3.2.25 \(\int (a+b \text {sech}^2(c+d x))^3 \tanh ^3(c+d x) \, dx\) [125]

Optimal. Leaf size=103 \[ \frac {a^3 \log (\cosh (c+d x))}{d}-\frac {3 a^2 b \text {sech}^2(c+d x)}{2 d}-\frac {3 a b^2 \text {sech}^4(c+d x)}{4 d}-\frac {b^3 \text {sech}^6(c+d x)}{6 d}+\frac {\left (b+a \cosh ^2(c+d x)\right )^4 \text {sech}^8(c+d x)}{8 b d} \]

[Out]

a^3*ln(cosh(d*x+c))/d-3/2*a^2*b*sech(d*x+c)^2/d-3/4*a*b^2*sech(d*x+c)^4/d-1/6*b^3*sech(d*x+c)^6/d+1/8*(b+a*cos
h(d*x+c)^2)^4*sech(d*x+c)^8/b/d

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Rubi [A]
time = 0.07, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4223, 457, 79, 45} \begin {gather*} \frac {a^3 \log (\cosh (c+d x))}{d}-\frac {3 a^2 b \text {sech}^2(c+d x)}{2 d}-\frac {3 a b^2 \text {sech}^4(c+d x)}{4 d}+\frac {\text {sech}^8(c+d x) \left (a \cosh ^2(c+d x)+b\right )^4}{8 b d}-\frac {b^3 \text {sech}^6(c+d x)}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*Log[Cosh[c + d*x]])/d - (3*a^2*b*Sech[c + d*x]^2)/(2*d) - (3*a*b^2*Sech[c + d*x]^4)/(4*d) - (b^3*Sech[c +
d*x]^6)/(6*d) + ((b + a*Cosh[c + d*x]^2)^4*Sech[c + d*x]^8)/(8*b*d)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

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 )^3 \tanh ^3(c+d x) \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right ) \left (b+a x^2\right )^3}{x^9} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {(1-x) (b+a x)^3}{x^5} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {\left (b+a \cosh ^2(c+d x)\right )^4 \text {sech}^8(c+d x)}{8 b d}+\frac {\text {Subst}\left (\int \frac {(b+a x)^3}{x^4} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {\left (b+a \cosh ^2(c+d x)\right )^4 \text {sech}^8(c+d x)}{8 b d}+\frac {\text {Subst}\left (\int \left (\frac {b^3}{x^4}+\frac {3 a b^2}{x^3}+\frac {3 a^2 b}{x^2}+\frac {a^3}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {a^3 \log (\cosh (c+d x))}{d}-\frac {3 a^2 b \text {sech}^2(c+d x)}{2 d}-\frac {3 a b^2 \text {sech}^4(c+d x)}{4 d}-\frac {b^3 \text {sech}^6(c+d x)}{6 d}+\frac {\left (b+a \cosh ^2(c+d x)\right )^4 \text {sech}^8(c+d x)}{8 b d}\\ \end {align*}

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Mathematica [A]
time = 0.51, size = 128, normalized size = 1.24 \begin {gather*} \frac {\cosh ^6(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \left (24 a^3 \log (\cosh (c+d x))+12 a^2 (a-3 b) \text {sech}^2(c+d x)+18 a (a-b) b \text {sech}^4(c+d x)+4 (3 a-b) b^2 \text {sech}^6(c+d x)+3 b^3 \text {sech}^8(c+d x)\right )}{3 d (a+2 b+a \cosh (2 c+2 d x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[c + d*x]^6*(a + b*Sech[c + d*x]^2)^3*(24*a^3*Log[Cosh[c + d*x]] + 12*a^2*(a - 3*b)*Sech[c + d*x]^2 + 18*
a*(a - b)*b*Sech[c + d*x]^4 + 4*(3*a - b)*b^2*Sech[c + d*x]^6 + 3*b^3*Sech[c + d*x]^8))/(3*d*(a + 2*b + a*Cosh
[2*c + 2*d*x])^3)

