3.2.13 \(\int (a+b \text {sech}^2(c+d x))^2 \tanh ^3(c+d x) \, dx\) [113]

Optimal. Leaf size=77 \[ \frac {a^2 \log (\cosh (c+d x))}{d}+\frac {a (a-2 b) \text {sech}^2(c+d x)}{2 d}+\frac {(2 a-b) b \text {sech}^4(c+d x)}{4 d}+\frac {b^2 \text {sech}^6(c+d x)}{6 d} \]

[Out]

a^2*ln(cosh(d*x+c))/d+1/2*a*(a-2*b)*sech(d*x+c)^2/d+1/4*(2*a-b)*b*sech(d*x+c)^4/d+1/6*b^2*sech(d*x+c)^6/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 = {4223, 457, 77} \begin {gather*} \frac {a^2 \log (\cosh (c+d x))}{d}+\frac {b (2 a-b) \text {sech}^4(c+d x)}{4 d}+\frac {a (a-2 b) \text {sech}^2(c+d x)}{2 d}+\frac {b^2 \text {sech}^6(c+d x)}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*Log[Cosh[c + d*x]])/d + (a*(a - 2*b)*Sech[c + d*x]^2)/(2*d) + ((2*a - b)*b*Sech[c + d*x]^4)/(4*d) + (b^2*
Sech[c + d*x]^6)/(6*d)

Rule 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4223

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

Rubi steps

\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right )^2 \tanh ^3(c+d x) \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right ) \left (b+a x^2\right )^2}{x^7} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {(1-x) (b+a x)^2}{x^4} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\text {Subst}\left (\int \left (\frac {b^2}{x^4}+\frac {(2 a-b) b}{x^3}+\frac {a (a-2 b)}{x^2}-\frac {a^2}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {a^2 \log (\cosh (c+d x))}{d}+\frac {a (a-2 b) \text {sech}^2(c+d x)}{2 d}+\frac {(2 a-b) b \text {sech}^4(c+d x)}{4 d}+\frac {b^2 \text {sech}^6(c+d x)}{6 d}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 107, normalized size = 1.39 \begin {gather*} \frac {\cosh ^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \left (12 a^2 \log (\cosh (c+d x))+6 a (a-2 b) \text {sech}^2(c+d x)+3 (2 a-b) b \text {sech}^4(c+d x)+2 b^2 \text {sech}^6(c+d x)\right )}{3 d (a+2 b+a \cosh (2 c+2 d x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[c + d*x]^4*(a + b*Sech[c + d*x]^2)^2*(12*a^2*Log[Cosh[c + d*x]] + 6*a*(a - 2*b)*Sech[c + d*x]^2 + 3*(2*a
- b)*b*Sech[c + d*x]^4 + 2*b^2*Sech[c + d*x]^6))/(3*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^2)

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Maple [A]
time = 1.42, size = 94, normalized size = 1.22

method	result	size
derivativedivides	\(\frac {a^{2} \left (\ln \left (\cosh \left (d x +c \right )\right )-\frac {\left (\tanh ^{2}\left (d x +c \right )\right )}{2}\right )+2 a b \left (-\frac {\sinh ^{2}\left (d x +c \right )}{2 \cosh \left (d x +c \right )^{4}}-\frac {1}{4 \cosh \left (d x +c \right )^{4}}\right )+b^{2} \left (-\frac {\sinh ^{2}\left (d x +c \right )}{4 \cosh \left (d x +c \right )^{6}}-\frac {1}{12 \cosh \left (d x +c \right )^{6}}\right )}{d}\)	\(94\)
default	\(\frac {a^{2} \left (\ln \left (\cosh \left (d x +c \right )\right )-\frac {\left (\tanh ^{2}\left (d x +c \right )\right )}{2}\right )+2 a b \left (-\frac {\sinh ^{2}\left (d x +c \right )}{2 \cosh \left (d x +c \right )^{4}}-\frac {1}{4 \cosh \left (d x +c \right )^{4}}\right )+b^{2} \left (-\frac {\sinh ^{2}\left (d x +c \right )}{4 \cosh \left (d x +c \right )^{6}}-\frac {1}{12 \cosh \left (d x +c \right )^{6}}\right )}{d}\)	\(94\)
risch	\(-a^{2} x -\frac {2 a^{2} c}{d}+\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (3 a^{2} {\mathrm e}^{8 d x +8 c}-6 a b \,{\mathrm e}^{8 d x +8 c}+12 a^{2} {\mathrm e}^{6 d x +6 c}-12 a b \,{\mathrm e}^{6 d x +6 c}-6 b^{2} {\mathrm e}^{6 d x +6 c}+18 a^{2} {\mathrm e}^{4 d x +4 c}-12 a b \,{\mathrm e}^{4 d x +4 c}+4 b^{2} {\mathrm e}^{4 d x +4 c}+12 a^{2} {\mathrm e}^{2 d x +2 c}-12 a b \,{\mathrm e}^{2 d x +2 c}-6 b^{2} {\mathrm e}^{2 d x +2 c}+3 a^{2}-6 a b \right )}{3 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{6}}+\frac {a^{2} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d}\)	\(223\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^2*(ln(cosh(d*x+c))-1/2*tanh(d*x+c)^2)+2*a*b*(-1/2*sinh(d*x+c)^2/cosh(d*x+c)^4-1/4/cosh(d*x+c)^4)+b^2*(-
1/4*sinh(d*x+c)^2/cosh(d*x+c)^6-1/12/cosh(d*x+c)^6))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (71) = 142\).
time = 0.47, size = 333, normalized size = 4.32 \begin {gather*} \frac {a b \tanh \left (d x + c\right )^{4}}{2 \, d} + a^{2} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} - \frac {4}{3} \, b^{2} {\left (\frac {3 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}} - \frac {2 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}} + \frac {3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, 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)^3,x, algorithm="maxima")

