3.96 \(\int \frac{\text{sech}^2(c+d x)}{(a+b \text{sech}^2(c+d x))^3} \, dx\)
Optimal. Leaf size=108 \[ \frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 \sqrt{b} d (a+b)^{5/2}}+\frac{3 \tanh (c+d x)}{8 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}+\frac{\tanh (c+d x)}{4 d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2} \]
[Out]
(3*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*Sqrt[b]*(a + b)^(5/2)*d) + Tanh[c + d*x]/(4*(a + b)*d*(a +
b - b*Tanh[c + d*x]^2)^2) + (3*Tanh[c + d*x])/(8*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2))
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Rubi [A] time = 0.090436, antiderivative size = 108, 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 =
{4146, 199, 208} \[ \frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 \sqrt{b} d (a+b)^{5/2}}+\frac{3 \tanh (c+d x)}{8 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}+\frac{\tanh (c+d x)}{4 d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[Sech[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]
[Out]
(3*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*Sqrt[b]*(a + b)^(5/2)*d) + Tanh[c + d*x]/(4*(a + b)*d*(a +
b - b*Tanh[c + d*x]^2)^2) + (3*Tanh[c + d*x])/(8*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2))
Rule 4146
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fre
eFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/
2), x]^p, x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(c+d x)}{\left (a+b \text{sech}^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\tanh (c+d x)}{4 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 (a+b) d}\\ &=\frac{\tanh (c+d x)}{4 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac{3 \tanh (c+d x)}{8 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{8 (a+b)^2 d}\\ &=\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 \sqrt{b} (a+b)^{5/2} d}+\frac{\tanh (c+d x)}{4 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac{3 \tanh (c+d x)}{8 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 2.07828, size = 258, normalized size = 2.39 \[ \frac{\text{sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (-\frac{\text{sech}(2 c) \left (\left (5 a^2+16 a b+8 b^2\right ) \sinh (2 c)-a (5 a+2 b) \sinh (2 d x)\right ) (a \cosh (2 (c+d x))+a+2 b)}{a^2}+\frac{4 b (a+b) \text{sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{a^2}+\frac{3 (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b)^2 \tanh ^{-1}\left (\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right )}{\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right )}{64 d (a+b)^2 \left (a+b \text{sech}^2(c+d x)\right )^3} \]
Antiderivative was successfully verified.
[In]
Integrate[Sech[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]
[Out]
((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*((3*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sin
h[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*
(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (4*b*(a + b)*Sech[2*c]*((a + 2*b)*Sinh[
2*c] - a*Sinh[2*d*x]))/a^2 - ((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[2*c]*((5*a^2 + 16*a*b + 8*b^2)*Sinh[2*c] -
a*(5*a + 2*b)*Sinh[2*d*x]))/a^2))/(64*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3)
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Maple [B] time = 0.073, size = 612, normalized size = 5.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sech(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x)
[Out]
5/4/d/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b
)^2/(a+b)*tanh(1/2*d*x+1/2*c)^7+15/4/d/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^
2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)^2*tanh(1/2*d*x+1/2*c)^5*a+3/4/d/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2
*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*b/(a+b)^2*tanh(1/2*d*x+1/2*c)^5+15/4/
d/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/
(a+b)^2*tanh(1/2*d*x+1/2*c)^3*a+3/4/d/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2
*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*b/(a+b)^2*tanh(1/2*d*x+1/2*c)^3+5/4/d/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*
d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)*tanh(1/2*d*x+1/2*c)+3/16/d/(a^2+