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Maple [A]
time = 1.72, size = 131, normalized size = 1.27

method	result	size
derivativedivides	\(\frac {a^{3} \left (\ln \left (\cosh \left (d x +c \right )\right )-\frac {\left (\tanh ^{2}\left (d x +c \right )\right )}{2}\right )+3 a^{2} b \left (-\frac {\sinh ^{2}\left (d x +c \right )}{2 \cosh \left (d x +c \right )^{4}}-\frac {1}{4 \cosh \left (d x +c \right )^{4}}\right )+3 a \,b^{2} \left (-\frac {\sinh ^{2}\left (d x +c \right )}{4 \cosh \left (d x +c \right )^{6}}-\frac {1}{12 \cosh \left (d x +c \right )^{6}}\right )+b^{3} \left (-\frac {\sinh ^{2}\left (d x +c \right )}{6 \cosh \left (d x +c \right )^{8}}-\frac {1}{24 \cosh \left (d x +c \right )^{8}}\right )}{d}\)	\(131\)
default	\(\frac {a^{3} \left (\ln \left (\cosh \left (d x +c \right )\right )-\frac {\left (\tanh ^{2}\left (d x +c \right )\right )}{2}\right )+3 a^{2} b \left (-\frac {\sinh ^{2}\left (d x +c \right )}{2 \cosh \left (d x +c \right )^{4}}-\frac {1}{4 \cosh \left (d x +c \right )^{4}}\right )+3 a \,b^{2} \left (-\frac {\sinh ^{2}\left (d x +c \right )}{4 \cosh \left (d x +c \right )^{6}}-\frac {1}{12 \cosh \left (d x +c \right )^{6}}\right )+b^{3} \left (-\frac {\sinh ^{2}\left (d x +c \right )}{6 \cosh \left (d x +c \right )^{8}}-\frac {1}{24 \cosh \left (d x +c \right )^{8}}\right )}{d}\)	\(131\)
risch	\(-a^{3} x -\frac {2 a^{3} c}{d}+\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (3 a^{3} {\mathrm e}^{12 d x +12 c}-9 a^{2} b \,{\mathrm e}^{12 d x +12 c}+18 a^{3} {\mathrm e}^{10 d x +10 c}-36 a^{2} b \,{\mathrm e}^{10 d x +10 c}-18 a \,b^{2} {\mathrm e}^{10 d x +10 c}+45 a^{3} {\mathrm e}^{8 d x +8 c}-63 a^{2} b \,{\mathrm e}^{8 d x +8 c}-24 a \,b^{2} {\mathrm e}^{8 d x +8 c}-16 b^{3} {\mathrm e}^{8 d x +8 c}+60 a^{3} {\mathrm e}^{6 d x +6 c}-72 a^{2} b \,{\mathrm e}^{6 d x +6 c}-12 a \,b^{2} {\mathrm e}^{6 d x +6 c}+16 b^{3} {\mathrm e}^{6 d x +6 c}+45 a^{3} {\mathrm e}^{4 d x +4 c}-63 a^{2} b \,{\mathrm e}^{4 d x +4 c}-24 a \,b^{2} {\mathrm e}^{4 d x +4 c}-16 b^{3} {\mathrm e}^{4 d x +4 c}+18 a^{3} {\mathrm e}^{2 d x +2 c}-36 a^{2} b \,{\mathrm e}^{2 d x +2 c}-18 a \,b^{2} {\mathrm e}^{2 d x +2 c}+3 a^{3}-9 a^{2} b \right )}{3 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{8}}+\frac {a^{3} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d}\)	\(366\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^3*(ln(cosh(d*x+c))-1/2*tanh(d*x+c)^2)+3*a^2*b*(-1/2*sinh(d*x+c)^2/cosh(d*x+c)^4-1/4/cosh(d*x+c)^4)+3*a*
b^2*(-1/4*sinh(d*x+c)^2/cosh(d*x+c)^6-1/12/cosh(d*x+c)^6)+b^3*(-1/6*sinh(d*x+c)^2/cosh(d*x+c)^8-1/24/cosh(d*x+
c)^8))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 652 vs. \(2 (95) = 190\).
time = 0.48, size = 652, normalized size = 6.33 \begin {gather*} \frac {3 \, a^{2} b \tanh \left (d x + c\right )^{4}}{4 \, d} + a^{3} {\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 )} - 4 \, a 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 )} - \frac {32}{3} \, b^{3} {\left (\frac {e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}} - \frac {e^{\left (-8 \, d x - 8 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}} + \frac {e^{\left (-10 \, d x - 10 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, 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)^3,x, algorithm="maxima")

[Out]

