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

Optimal. Leaf size=71 \[ -\frac{3 a^2 b \text{sech}^2(c+d x)}{2 d}+\frac{a^3 \log (\cosh (c+d x))}{d}-\frac{3 a b^2 \text{sech}^4(c+d x)}{4 d}-\frac{b^3 \text{sech}^6(c+d x)}{6 d} \]

[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)

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Rubi [A]  time = 0.0581538, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4138, 266, 43} \[ -\frac{3 a^2 b \text{sech}^2(c+d x)}{2 d}+\frac{a^3 \log (\cosh (c+d x))}{d}-\frac{3 a b^2 \text{sech}^4(c+d x)}{4 d}-\frac{b^3 \text{sech}^6(c+d x)}{6 d} \]

Antiderivative was successfully verified.

[In]

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

Rule 4138

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]

Rule 266

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

Rule 43

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])

Rubi steps

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

Mathematica [A]  time = 0.291873, size = 100, normalized size = 1.41 \[ \frac{2 \text{sech}^6(c+d x) \left (a \cosh ^2(c+d x)+b\right )^3 \left (-18 a^2 b \cosh ^4(c+d x)+12 a^3 \cosh ^6(c+d x) \log (\cosh (c+d x))-9 a b^2 \cosh ^2(c+d x)-2 b^3\right )}{3 d (a \cosh (2 (c+d x))+a+2 b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b + a*Cosh[c + d*x]^2)^3*(-2*b^3 - 9*a*b^2*Cosh[c + d*x]^2 - 18*a^2*b*Cosh[c + d*x]^4 + 12*a^3*Cosh[c + d*
x]^6*Log[Cosh[c + d*x]])*Sech[c + d*x]^6)/(3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)

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Maple [A]  time = 0.013, size = 67, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3} \left ({\rm sech} \left (dx+c\right ) \right ) ^{6}}{6\,d}}-{\frac{3\,a{b}^{2} \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}}{4\,d}}-{\frac{3\,{a}^{2}b \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{2\,d}}-{\frac{{a}^{3}\ln \left ({\rm sech} \left (dx+c\right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12722, size = 115, normalized size = 1.62 \begin{align*} \frac{3 \, a^{2} b \tanh \left (d x + c\right )^{2}}{2 \, d} + \frac{a^{3} \log \left (\cosh \left (d x + c\right )\right )}{d} - \frac{12 \, a b^{2}}{d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4}} - \frac{32 \, b^{3}}{3 \, d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a^2*b*tanh(d*x + c)^2/d + a^3*log(cosh(d*x + c))/d - 12*a*b^2/(d*(e^(d*x + c) + e^(-d*x - c))^4) - 32/3*b^
3/(d*(e^(d*x + c) + e^(-d*x - c))^6)