2*a*b+b^2)/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))
-3/16/d/(a^2+2*a*b+b^2)/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)
+(a+b)^(1/2))
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.79423, size = 11784, normalized size = 109.11 \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)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
[Out]
[-1/16*(4*(5*a^4*b + 21*a^3*b^2 + 24*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)^6 + 24*(5*a^4*b + 21*a^3*b^2 + 24*a^2*b^
3 + 8*a*b^4)*cosh(d*x + c)*sinh(d*x + c)^5 + 4*(5*a^4*b + 21*a^3*b^2 + 24*a^2*b^3 + 8*a*b^4)*sinh(d*x + c)^6 +
20*a^4*b + 28*a^3*b^2 + 8*a^2*b^3 + 4*(15*a^4*b + 61*a^3*b^2 + 102*a^2*b^3 + 72*a*b^4 + 16*b^5)*cosh(d*x + c)
^4 + 4*(15*a^4*b + 61*a^3*b^2 + 102*a^2*b^3 + 72*a*b^4 + 16*b^5 + 15*(5*a^4*b + 21*a^3*b^2 + 24*a^2*b^3 + 8*a*
b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(5*(5*a^4*b + 21*a^3*b^2 + 24*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)^3 +
(15*a^4*b + 61*a^3*b^2 + 102*a^2*b^3 + 72*a*b^4 + 16*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(15*a^4*b + 47*a^
3*b^2 + 40*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)^2 + 4*(15*a^4*b + 47*a^3*b^2 + 40*a^2*b^3 + 8*a*b^4 + 15*(5*a^4*b
+ 21*a^3*b^2 + 24*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)^4 + 6*(15*a^4*b + 61*a^3*b^2 + 102*a^2*b^3 + 72*a*b^4 + 16*
b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 3*(a^4*cosh(d*x + c)^8 + 8*a^4*cosh(d*x + c)*sinh(d*x + c)^7 + a^4*sin
h(d*x + c)^8 + 4*(a^4 + 2*a^3*b)*cosh(d*x + c)^6 + 4*(7*a^4*cosh(d*x + c)^2 + a^4 + 2*a^3*b)*sinh(d*x + c)^6 +
8*(7*a^4*cosh(d*x + c)^3 + 3*(a^4 + 2*a^3*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^4 + 8*a^3*b + 8*a^2*b^2)
*cosh(d*x + c)^4 + 2*(35*a^4*cosh(d*x + c)^4 + 3*a^4 + 8*a^3*b + 8*a^2*b^2 + 30*(a^4 + 2*a^3*b)*cosh(d*x + c)^
2)*sinh(d*x + c)^4 + a^4 + 8*(7*a^4*cosh(d*x + c)^5 + 10*(a^4 + 2*a^3*b)*cosh(d*x + c)^3 + (3*a^4 + 8*a^3*b +
8*a^2*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^4 + 2*a^3*b)*cosh(d*x + c)^2 + 4*(7*a^4*cosh(d*x + c)^6 + 15*
(a^4 + 2*a^3*b)*cosh(d*x + c)^4 + a^4 + 2*a^3*b + 3*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x +
c)^2 + 8*(a^4*cosh(d*x + c)^7 + 3*(a^4 + 2*a^3*b)*cosh(d*x + c)^5 + (3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c
)^3 + (a^4 + 2*a^3*b)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a*b + b^2)*log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x
+ c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c)^2 + a^2
+ 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b + 8*b^2 + 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*
x + c) - 4*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a + 2*b)*sqrt(a*b + b^2)
)/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2
*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x +
c) + a)) + 8*(3*(5*a^4*b + 21*a^3*b^2 + 24*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)^5 + 2*(15*a^4*b + 61*a^3*b^2 + 102
*a^2*b^3 + 72*a*b^4 + 16*b^5)*cosh(d*x + c)^3 + (15*a^4*b + 47*a^3*b^2 + 40*a^2*b^3 + 8*a*b^4)*cosh(d*x + c))*
sinh(d*x + c))/((a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*d*cosh(d*x + c)^8 + 8*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3
+ a^4*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*d*sinh(d*x + c)^8 + 4*
(a^7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)*d*cosh(d*x + c)^6 + 4*(7*(a^7*b + 3*a^6*b^2 + 3*a^5*b^
3 + a^4*b^4)*d*cosh(d*x + c)^2 + (a^7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)*d)*sinh(d*x + c)^6 +
2*(3*a^7*b + 17*a^6*b^2 + 41*a^5*b^3 + 51*a^4*b^4 + 32*a^3*b^5 + 8*a^2*b^6)*d*cosh(d*x + c)^4 + 8*(7*(a^7*b +
3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*d*cosh(d*x + c)^3 + 3*(a^7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)
*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*d*cosh(d*x + c)^4 + 30*(a^
7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)*d*cosh(d*x + c)^2 + (3*a^7*b + 17*a^6*b^2 + 41*a^5*b^3 +
51*a^4*b^4 + 32*a^3*b^5 + 8*a^2*b^6)*d)*sinh(d*x + c)^4 + 4*(a^7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3
*b^5)*d*cosh(d*x + c)^2 + 8*(7*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*d*cosh(d*x + c)^5 + 10*(a^7*b + 5*a^6
*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)*d*cosh(d*x + c)^3 + (3*a^7*b + 17*a^6*b^2 + 41*a^5*b^3 + 51*a^4*b^4
