3.63 \(\int \text{sech}^3(c+d x) (a+b \text{sech}^2(c+d x))^2 \, dx\)
Optimal. Leaf size=128 \[ \frac{\left (8 a^2+12 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac{\left (8 a^2+12 a b+5 b^2\right ) \tanh (c+d x) \text{sech}(c+d x)}{16 d}+\frac{b (8 a+5 b) \tanh (c+d x) \text{sech}^3(c+d x)}{24 d}+\frac{b \tanh (c+d x) \text{sech}^5(c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{6 d} \]
[Out]
((8*a^2 + 12*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]])/(16*d) + ((8*a^2 + 12*a*b + 5*b^2)*Sech[c + d*x]*Tanh[c + d*x
])/(16*d) + (b*(8*a + 5*b)*Sech[c + d*x]^3*Tanh[c + d*x])/(24*d) + (b*Sech[c + d*x]^5*(a + b + a*Sinh[c + d*x]
^2)*Tanh[c + d*x])/(6*d)
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Rubi [A] time = 0.149448, antiderivative size = 128, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used =
{4147, 413, 385, 199, 203} \[ \frac{\left (8 a^2+12 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac{\left (8 a^2+12 a b+5 b^2\right ) \tanh (c+d x) \text{sech}(c+d x)}{16 d}+\frac{b (8 a+5 b) \tanh (c+d x) \text{sech}^3(c+d x)}{24 d}+\frac{b \tanh (c+d x) \text{sech}^5(c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{6 d} \]
Antiderivative was successfully verified.
[In]
Int[Sech[c + d*x]^3*(a + b*Sech[c + d*x]^2)^2,x]
[Out]
((8*a^2 + 12*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]])/(16*d) + ((8*a^2 + 12*a*b + 5*b^2)*Sech[c + d*x]*Tanh[c + d*x
])/(16*d) + (b*(8*a + 5*b)*Sech[c + d*x]^3*Tanh[c + d*x])/(24*d) + (b*Sech[c + d*x]^5*(a + b + a*Sinh[c + d*x]
^2)*Tanh[c + d*x])/(6*d)
Rule 4147
Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fr
eeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^
((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n
/2] && IntegerQ[p]
Rule 413
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((a*d - c*b)*x*(a + b*x^n)^
(p + 1)*(c + d*x^n)^(q - 1))/(a*b*n*(p + 1)), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]
Rule 385
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])
Rule 199
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rubi steps
\begin{align*} \int \text{sech}^3(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b+a x^2\right )^2}{\left (1+x^2\right )^4} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{b \text{sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{6 d}+\frac{\operatorname{Subst}\left (\int \frac{(a+b) (6 a+5 b)+3 a (2 a+b) x^2}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{6 d}\\ &=\frac{b (8 a+5 b) \text{sech}^3(c+d x) \tanh (c+d x)}{24 d}+\frac{b \text{sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{6 d}+\frac{\left (8 a^2+12 a b+5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{8 d}\\ &=\frac{\left (8 a^2+12 a b+5 b^2\right ) \text{sech}(c+d x) \tanh (c+d x)}{16 d}+\frac{b (8 a+5 b) \text{sech}^3(c+d x) \tanh (c+d x)}{24 d}+\frac{b \text{sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{6 d}+\frac{\left (8 a^2+12 a b+5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{16 d}\\ &=\frac{\left (8 a^2+12 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac{\left (8 a^2+12 a b+5 b^2\right ) \text{sech}(c+d x) \tanh (c+d x)}{16 d}+\frac{b (8 a+5 b) \text{sech}^3(c+d x) \tanh (c+d x)}{24 d}+\frac{b \text{sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.237709, size = 104, normalized size = 0.81 \[ \frac{3 \left (8 a^2+12 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))+3 \left (8 a^2+12 a b+5 b^2\right ) \tanh (c+d x) \text{sech}(c+d x)+2 b (12 a+5 b) \tanh (c+d x) \text{sech}^3(c+d x)+8 b^2 \tanh (c+d x) \text{sech}^5(c+d x)}{48 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Sech[c + d*x]^3*(a + b*Sech[c + d*x]^2)^2,x]
[Out]
(3*(8*a^2 + 12*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]] + 3*(8*a^2 + 12*a*b + 5*b^2)*Sech[c + d*x]*Tanh[c + d*x] + 2
*b*(12*a + 5*b)*Sech[c + d*x]^3*Tanh[c + d*x] + 8*b^2*Sech[c + d*x]^5*Tanh[c + d*x])/(48*d)
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Maple [A] time = 0.026, size = 169, normalized size = 1.3 \begin{align*}{\frac{{a}^{2}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{2}\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}+{\frac{ab\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{2\,d}}+{\frac{3\,ab{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{4\,d}}+{\frac{3\,ab\arctan \left ({{\rm e}^{dx+c}} \right ) }{2\,d}}+{\frac{{b}^{2}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{5}}{6\,d}}+{\frac{5\,{b}^{2}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{24\,d}}+{\frac{5\,{b}^{2}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{16\,d}}+{\frac{5\,{b}^{2}\arctan \left ({{\rm e}^{dx+c}} \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sech(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x)