[Out]

1/2*a*b*tanh(d*x + c)^4/d + a^2*(x + c/d + log(e^(-2*d*x - 2*c) + 1)/d + 2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x -
2*c) + e^(-4*d*x - 4*c) + 1))) - 4/3*b^2*(3*e^(-4*d*x - 4*c)/(d*(6*e^(-2*d*x - 2*c) + 15*e^(-4*d*x - 4*c) + 20
*e^(-6*d*x - 6*c) + 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) + e^(-12*d*x - 12*c) + 1)) - 2*e^(-6*d*x - 6*c)
/(d*(6*e^(-2*d*x - 2*c) + 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) + 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*
c) + e^(-12*d*x - 12*c) + 1)) + 3*e^(-8*d*x - 8*c)/(d*(6*e^(-2*d*x - 2*c) + 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x
- 6*c) + 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) + e^(-12*d*x - 12*c) + 1)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2591 vs. \(2 (71) = 142\).
time = 0.38, size = 2591, normalized size = 33.65 \begin {gather*} \text {Too large to display} \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)^3,x, algorithm="fricas")

[Out]

-1/3*(3*a^2*d*x*cosh(d*x + c)^12 + 36*a^2*d*x*cosh(d*x + c)*sinh(d*x + c)^11 + 3*a^2*d*x*sinh(d*x + c)^12 + 6*
(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^10 + 6*(33*a^2*d*x*cosh(d*x + c)^2 + 3*a^2*d*x - a^2 + 2*a*b)*sinh(d*x
+ c)^10 + 60*(11*a^2*d*x*cosh(d*x + c)^3 + (3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^9 + 3*(15*a
^2*d*x - 8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^8 + 3*(495*a^2*d*x*cosh(d*x + c)^4 + 15*a^2*d*x + 90*(3*a^2*d*x
- a^2 + 2*a*b)*cosh(d*x + c)^2 - 8*a^2 + 8*a*b + 4*b^2)*sinh(d*x + c)^8 + 24*(99*a^2*d*x*cosh(d*x + c)^5 + 30*
(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^3 + (15*a^2*d*x - 8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^
7 + 4*(15*a^2*d*x - 9*a^2 + 6*a*b - 2*b^2)*cosh(d*x + c)^6 + 4*(693*a^2*d*x*cosh(d*x + c)^6 + 315*(3*a^2*d*x -
a^2 + 2*a*b)*cosh(d*x + c)^4 + 15*a^2*d*x + 21*(15*a^2*d*x - 8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^2 - 9*a^2 +
6*a*b - 2*b^2)*sinh(d*x + c)^6 + 24*(99*a^2*d*x*cosh(d*x + c)^7 + 63*(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^
5 + 7*(15*a^2*d*x - 8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^3 + (15*a^2*d*x - 9*a^2 + 6*a*b - 2*b^2)*cosh(d*x + c
))*sinh(d*x + c)^5 + 3*(15*a^2*d*x - 8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^4 + 3*(495*a^2*d*x*cosh(d*x + c)^8 +
420*(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^6 + 70*(15*a^2*d*x - 8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^4 + 15*
a^2*d*x + 20*(15*a^2*d*x - 9*a^2 + 6*a*b - 2*b^2)*cosh(d*x + c)^2 - 8*a^2 + 8*a*b + 4*b^2)*sinh(d*x + c)^4 + 3
*a^2*d*x + 4*(165*a^2*d*x*cosh(d*x + c)^9 + 180*(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^7 + 42*(15*a^2*d*x - 8
*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^5 + 20*(15*a^2*d*x - 9*a^2 + 6*a*b - 2*b^2)*cosh(d*x + c)^3 + 3*(15*a^2*d*
x - 8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^2 + 6*(3
3*a^2*d*x*cosh(d*x + c)^10 + 45*(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^8 + 14*(15*a^2*d*x - 8*a^2 + 8*a*b + 4
*b^2)*cosh(d*x + c)^6 + 10*(15*a^2*d*x - 9*a^2 + 6*a*b - 2*b^2)*cosh(d*x + c)^4 + 3*a^2*d*x + 3*(15*a^2*d*x -
