3.126 \(\int (a+b \text{sech}^2(c+d x))^3 \tanh ^2(c+d x) \, dx\)
Optimal. Leaf size=92 \[ \frac{b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac{a^3 \tanh (c+d x)}{d}+a^3 x-\frac{b^2 (3 a+2 b) \tanh ^5(c+d x)}{5 d}+\frac{b^3 \tanh ^7(c+d x)}{7 d} \]
[Out]
a^3*x - (a^3*Tanh[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)
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Rubi [A] time = 0.110217, antiderivative size = 92, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used =
{4141, 1802, 206} \[ \frac{b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac{a^3 \tanh (c+d x)}{d}+a^3 x-\frac{b^2 (3 a+2 b) \tanh ^5(c+d x)}{5 d}+\frac{b^3 \tanh ^7(c+d x)}{7 d} \]
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 4141
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])
Rule 1802
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \left (a+b \text{sech}^2(c+d x)\right )^3 \tanh ^2(c+d x) \, dx &=\frac{\operatorname{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{\operatorname{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 \operatorname{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*}
Mathematica [B] time = 1.80319, size = 479, normalized size = 5.21 \[ \frac{\text{sech}(c) \text{sech}^7(c+d x) \left (-3990 a^2 b \sinh (2 c+d x)+1890 a^2 b \sinh (2 c+3 d x)-2520 a^2 b \sinh (4 c+3 d x)+840 a^2 b \sinh (4 c+5 d x)-630 a^2 b \sinh (6 c+5 d x)+210 a^2 b \sinh (6 c+7 d x)+3360 a^2 b \sinh (d x)+3150 a^3 \sinh (2 c+d x)-3150 a^3 \sinh (2 c+3 d x)+1260 a^3 \sinh (4 c+3 d x)-1260 a^3 \sinh (4 c+5 d x)+210 a^3 \sinh (6 c+5 d x)-210 a^3 \sinh (6 c+7 d x)+3675 a^3 d x \cosh (2 c+d x)+2205 a^3 d x \cosh (2 c+3 d x)+2205 a^3 d x \cosh (4 c+3 d x)+735 a^3 d x \cosh (4 c+5 d x)+735 a^3 d x \cosh (6 c+5 d x)+105 a^3 d x \cosh (6 c+7 d x)+105 a^3 d x \cosh (8 c+7 d x)-4200 a^3 \sinh (d x)+3675 a^3 d x \cosh (d x)-2100 a b^2 \sinh (2 c+d x)+504 a b^2 \sinh (2 c+3 d x)-1260 a b^2 \sinh (4 c+3 d x)+588 a b^2 \sinh (4 c+5 d x)+84 a b^2 \sinh (6 c+7 d x)+840 a b^2 \sinh (d x)-1120 b^3 \sinh (2 c+d x)+336 b^3 \sinh (2 c+3 d x)+112 b^3 \sinh (4 c+5 d x)+16 b^3 \sinh (6 c+7 d x)-560 b^3 \sinh (d x)\right )}{13440 d} \]
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] time = 0.042, size = 180, normalized size = 2. \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( dx+c-\tanh \left ( dx+c \right ) \right ) +3\,{a}^{2}b \left ( -1/2\,{\frac{\sinh \left ( dx+c \right ) }{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+1/2\, \left ( 2/3+1/3\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2} \right ) \tanh \left ( dx+c \right ) \right ) +3\,a{b}^{2} \left ( -1/4\,{\frac{\sinh \left ( dx+c \right ) }{ \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+1/4\, \left ({\frac{8}{15}}+1/5\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}+{\frac{4\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{15}} \right ) \tanh \left ( dx+c \right ) \right ) +{b}^{3} \left ( -{\frac{\sinh \left ( dx+c \right ) }{6\, \left ( \cosh \left ( dx+c \right ) \right ) ^{7}}}+{\frac{\tanh \left ( dx+c \right ) }{6} \left ({\frac{16}{35}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{6}}{7}}+{\frac{6\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}}{35}}+{\frac{8\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{35}} \right ) } \right ) \right ) } \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)^2,x)
[Out]
1/d*(a^3*(d*x+c-tanh(d*x+c))+3*a^2*b*(-1/2*sinh(d*x+c)/cosh(d*x+c)^3+1/2*(2/3+1/3*sech(d*x+c)^2)*tanh(d*x+c))+
3*a*b^2*(-1/4*sinh(d*x+c)/cosh(d*x+c)^5+1/4*(8/15+1/5*sech(d*x+c)^4+4/15*sech(d*x+c)^2)*tanh(d*x+c))+b^3*(-1/6
*sinh(d*x+c)/cosh(d*x+c)^7+1/6*(16/35+1/7*sech(d*x+c)^6+6/35*sech(d*x+c)^4+8/35*sech(d*x+c)^2)*tanh(d*x+c)))
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Maxima [B] time = 1.24013, size = 1064, normalized size = 11.57 \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)^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] time = 2.18119, size = 2249, normalized size = 24.45 \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)^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., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{sech}^{2}{\left (c + d x \right )}\right )^{3} \tanh ^{2}{\left (c + d x \right )}\, dx \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)**2,x)
[Out]
Integral((a + b*sech(c + d*x)**2)**3*tanh(c + d*x)**2, x)
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Giac [B] time = 1.30873, size = 481, normalized size = 5.23 \begin{align*} \frac{105 \, a^{3} d x + \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{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)^2,x, algorithm="giac")
[Out]
1/105*(105*a^3*d*x + 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