - 105*a^2*b - 42*a*b^2 - 8*b^3)*cosh(d*x + c) - 7*((105*a^3 - 105*a^2*b - 42*a*b^2 - 8*b^3)*co
sh(d*x + c)^6 + 5*(75*a^3 - 15*a^2*b - 42*a*b^2 - 8*b^3)*cosh(d*x + c)^4 + 75*a^3 + 45*a^2*b + 90*a*b^2 + 120*
b^3 + 9*(45*a^3 + 15*a^2*b + 18*a*b^2 - 8*b^3)*cosh(d*x + c)^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 )^{3} \tanh ^{2}{\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)**3*tanh(d*x+c)**2,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (86) = 172\).
time = 0.43, size = 359, normalized size = 3.90 \begin {gather*} \frac {105 \, {\left (d x + c\right )} a^{3} + \frac {2 \, {\left (105 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} - 315 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 630 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} - 1260 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 630 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} - 1995 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 1050 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 560 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 2100 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 1680 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 420 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 280 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1575 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 945 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 252 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 168 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 630 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 420 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 294 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 56 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} - 105 \, a^{2} b - 42 \, a b^{2} - 8 \, b^{3}\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)^3*tanh(d*x+c)^2,x, algorithm="giac")

[Out]

1/105*(105*(d*x + c)*a^3 + 2*(105*a^3*e^(12*d*x + 12*c) - 315*a^2*b*e^(12*d*x + 12*c) + 630*a^3*e^(10*d*x + 10
*c) - 1260*a^2*b*e^(10*d*x + 10*c) - 630*a*b^2*e^(10*d*x + 10*c) + 1575*a^3*e^(8*d*x + 8*c) - 1995*a^2*b*e^(8*
d*x + 8*c) - 1050*a*b^2*e^(8*d*x + 8*c) - 560*b^3*e^(8*d*x + 8*c) + 2100*a^3*e^(6*d*x + 6*c) - 1680*a^2*b*e^(6
*d*x + 6*c) - 420*a*b^2*e^(6*d*x + 6*c) + 280*b^3*e^(6*d*x + 6*c) + 1575*a^3*e^(4*d*x + 4*c) - 945*a^2*b*e^(4*
d*x + 4*c) - 252*a*b^2*e^(4*d*x + 4*c) - 168*b^3*e^(4*d*x + 4*c) + 630*a^3*e^(2*d*x + 2*c) - 420*a^2*b*e^(2*d*
x + 2*c) - 294*a*b^2*e^(2*d*x + 2*c) - 56*b^3*e^(2*d*x + 2*c) + 105*a^3 - 105*a^2*b - 42*a*b^2 - 8*b^3)/(e^(2*
d*x + 2*c) + 1)^7)/d

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Mupad [B]
time = 0.21, size = 1133, normalized size = 12.32 \begin {gather*} a^3\,x-\frac {\frac {2\,\left (-a^3+a^2\,b+2\,a\,b^2\right )}{7\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (15\,a^3+3\,a^2\,b-16\,b^3\right )}{21\,d}-\frac {4\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (15\,a^3+3\,a^2\,b-16\,b^3\right )}{21\,d}+\frac {10\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (-a^3+a^2\,b+2\,a\,b^2\right )}{7\,d}-\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (5\,a^3+3\,a^2\,b+6\,a\,b^2+8\,b^3\right )}{7\,d}+\frac {2\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (3\,a^2\,b-a^3\right )}{7\,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 {2\,\left (3\,a^2\,b-a^3\right )}{7\,d}-\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (15\,a^3+3\,a^2\,b-16\,b^3\right )}{7\,d}-\frac {2\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (15\,a^3+3\,a^2\,b-16\,b^3\right )}{7\,d}+\frac {12\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (-a^3+a^2\,b+2\,a\,b^2\right )}{7\,d}+\frac {12\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (-a^3+a^2\,b+2\,a\,b^2\right )}{7\,d}-\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (5\,a^3+3\,a^2\,b+6\,a\,b^2+8\,b^3\right )}{7\,d}+\frac {2\,{\mathrm {e}}^{12\,c+12\,d\,x}\,\left (3\,a^2\,b-a^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 {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (-a^3+a^2\,b+2\,a\,b^2\right )}{7\,d}-\frac {2\,\left (15\,a^3+3\,a^2\,b-16\,b^3\right )}{105\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (3\,a^2\,b-a^3\right )}{7\,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 (5\,a^3+3\,a^2\,b+6\,a\,b^2+8\,b^3\right )}{35\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (15\,a^3+3\,a^2\,b-16\,b^3\right )}{35\,d}-\frac {6\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (-a^3+a^2\,b+2\,a\,b^2\right )}{7\,d}-\frac {2\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (3\,a^2\,b-a^3\right )}{7\,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 {2\,\left (15\,a^3+3\,a^2\,b-16\,b^3\right )}{105\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (15\,a^3+3\,a^2\,b-16\,b^3\right )}{35\,d}-\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (-a^3+a^2\,b+2\,a\,b^2\right )}{7\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (5\,a^3+3\,a^2\,b+6\,a\,b^2+8\,b^3\right )}{35\,d}-\frac {2\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (3\,a^2\,b-a^3\right )}{7\,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 (-a^3+a^2\,b+2\,a\,b^2\right )}{7\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a^2\,b-a^3\right )}{7\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {2\,\left (3\,a^2\,b-a^3\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)^2*(a + b/cosh(c + d*x)^2)^3,x)