[Out]
1/2/d*a^2*sech(d*x+c)*tanh(d*x+c)+1/d*a^2*arctan(exp(d*x+c))+1/2/d*a*b*tanh(d*x+c)*sech(d*x+c)^3+3/4/d*a*b*sec
h(d*x+c)*tanh(d*x+c)+3/2/d*a*b*arctan(exp(d*x+c))+1/6/d*b^2*tanh(d*x+c)*sech(d*x+c)^5+5/24/d*b^2*tanh(d*x+c)*s
ech(d*x+c)^3+5/16*b^2*sech(d*x+c)*tanh(d*x+c)/d+5/8/d*b^2*arctan(exp(d*x+c))
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Maxima [B] time = 1.70017, size = 470, normalized size = 3.67 \begin{align*} -\frac{1}{24} \, b^{2}{\left (\frac{15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{15 \, e^{\left (-d x - c\right )} + 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} - 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} - 15 \, e^{\left (-11 \, d x - 11 \, 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{1}{2} \, a b{\left (\frac{3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{3 \, e^{\left (-d x - c\right )} + 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} - 3 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d{\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - a^{2}{\left (\frac{\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")
[Out]
-1/24*b^2*(15*arctan(e^(-d*x - c))/d - (15*e^(-d*x - c) + 85*e^(-3*d*x - 3*c) + 198*e^(-5*d*x - 5*c) - 198*e^(
-7*d*x - 7*c) - 85*e^(-9*d*x - 9*c) - 15*e^(-11*d*x - 11*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))) - 1/2*a*b*(3*arctan
(e^(-d*x - c))/d - (3*e^(-d*x - c) + 11*e^(-3*d*x - 3*c) - 11*e^(-5*d*x - 5*c) - 3*e^(-7*d*x - 7*c))/(d*(4*e^(
-2*d*x - 2*c) + 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1))) - a^2*(arctan(e^(-d*x - c))/
d - (e^(-d*x - c) - e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1)))
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Fricas [B] time = 2.49081, size = 7609, normalized size = 59.45 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")
[Out]
1/24*(3*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^11 + 33*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)*sinh(d*x + c)^10
+ 3*(8*a^2 + 12*a*b + 5*b^2)*sinh(d*x + c)^11 + (72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c)^9 + (165*(8*a^2 + 1
2*a*b + 5*b^2)*cosh(d*x + c)^2 + 72*a^2 + 204*a*b + 85*b^2)*sinh(d*x + c)^9 + 9*(55*(8*a^2 + 12*a*b + 5*b^2)*c
osh(d*x + c)^3 + (72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c))*sinh(d*x + c)^8 + 6*(8*a^2 + 28*a*b + 33*b^2)*cosh
(d*x + c)^7 + 6*(165*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^4 + 6*(72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c)^2
+ 8*a^2 + 28*a*b + 33*b^2)*sinh(d*x + c)^7 + 42*(33*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^5 + 2*(72*a^2 + 204
*a*b + 85*b^2)*cosh(d*x + c)^3 + (8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c))*sinh(d*x + c)^6 - 6*(8*a^2 + 28*a*b
+ 33*b^2)*cosh(d*x + c)^5 + 6*(231*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^6 + 21*(72*a^2 + 204*a*b + 85*b^2)*c
osh(d*x + c)^4 + 21*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^2 - 8*a^2 - 28*a*b - 33*b^2)*sinh(d*x + c)^5 + 6*(
165*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^7 + 21*(72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c)^5 + 35*(8*a^2 + 28
*a*b + 33*b^2)*cosh(d*x + c)^3 - 5*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 - (72*a^2 + 204*a*
b + 85*b^2)*cosh(d*x + c)^3 + (495*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^8 + 84*(72*a^2 + 204*a*b + 85*b^2)*c
osh(d*x + c)^6 + 210*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^4 - 60*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^2
- 72*a^2 - 204*a*b - 85*b^2)*sinh(d*x + c)^3 + 3*(55*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^9 + 12*(72*a^2 + 2
04*a*b + 85*b^2)*cosh(d*x + c)^7 + 42*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^5 - 20*(8*a^2 + 28*a*b + 33*b^2)
*cosh(d*x + c)^3 - (72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + 3*((8*a^2 + 12*a*b + 5*b^2)*co
sh(d*x + c)^12 + 12*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)*sinh(d*x + c)^11 + (8*a^2 + 12*a*b + 5*b^2)*sinh(d*
x + c)^12 + 6*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^10 + 6*(11*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^2 + 8*a
^2 + 12*a*b + 5*b^2)*sinh(d*x + c)^10 + 20*(11*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^3 + 3*(8*a^2 + 12*a*b +
5*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 + 15*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^8 + 15*(33*(8*a^2 + 12*a*b +