8*a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^2 - a^2 + 2*a*b)*sinh(d*x + c)^2 - 3*(a^2*cosh(d*x + c)^12 + 12*a^2*cosh(
d*x + c)*sinh(d*x + c)^11 + a^2*sinh(d*x + c)^12 + 6*a^2*cosh(d*x + c)^10 + 6*(11*a^2*cosh(d*x + c)^2 + a^2)*s
inh(d*x + c)^10 + 15*a^2*cosh(d*x + c)^8 + 20*(11*a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c))*sinh(d*x + c)^9 +
15*(33*a^2*cosh(d*x + c)^4 + 18*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^8 + 20*a^2*cosh(d*x + c)^6 + 24*(33*
a^2*cosh(d*x + c)^5 + 30*a^2*cosh(d*x + c)^3 + 5*a^2*cosh(d*x + c))*sinh(d*x + c)^7 + 4*(231*a^2*cosh(d*x + c)
^6 + 315*a^2*cosh(d*x + c)^4 + 105*a^2*cosh(d*x + c)^2 + 5*a^2)*sinh(d*x + c)^6 + 15*a^2*cosh(d*x + c)^4 + 24*
(33*a^2*cosh(d*x + c)^7 + 63*a^2*cosh(d*x + c)^5 + 35*a^2*cosh(d*x + c)^3 + 5*a^2*cosh(d*x + c))*sinh(d*x + c)
^5 + 15*(33*a^2*cosh(d*x + c)^8 + 84*a^2*cosh(d*x + c)^6 + 70*a^2*cosh(d*x + c)^4 + 20*a^2*cosh(d*x + c)^2 + a
^2)*sinh(d*x + c)^4 + 6*a^2*cosh(d*x + c)^2 + 20*(11*a^2*cosh(d*x + c)^9 + 36*a^2*cosh(d*x + c)^7 + 42*a^2*cos
h(d*x + c)^5 + 20*a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(11*a^2*cosh(d*x + c)^10 + 45
*a^2*cosh(d*x + c)^8 + 70*a^2*cosh(d*x + c)^6 + 50*a^2*cosh(d*x + c)^4 + 15*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*
x + c)^2 + a^2 + 12*(a^2*cosh(d*x + c)^11 + 5*a^2*cosh(d*x + c)^9 + 10*a^2*cosh(d*x + c)^7 + 10*a^2*cosh(d*x +
c)^5 + 5*a^2*cosh(d*x + c)^3 + a^2*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*
x + c))) + 12*(3*a^2*d*x*cosh(d*x + c)^11 + 5*(3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c)^9 + 2*(15*a^2*d*x - 8*a^
2 + 8*a*b + 4*b^2)*cosh(d*x + c)^7 + 2*(15*a^2*d*x - 9*a^2 + 6*a*b - 2*b^2)*cosh(d*x + c)^5 + (15*a^2*d*x - 8*
a^2 + 8*a*b + 4*b^2)*cosh(d*x + c)^3 + (3*a^2*d*x - a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c
)^12 + 12*d*cosh(d*x + c)*sinh(d*x + c)^11 + d*sinh(d*x + c)^12 + 6*d*cosh(d*x + c)^10 + 6*(11*d*cosh(d*x + c)
^2 + d)*sinh(d*x + c)^10 + 20*(11*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^9 + 15*d*cosh(d*x + c)^
8 + 15*(33*d*cosh(d*x + c)^4 + 18*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^8 + 24*(33*d*cosh(d*x + c)^5 + 30*d*cos
h(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)^7 + 20*d*cosh(d*x + c)^6 + 4*(231*d*cosh(d*x + c)^6 + 315*d*co
sh(d*x + c)^4 + 105*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^6 + 24*(33*d*cosh(d*x + c)^7 + 63*d*cosh(d*x + c)^5
+ 35*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)^5 + 15*d*cosh(d*x + c)^4 + 15*(33*d*cosh(d*x + c)^8
+ 84*d*cosh(d*x + c)^6 + 70*d*cosh(d*x + c)^4 + 20*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^4 + 20*(11*d*cosh(d*x
+ c)^9 + 36*d*cosh(d*x + c)^7 + 42*d*cosh(d*x + c)^5 + 20*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c
)^3 + 6*d*cosh(d*x + c)^2 + 6*(11*d*cosh(d*x + c)^10 + 45*d*cosh(d*x + c)^8 + 70*d*cosh(d*x + c)^6 + 50*d*cosh
(d*x + c)^4 + 15*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 12*(d*cosh(d*x + c)^11 + 5*d*cosh(d*x + c)^9 + 10*d*
cosh(d*x + c)^7 + 10*d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c) + d)