[Out]

a^3*x - ((2*(2*a*b^2 + a^2*b - a^3))/(7*d) - (2*exp(2*c + 2*d*x)*(3*a^2*b + 15*a^3 - 16*b^3))/(21*d) - (4*exp(
6*c + 6*d*x)*(3*a^2*b + 15*a^3 - 16*b^3))/(21*d) + (10*exp(8*c + 8*d*x)*(2*a*b^2 + a^2*b - a^3))/(7*d) - (4*ex
p(4*c + 4*d*x)*(6*a*b^2 + 3*a^2*b + 5*a^3 + 8*b^3))/(7*d) + (2*exp(10*c + 10*d*x)*(3*a^2*b - a^3))/(7*d))/(6*e
xp(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) - ((2*(3*a^2*b - a^3))/(7*d) - (2*exp(4*c + 4*d*x)*(3*a^2*b + 15*a^3 - 16*b^3))/(7*d) - (
2*exp(8*c + 8*d*x)*(3*a^2*b + 15*a^3 - 16*b^3))/(7*d) + (12*exp(2*c + 2*d*x)*(2*a*b^2 + a^2*b - a^3))/(7*d) +
(12*exp(10*c + 10*d*x)*(2*a*b^2 + a^2*b - a^3))/(7*d) - (8*exp(6*c + 6*d*x)*(6*a*b^2 + 3*a^2*b + 5*a^3 + 8*b^3
))/(7*d) + (2*exp(12*c + 12*d*x)*(3*a^2*b - a^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 + 10*d*x) + 7*exp(12*c + 12*d*x) + exp(14*c + 14*d*x) + 1) - ((
4*exp(2*c + 2*d*x)*(2*a*b^2 + a^2*b - a^3))/(7*d) - (2*(3*a^2*b + 15*a^3 - 16*b^3))/(105*d) + (2*exp(4*c + 4*d
*x)*(3*a^2*b - a^3))/(7*d))/(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1) + ((2*(6*a*b^2 +
3*a^2*b + 5*a^3 + 8*b^3))/(35*d) + (2*exp(2*c + 2*d*x)*(3*a^2*b + 15*a^3 - 16*b^3))/(35*d) - (6*exp(4*c + 4*d*
x)*(2*a*b^2 + a^2*b - a^3))/(7*d) - (2*exp(6*c + 6*d*x)*(3*a^2*b - a^3))/(7*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) + ((2*(3*a^2*b + 15*a^3 - 16*b^3))/(105*d) + (4*exp(4*
c + 4*d*x)*(3*a^2*b + 15*a^3 - 16*b^3))/(35*d) - (8*exp(6*c + 6*d*x)*(2*a*b^2 + a^2*b - a^3))/(7*d) + (8*exp(2
*c + 2*d*x)*(6*a*b^2 + 3*a^2*b + 5*a^3 + 8*b^3))/(35*d) - (2*exp(8*c + 8*d*x)*(3*a^2*b - a^3))/(7*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*(
2*a*b^2 + a^2*b - a^3))/(7*d) + (2*exp(2*c + 2*d*x)*(3*a^2*b - a^3))/(7*d))/(2*exp(2*c + 2*d*x) + exp(4*c + 4*
d*x) + 1) - (2*(3*a^2*b - a^3))/(7*d*(exp(2*c + 2*d*x) + 1))

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