3/4*a^2*b*tanh(d*x + c)^4/d + a^3*(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*a*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 - 1
0*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))) - 32/3*b^3*(e^(-6*d*x - 6*c
)/(d*(8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 1
0*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)) - e^(-8*d*x - 8*c)/(d*(8*e^(-2*
d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 10*c) + 28*e^(-
12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)) + e^(-10*d*x - 10*c)/(d*(8*e^(-2*d*x - 2*c) +
28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 10*c) + 28*e^(-12*d*x - 12*
c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4658 vs. \(2 (95) = 190\).
time = 0.43, size = 4658, normalized size = 45.22 \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)^3*tanh(d*x+c)^3,x, algorithm="fricas")

[Out]

-1/3*(3*a^3*d*x*cosh(d*x + c)^16 + 48*a^3*d*x*cosh(d*x + c)*sinh(d*x + c)^15 + 3*a^3*d*x*sinh(d*x + c)^16 + 6*
(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^14 + 6*(60*a^3*d*x*cosh(d*x + c)^2 + 4*a^3*d*x - a^3 + 3*a^2*b)*sinh
(d*x + c)^14 + 84*(20*a^3*d*x*cosh(d*x + c)^3 + (4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^13 +
12*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^12 + 6*(910*a^3*d*x*cosh(d*x + c)^4 + 14*a^3*d*x - 6*
a^3 + 12*a^2*b + 6*a*b^2 + 91*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^12 + 24*(546*a^3*d*x*
cosh(d*x + c)^5 + 91*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^3 + 6*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*c
osh(d*x + c))*sinh(d*x + c)^11 + 2*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^10 + 2*(
12012*a^3*d*x*cosh(d*x + c)^6 + 84*a^3*d*x + 3003*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^4 - 45*a^3 + 63*a^
2*b + 24*a*b^2 + 16*b^3 + 396*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 4*(8
580*a^3*d*x*cosh(d*x + c)^7 + 3003*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^5 + 660*(7*a^3*d*x - 3*a^3 + 6*a^
2*b + 3*a*b^2)*cosh(d*x + c)^3 + 5*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c))*sinh(d*
x + c)^9 + 2*(105*a^3*d*x - 60*a^3 + 72*a^2*b + 12*a*b^2 - 16*b^3)*cosh(d*x + c)^8 + 2*(19305*a^3*d*x*cosh(d*x
+ c)^8 + 9009*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^6 + 105*a^3*d*x + 2970*(7*a^3*d*x - 3*a^3 + 6*a^2*b +
3*a*b^2)*cosh(d*x + c)^4 - 60*a^3 + 72*a^2*b + 12*a*b^2 - 16*b^3 + 45*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*
b^2 + 16*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 16*(2145*a^3*d*x*cosh(d*x + c)^9 + 1287*(4*a^3*d*x - a^3 + 3*
a^2*b)*cosh(d*x + c)^7 + 594*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^5 + 15*(84*a^3*d*x - 45*a^3
+ 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^3 + (105*a^3*d*x - 60*a^3 + 72*a^2*b + 12*a*b^2 - 16*b^3)*cosh(
d*x + c))*sinh(d*x + c)^7 + 2*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^6 + 2*(12012*
a^3*d*x*cosh(d*x + c)^10 + 9009*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^8 + 5544*(7*a^3*d*x - 3*a^3 + 6*a^2*