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Sympy [A]
time = 0.86, size = 129, normalized size = 1.68 \begin {gather*} \begin {cases} a^{2} x - \frac {a^{2} \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} - \frac {a^{2} \tanh ^{2}{\left (c + d x \right )}}{2 d} - \frac {a b \tanh ^{2}{\left (c + d x \right )} \operatorname {sech}^{2}{\left (c + d x \right )}}{2 d} - \frac {a b \operatorname {sech}^{2}{\left (c + d x \right )}}{2 d} - \frac {b^{2} \tanh ^{2}{\left (c + d x \right )} \operatorname {sech}^{4}{\left (c + d x \right )}}{6 d} - \frac {b^{2} \operatorname {sech}^{4}{\left (c + d x \right )}}{12 d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {sech}^{2}{\left (c \right )}\right )^{2} \tanh ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \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)**3,x)

[Out]

Piecewise((a**2*x - a**2*log(tanh(c + d*x) + 1)/d - a**2*tanh(c + d*x)**2/(2*d) - a*b*tanh(c + d*x)**2*sech(c
+ d*x)**2/(2*d) - a*b*sech(c + d*x)**2/(2*d) - b**2*tanh(c + d*x)**2*sech(c + d*x)**4/(6*d) - b**2*sech(c + d*
x)**4/(12*d), Ne(d, 0)), (x*(a + b*sech(c)**2)**2*tanh(c)**3, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (71) = 142\).
time = 0.45, size = 244, normalized size = 3.17 \begin {gather*} -\frac {60 \, {\left (d x + c\right )} a^{2} - 60 \, a^{2} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + \frac {147 \, a^{2} e^{\left (12 \, d x + 12 \, c\right )} + 762 \, a^{2} e^{\left (10 \, d x + 10 \, c\right )} + 240 \, a b e^{\left (10 \, d x + 10 \, c\right )} + 1725 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 480 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 240 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 2220 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 480 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 160 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1725 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 480 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 240 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 762 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 240 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 147 \, a^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{6}}}{60 \, 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)^3,x, algorithm="giac")

[Out]

-1/60*(60*(d*x + c)*a^2 - 60*a^2*log(e^(2*d*x + 2*c) + 1) + (147*a^2*e^(12*d*x + 12*c) + 762*a^2*e^(10*d*x + 1
0*c) + 240*a*b*e^(10*d*x + 10*c) + 1725*a^2*e^(8*d*x + 8*c) + 480*a*b*e^(8*d*x + 8*c) + 240*b^2*e^(8*d*x + 8*c
) + 2220*a^2*e^(6*d*x + 6*c) + 480*a*b*e^(6*d*x + 6*c) - 160*b^2*e^(6*d*x + 6*c) + 1725*a^2*e^(4*d*x + 4*c) +
480*a*b*e^(4*d*x + 4*c) + 240*b^2*e^(4*d*x + 4*c) + 762*a^2*e^(2*d*x + 2*c) + 240*a*b*e^(2*d*x + 2*c) + 147*a^
2)/(e^(2*d*x + 2*c) + 1)^6)/d

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Mupad [B]
time = 1.52, size = 349, normalized size = 4.53 \begin {gather*} \frac {4\,\left (2\,a\,b-9\,b^2\right )}{d\,\left (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\right )}-\frac {32\,b^2}{3\,d\,\left (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\right )}-\frac {2\,\left (a^2-6\,a\,b+2\,b^2\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {2\,\left (2\,a\,b-a^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,\left (6\,a\,b-7\,b^2\right )}{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 )}-a^2\,x+\frac {a^2\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1\right )}{d}+\frac {32\,b^2}{d\,\left (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\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(c + d*x)^3*(a + b/cosh(c + d*x)^2)^2,x)

[Out]

(4*(2*a*b - 9*b^2))/(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))
- (32*b^2)/(3*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)) - (2*(a^2 - 6*a*b + 2*b^2))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x
) + 1)) - (2*(2*a*b - a^2))/(d*(exp(2*c + 2*d*x) + 1)) - (8*(6*a*b - 7*b^2))/(3*d*(3*exp(2*c + 2*d*x) + 3*exp(
4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) - a^2*x + (a^2*log(exp(2*c)*exp(2*d*x) + 1))/d + (32*b^2)/(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))

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