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Fricas [B]  time = 2.75446, size = 6511, normalized size = 91.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*a^3*d*x*cosh(d*x + c)^12 + 36*a^3*d*x*cosh(d*x + c)*sinh(d*x + c)^11 + 3*a^3*d*x*sinh(d*x + c)^12 + 18
*(a^3*d*x + a^2*b)*cosh(d*x + c)^10 + 18*(11*a^3*d*x*cosh(d*x + c)^2 + a^3*d*x + a^2*b)*sinh(d*x + c)^10 + 60*
(11*a^3*d*x*cosh(d*x + c)^3 + 3*(a^3*d*x + a^2*b)*cosh(d*x + c))*sinh(d*x + c)^9 + 9*(5*a^3*d*x + 8*a^2*b + 4*
a*b^2)*cosh(d*x + c)^8 + 9*(165*a^3*d*x*cosh(d*x + c)^4 + 5*a^3*d*x + 8*a^2*b + 4*a*b^2 + 90*(a^3*d*x + a^2*b)
*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 72*(33*a^3*d*x*cosh(d*x + c)^5 + 30*(a^3*d*x + a^2*b)*cosh(d*x + c)^3 + (5
*a^3*d*x + 8*a^2*b + 4*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 + 4*(15*a^3*d*x + 27*a^2*b + 18*a*b^2 + 8*b^3)*co
sh(d*x + c)^6 + 4*(693*a^3*d*x*cosh(d*x + c)^6 + 15*a^3*d*x + 945*(a^3*d*x + a^2*b)*cosh(d*x + c)^4 + 27*a^2*b
+ 18*a*b^2 + 8*b^3 + 63*(5*a^3*d*x + 8*a^2*b + 4*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 24*(99*a^3*d*x*cos
h(d*x + c)^7 + 189*(a^3*d*x + a^2*b)*cosh(d*x + c)^5 + 21*(5*a^3*d*x + 8*a^2*b + 4*a*b^2)*cosh(d*x + c)^3 + (1
5*a^3*d*x + 27*a^2*b + 18*a*b^2 + 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*a^3*d*x + 9*(5*a^3*d*x + 8*a^2*b +
4*a*b^2)*cosh(d*x + c)^4 + 3*(495*a^3*d*x*cosh(d*x + c)^8 + 1260*(a^3*d*x + a^2*b)*cosh(d*x + c)^6 + 15*a^3*d
*x + 210*(5*a^3*d*x + 8*a^2*b + 4*a*b^2)*cosh(d*x + c)^4 + 24*a^2*b + 12*a*b^2 + 20*(15*a^3*d*x + 27*a^2*b + 1
8*a*b^2 + 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(165*a^3*d*x*cosh(d*x + c)^9 + 540*(a^3*d*x + a^2*b)*cos
h(d*x + c)^7 + 126*(5*a^3*d*x + 8*a^2*b + 4*a*b^2)*cosh(d*x + c)^5 + 20*(15*a^3*d*x + 27*a^2*b + 18*a*b^2 + 8*
b^3)*cosh(d*x + c)^3 + 9*(5*a^3*d*x + 8*a^2*b + 4*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 18*(a^3*d*x + a^2*b)
*cosh(d*x + c)^2 + 6*(33*a^3*d*x*cosh(d*x + c)^10 + 135*(a^3*d*x + a^2*b)*cosh(d*x + c)^8 + 42*(5*a^3*d*x + 8*
a^2*b + 4*a*b^2)*cosh(d*x + c)^6 + 3*a^3*d*x + 10*(15*a^3*d*x + 27*a^2*b + 18*a*b^2 + 8*b^3)*cosh(d*x + c)^4 +
3*a^2*b + 9*(5*a^3*d*x + 8*a^2*b + 4*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 3*(a^3*cosh(d*x + c)^12 + 12*a
^3*cosh(d*x + c)*sinh(d*x + c)^11 + a^3*sinh(d*x + c)^12 + 6*a^3*cosh(d*x + c)^10 + 15*a^3*cosh(d*x + c)^8 + 6
*(11*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^10 + 20*(11*a^3*cosh(d*x + c)^3 + 3*a^3*cosh(d*x + c))*sinh(d*x
+ c)^9 + 20*a^3*cosh(d*x + c)^6 + 15*(33*a^3*cosh(d*x + c)^4 + 18*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^8 +
24*(33*a^3*cosh(d*x + c)^5 + 30*a^3*cosh(d*x + c)^3 + 5*a^3*cosh(d*x + c))*sinh(d*x + c)^7 + 15*a^3*cosh(d*x
+ c)^4 + 4*(231*a^3*cosh(d*x + c)^6 + 315*a^3*cosh(d*x + c)^4 + 105*a^3*cosh(d*x + c)^2 + 5*a^3)*sinh(d*x + c)
^6 + 24*(33*a^3*cosh(d*x + c)^7 + 63*a^3*cosh(d*x + c)^5 + 35*a^3*cosh(d*x + c)^3 + 5*a^3*cosh(d*x + c))*sinh(
d*x + c)^5 + 6*a^3*cosh(d*x + c)^2 + 15*(33*a^3*cosh(d*x + c)^8 + 84*a^3*cosh(d*x + c)^6 + 70*a^3*cosh(d*x + c