5*b^2)*cosh(d*x + c)^4 + 18*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^2 + 8*a^2 + 12*a*b + 5*b^2)*sinh(d*x + c)^
8 + 24*(33*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^5 + 30*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^3 + 5*(8*a^2 +
12*a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 + 20*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^6 + 4*(231*(8*a^2
+ 12*a*b + 5*b^2)*cosh(d*x + c)^6 + 315*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^4 + 105*(8*a^2 + 12*a*b + 5*b^2
)*cosh(d*x + c)^2 + 40*a^2 + 60*a*b + 25*b^2)*sinh(d*x + c)^6 + 24*(33*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^
7 + 63*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^5 + 35*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^3 + 5*(8*a^2 + 12*
a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^4 + 15*(33*(8*a^2 + 12
*a*b + 5*b^2)*cosh(d*x + c)^8 + 84*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^6 + 70*(8*a^2 + 12*a*b + 5*b^2)*cosh
(d*x + c)^4 + 20*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^2 + 8*a^2 + 12*a*b + 5*b^2)*sinh(d*x + c)^4 + 20*(11*(
8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^9 + 36*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^7 + 42*(8*a^2 + 12*a*b + 5
*b^2)*cosh(d*x + c)^5 + 20*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^3 + 3*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)
)*sinh(d*x + c)^3 + 6*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^2 + 6*(11*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^
10 + 45*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^8 + 70*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^6 + 50*(8*a^2 + 1
2*a*b + 5*b^2)*cosh(d*x + c)^4 + 15*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^2 + 8*a^2 + 12*a*b + 5*b^2)*sinh(d*
x + c)^2 + 8*a^2 + 12*a*b + 5*b^2 + 12*((8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^11 + 5*(8*a^2 + 12*a*b + 5*b^2)
*cosh(d*x + c)^9 + 10*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^7 + 10*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^5 +
5*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^3 + (8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c))*arctan(co
sh(d*x + c) + sinh(d*x + c)) - 3*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c) + 3*(11*(8*a^2 + 12*a*b + 5*b^2)*cosh(
d*x + c)^10 + 3*(72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c)^8 + 14*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^6 - 1
0*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^4 - (72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c)^2 - 8*a^2 - 12*a*b - 5
*b^2)*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))*s
inh(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*co
sh(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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{sech}^{2}{\left (c + d x \right )}\right )^{2} \operatorname{sech}^{3}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)**3*(a+b*sech(d*x+c)**2)**2,x)
[Out]
Integral((a + b*sech(c + d*x)**2)**2*sech(c + d*x)**3, x)
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Giac [B] time = 1.13563, size = 397, normalized size = 3.1 \begin{align*} \frac{{\left (\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )}{\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )}}{32 \, d} + \frac{24 \, a^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 36 \, a b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 15 \, b^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 192 \, a^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 384 \, a b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 160 \, b^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 384 \, a^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 960 \, a b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 528 \, b^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{24 \,{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")
[Out]
1/32*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(8*a^2 + 12*a*b + 5*b^2)/d + 1/24*(24*a^2*(e^(d*x
+ c) - e^(-d*x - c))^5 + 36*a*b*(e^(d*x + c) - e^(-d*x - c))^5 + 15*b^2*(e^(d*x + c) - e^(-d*x - c))^5 + 192*
a^2*(e^(d*x + c) - e^(-d*x - c))^3 + 384*a*b*(e^(d*x + c) - e^(-d*x - c))^3 + 160*b^2*(e^(d*x + c) - e^(-d*x -
c))^3 + 384*a^2*(e^(d*x + c) - e^(-d*x - c)) + 960*a*b*(e^(d*x + c) - e^(-d*x - c)) + 528*b^2*(e^(d*x + c) -
e^(-d*x - c)))/(((e^(d*x + c) - e^(-d*x - c))^2 + 4)^3*d)