b + 3*a*b^2)*cosh(d*x + c)^6 + 84*a^3*d*x + 210*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x
+ c)^4 - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3 + 28*(105*a^3*d*x - 60*a^3 + 72*a^2*b + 12*a*b^2 - 16*b^3)*cosh
(d*x + c)^2)*sinh(d*x + c)^6 + 4*(3276*a^3*d*x*cosh(d*x + c)^11 + 3003*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x +
c)^9 + 2376*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^7 + 126*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24
*a*b^2 + 16*b^3)*cosh(d*x + c)^5 + 28*(105*a^3*d*x - 60*a^3 + 72*a^2*b + 12*a*b^2 - 16*b^3)*cosh(d*x + c)^3 +
3*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*a^3*d*x + 12*(7*a^3*
d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^4 + 2*(2730*a^3*d*x*cosh(d*x + c)^12 + 3003*(4*a^3*d*x - a^3 +
3*a^2*b)*cosh(d*x + c)^10 + 2970*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^8 + 210*(84*a^3*d*x - 4
5*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^6 + 42*a^3*d*x + 70*(105*a^3*d*x - 60*a^3 + 72*a^2*b + 12*
a*b^2 - 16*b^3)*cosh(d*x + c)^4 - 18*a^3 + 36*a^2*b + 18*a*b^2 + 15*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2
+ 16*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(210*a^3*d*x*cosh(d*x + c)^13 + 273*(4*a^3*d*x - a^3 + 3*a^2*b
)*cosh(d*x + c)^11 + 330*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^9 + 30*(84*a^3*d*x - 45*a^3 + 6
3*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^7 + 14*(105*a^3*d*x - 60*a^3 + 72*a^2*b + 12*a*b^2 - 16*b^3)*cosh(d
*x + c)^5 + 5*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^3 + 6*(7*a^3*d*x - 3*a^3 + 6*
a^2*b + 3*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^2 + 2*(180*a^3*d
*x*cosh(d*x + c)^14 + 273*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^12 + 396*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*
a*b^2)*cosh(d*x + c)^10 + 45*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^8 + 28*(105*a^
3*d*x - 60*a^3 + 72*a^2*b + 12*a*b^2 - 16*b^3)*cosh(d*x + c)^6 + 12*a^3*d*x + 15*(84*a^3*d*x - 45*a^3 + 63*a^2
*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^4 - 3*a^3 + 9*a^2*b + 36*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*
x + c)^2)*sinh(d*x + c)^2 - 3*(a^3*cosh(d*x + c)^16 + 16*a^3*cosh(d*x + c)*sinh(d*x + c)^15 + a^3*sinh(d*x + c
)^16 + 8*a^3*cosh(d*x + c)^14 + 28*a^3*cosh(d*x + c)^12 + 8*(15*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^14 +
112*(5*a^3*cosh(d*x + c)^3 + a^3*cosh(d*x + c))*sinh(d*x + c)^13 + 56*a^3*cosh(d*x + c)^10 + 28*(65*a^3*cosh(d
*x + c)^4 + 26*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^12 + 112*(39*a^3*cosh(d*x + c)^5 + 26*a^3*cosh(d*x + c
)^3 + 3*a^3*cosh(d*x + c))*sinh(d*x + c)^11 + 70*a^3*cosh(d*x + c)^8 + 56*(143*a^3*cosh(d*x + c)^6 + 143*a^3*c
osh(d*x + c)^4 + 33*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^10 + 16*(715*a^3*cosh(d*x + c)^7 + 1001*a^3*cosh(
d*x + c)^5 + 385*a^3*cosh(d*x + c)^3 + 35*a^3*cosh(d*x + c))*sinh(d*x + c)^9 + 56*a^3*cosh(d*x + c)^6 + 2*(643
5*a^3*cosh(d*x + c)^8 + 12012*a^3*cosh(d*x + c)...