)^4 + 20*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^4 + 20*(11*a^3*cosh(d*x + c)^9 + 36*a^3*cosh(d*x + c)^7 + 42
*a^3*cosh(d*x + c)^5 + 20*a^3*cosh(d*x + c)^3 + 3*a^3*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 + 6*(11*a^3*cosh(d*
x + c)^10 + 45*a^3*cosh(d*x + c)^8 + 70*a^3*cosh(d*x + c)^6 + 50*a^3*cosh(d*x + c)^4 + 15*a^3*cosh(d*x + c)^2
+ a^3)*sinh(d*x + c)^2 + 12*(a^3*cosh(d*x + c)^11 + 5*a^3*cosh(d*x + c)^9 + 10*a^3*cosh(d*x + c)^7 + 10*a^3*co
sh(d*x + c)^5 + 5*a^3*cosh(d*x + c)^3 + a^3*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^3*d*x*cosh(d*x + c)^11 + 15*(a^3*d*x + a^2*b)*cosh(d*x + c)^9 + 6*(5*a^3*d*x + 8*a^
2*b + 4*a*b^2)*cosh(d*x + c)^7 + 2*(15*a^3*d*x + 27*a^2*b + 18*a*b^2 + 8*b^3)*cosh(d*x + c)^5 + 3*(5*a^3*d*x +
8*a^2*b + 4*a*b^2)*cosh(d*x + c)^3 + 3*(a^3*d*x + a^2*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*cosh(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*cosh(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 = 8.92085, size = 87, normalized size = 1.23 \begin{align*} \begin{cases} a^{3} x - \frac{a^{3} \log{\left (\tanh{\left (c + d x \right )} + 1 \right )}}{d} - \frac{3 a^{2} b \operatorname{sech}^{2}{\left (c + d x \right )}}{2 d} - \frac{3 a b^{2} \operatorname{sech}^{4}{\left (c + d x \right )}}{4 d} - \frac{b^{3} \operatorname{sech}^{6}{\left (c + d x \right )}}{6 d} & \text{for}\: d \neq 0 \\x \left (a + b \operatorname{sech}^{2}{\left (c \right )}\right )^{3} \tanh{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.23772, size = 362, normalized size = 5.1 \begin{align*} -\frac{60 \, a^{3} d x - 60 \, a^{3} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + \frac{147 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 882 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 360 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 2205 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 1440 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 720 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 2940 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 2160 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 1440 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 640 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 2205 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 1440 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 720 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 882 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 360 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 147 \, a^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{6}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(60*a^3*d*x - 60*a^3*log(e^(2*d*x + 2*c) + 1) + (147*a^3*e^(12*d*x + 12*c) + 882*a^3*e^(10*d*x + 10*c) +
360*a^2*b*e^(10*d*x + 10*c) + 2205*a^3*e^(8*d*x + 8*c) + 1440*a^2*b*e^(8*d*x + 8*c) + 720*a*b^2*e^(8*d*x + 8*
c) + 2940*a^3*e^(6*d*x + 6*c) + 2160*a^2*b*e^(6*d*x + 6*c) + 1440*a*b^2*e^(6*d*x + 6*c) + 640*b^3*e^(6*d*x + 6
*c) + 2205*a^3*e^(4*d*x + 4*c) + 1440*a^2*b*e^(4*d*x + 4*c) + 720*a*b^2*e^(4*d*x + 4*c) + 882*a^3*e^(2*d*x + 2
*c) + 360*a^2*b*e^(2*d*x + 2*c) + 147*a^3)/(e^(2*d*x + 2*c) + 1)^6)/d