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Sympy [A]
time = 1.72, size = 178, normalized size = 1.73 \begin {gather*} \begin {cases} a^{3} x - \frac {a^{3} \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} - \frac {a^{3} \tanh ^{2}{\left (c + d x \right )}}{2 d} - \frac {3 a^{2} b \tanh ^{2}{\left (c + d x \right )} \operatorname {sech}^{2}{\left (c + d x \right )}}{4 d} - \frac {3 a^{2} b \operatorname {sech}^{2}{\left (c + d x \right )}}{4 d} - \frac {a b^{2} \tanh ^{2}{\left (c + d x \right )} \operatorname {sech}^{4}{\left (c + d x \right )}}{2 d} - \frac {a b^{2} \operatorname {sech}^{4}{\left (c + d x \right )}}{4 d} - \frac {b^{3} \tanh ^{2}{\left (c + d x \right )} \operatorname {sech}^{6}{\left (c + d x \right )}}{8 d} - \frac {b^{3} \operatorname {sech}^{6}{\left (c + d x \right )}}{24 d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {sech}^{2}{\left (c \right )}\right )^{3} \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)**3*tanh(d*x+c)**3,x)

[Out]

Piecewise((a**3*x - a**3*log(tanh(c + d*x) + 1)/d - a**3*tanh(c + d*x)**2/(2*d) - 3*a**2*b*tanh(c + d*x)**2*se
ch(c + d*x)**2/(4*d) - 3*a**2*b*sech(c + d*x)**2/(4*d) - a*b**2*tanh(c + d*x)**2*sech(c + d*x)**4/(2*d) - a*b*
*2*sech(c + d*x)**4/(4*d) - b**3*tanh(c + d*x)**2*sech(c + d*x)**6/(8*d) - b**3*sech(c + d*x)**6/(24*d), Ne(d,
0)), (x*(a + b*sech(c)**2)**3*tanh(c)**3, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (95) = 190\).
time = 0.48, size = 387, normalized size = 3.76 \begin {gather*} -\frac {840 \, {\left (d x + c\right )} a^{3} - 840 \, a^{3} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + \frac {2283 \, a^{3} e^{\left (16 \, d x + 16 \, c\right )} + 16584 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} + 5040 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 53844 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 20160 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 10080 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 102648 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 35280 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 13440 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 8960 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 126210 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 40320 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 6720 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 8960 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 102648 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 35280 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 13440 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 8960 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 53844 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 20160 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 10080 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 16584 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5040 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 2283 \, a^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{840 \, 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)^3,x, algorithm="giac")

[Out]

-1/840*(840*(d*x + c)*a^3 - 840*a^3*log(e^(2*d*x + 2*c) + 1) + (2283*a^3*e^(16*d*x + 16*c) + 16584*a^3*e^(14*d
*x + 14*c) + 5040*a^2*b*e^(14*d*x + 14*c) + 53844*a^3*e^(12*d*x + 12*c) + 20160*a^2*b*e^(12*d*x + 12*c) + 1008
0*a*b^2*e^(12*d*x + 12*c) + 102648*a^3*e^(10*d*x + 10*c) + 35280*a^2*b*e^(10*d*x + 10*c) + 13440*a*b^2*e^(10*d
*x + 10*c) + 8960*b^3*e^(10*d*x + 10*c) + 126210*a^3*e^(8*d*x + 8*c) + 40320*a^2*b*e^(8*d*x + 8*c) + 6720*a*b^
2*e^(8*d*x + 8*c) - 8960*b^3*e^(8*d*x + 8*c) + 102648*a^3*e^(6*d*x + 6*c) + 35280*a^2*b*e^(6*d*x + 6*c) + 1344
0*a*b^2*e^(6*d*x + 6*c) + 8960*b^3*e^(6*d*x + 6*c) + 53844*a^3*e^(4*d*x + 4*c) + 20160*a^2*b*e^(4*d*x + 4*c) +
10080*a*b^2*e^(4*d*x + 4*c) + 16584*a^3*e^(2*d*x + 2*c) + 5040*a^2*b*e^(2*d*x + 2*c) + 2283*a^3)/(e^(2*d*x +
2*c) + 1)^8)/d

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Mupad [B]
time = 1.64, size = 573, normalized size = 5.56 \begin {gather*} \frac {32\,\left (3\,a\,b^2-5\,b^3\right )}{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 )}-a^3\,x-\frac {128\,b^3}{d\,\left (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\right )}-\frac {32\,\left (3\,a\,b^2-19\,b^3\right )}{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 {32\,b^3}{d\,\left (8\,{\mathrm {e}}^{2\,c+2\,d\,x}+28\,{\mathrm {e}}^{4\,c+4\,d\,x}+56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}+56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}+8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1\right )}-\frac {8\,\left (9\,a^2\,b-21\,a\,b^2+4\,b^3\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 )}+\frac {4\,\left (3\,a^2\,b-27\,a\,b^2+16\,b^3\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 {2\,\left (3\,a^2\,b-a^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {a^3\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1\right )}{d}-\frac {2\,\left (a^3-9\,a^2\,b+6\,a\,b^2\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,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)^3,x)

[Out]

(32*(3*a*b^2 - 5*b^3))/(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)) - a^3*x - (128*b^3)/(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)) - (32*(3*
a*b^2 - 19*b^3))/(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)) + (32*b^3)/(d*(8*exp(2*c + 2*d*x) + 28*exp(4*c + 4*d*x) + 56*e
xp(6*c + 6*d*x) + 70*exp(8*c + 8*d*x) + 56*exp(10*c + 10*d*x) + 28*exp(12*c + 12*d*x) + 8*exp(14*c + 14*d*x) +
exp(16*c + 16*d*x) + 1)) - (8*(9*a^2*b - 21*a*b^2 + 4*b^3))/(3*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + e
xp(6*c + 6*d*x) + 1)) + (4*(3*a^2*b - 27*a*b^2 + 16*b^3))/(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 - a^3))/(d*(exp(2*c + 2*d*x) + 1)) + (a^3*log(exp(2*c)*exp
(2*d*x) + 1))/d - (2*(6*a*b^2 - 9*a^2*b + a^3))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))

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