+ 32*a^3*b^5 + 8*a^2*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*d*
cosh(d*x + c)^6 + 15*(a^7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)*d*cosh(d*x + c)^4 + 3*(3*a^7*b +
17*a^6*b^2 + 41*a^5*b^3 + 51*a^4*b^4 + 32*a^3*b^5 + 8*a^2*b^6)*d*cosh(d*x + c)^2 + (a^7*b + 5*a^6*b^2 + 9*a^5*
b^3 + 7*a^4*b^4 + 2*a^3*b^5)*d)*sinh(d*x + c)^2 + (a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*d + 8*((a^7*b + 3*
a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*d*cosh(d*x + c)^7 + 3*(a^7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)*d
*cosh(d*x + c)^5 + (3*a^7*b + 17*a^6*b^2 + 41*a^5*b^3 + 51*a^4*b^4 + 32*a^3*b^5 + 8*a^2*b^6)*d*cosh(d*x + c)^3
+ (a^7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)), -1/8*(2*(5*a^4*b +
21*a^3*b^2 + 24*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)^6 + 12*(5*a^4*b + 21*a^3*b^2 + 24*a^2*b^3 + 8*a*b^4)*cosh(d*
x + c)*sinh(d*x + c)^5 + 2*(5*a^4*b + 21*a^3*b^2 + 24*a^2*b^3 + 8*a*b^4)*sinh(d*x + c)^6 + 10*a^4*b + 14*a^3*b
^2 + 4*a^2*b^3 + 2*(15*a^4*b + 61*a^3*b^2 + 102*a^2*b^3 + 72*a*b^4 + 16*b^5)*cosh(d*x + c)^4 + 2*(15*a^4*b + 6
1*a^3*b^2 + 102*a^2*b^3 + 72*a*b^4 + 16*b^5 + 15*(5*a^4*b + 21*a^3*b^2 + 24*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)^2
)*sinh(d*x + c)^4 + 8*(5*(5*a^4*b + 21*a^3*b^2 + 24*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)^3 + (15*a^4*b + 61*a^3*b^
2 + 102*a^2*b^3 + 72*a*b^4 + 16*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(15*a^4*b + 47*a^3*b^2 + 40*a^2*b^3 +
8*a*b^4)*cosh(d*x + c)^2 + 2*(15*a^4*b + 47*a^3*b^2 + 40*a^2*b^3 + 8*a*b^4 + 15*(5*a^4*b + 21*a^3*b^2 + 24*a^2
*b^3 + 8*a*b^4)*cosh(d*x + c)^4 + 6*(15*a^4*b + 61*a^3*b^2 + 102*a^2*b^3 + 72*a*b^4 + 16*b^5)*cosh(d*x + c)^2)
*sinh(d*x + c)^2 - 3*(a^4*cosh(d*x + c)^8 + 8*a^4*cosh(d*x + c)*sinh(d*x + c)^7 + a^4*sinh(d*x + c)^8 + 4*(a^4
+ 2*a^3*b)*cosh(d*x + c)^6 + 4*(7*a^4*cosh(d*x + c)^2 + a^4 + 2*a^3*b)*sinh(d*x + c)^6 + 8*(7*a^4*cosh(d*x +
c)^3 + 3*(a^4 + 2*a^3*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^4 + 2*
(35*a^4*cosh(d*x + c)^4 + 3*a^4 + 8*a^3*b + 8*a^2*b^2 + 30*(a^4 + 2*a^3*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 +
a^4 + 8*(7*a^4*cosh(d*x + c)^5 + 10*(a^4 + 2*a^3*b)*cosh(d*x + c)^3 + (3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x +
c))*sinh(d*x + c)^3 + 4*(a^4 + 2*a^3*b)*cosh(d*x + c)^2 + 4*(7*a^4*cosh(d*x + c)^6 + 15*(a^4 + 2*a^3*b)*cosh(
d*x + c)^4 + a^4 + 2*a^3*b + 3*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(a^4*cosh(d*
x + c)^7 + 3*(a^4 + 2*a^3*b)*cosh(d*x + c)^5 + (3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^3 + (a^4 + 2*a^3*b)
*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a*b - b^2)*arctan(1/2*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c
) + a*sinh(d*x + c)^2 + a + 2*b)*sqrt(-a*b - b^2)/(a*b + b^2)) + 4*(3*(5*a^4*b + 21*a^3*b^2 + 24*a^2*b^3 + 8*a
*b^4)*cosh(d*x + c)^5 + 2*(15*a^4*b + 61*a^3*b^2 + 102*a^2*b^3 + 72*a*b^4 + 16*b^5)*cosh(d*x + c)^3 + (15*a^4*
b + 47*a^3*b^2 + 40*a^2*b^3 + 8*a*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4
)*d*cosh(d*x + c)^8 + 8*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^7*b + 3
*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*d*sinh(d*x + c)^8 + 4*(a^7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)*
d*cosh(d*x + c)^6 + 4*(7*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*d*cosh(d*x + c)^2 + (a^7*b + 5*a^6*b^2 + 9*
a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)*d)*sinh(d*x + c)^6 + 2*(3*a^7*b + 17*a^6*b^2 + 41*a^5*b^3 + 51*a^4*b^4 + 32*a
^3*b^5 + 8*a^2*b^6)*d*cosh(d*x + c)^4 + 8*(7*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*d*cosh(d*x + c)^3 + 3*(
a^7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^7*b + 3*a^6
*b^2 + 3*a^5*b^3 + a^4*b^4)*d*cosh(d*x + c)^4 + 30*(a^7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)*d*c
osh(d*x + c)^2 + (3*a^7*b + 17*a^6*b^2 + 41*a^5*b^3 + 51*a^4*b^4 + 32*a^3*b^5 + 8*a^2*b^6)*d)*sinh(d*x + c)^4
+ 4*(a^7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)*d*cosh(d*x + c)^2 + 8*(7*(a^7*b + 3*a^6*b^2 + 3*a^
5*b^3 + a^4*b^4)*d*cosh(d*x + c)^5 + 10*(a^7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)*d*cosh(d*x + c
)^3 + (3*a^7*b + 17*a^6*b^2 + 41*a^5*b^3 + 51*a^4*b^4 + 32*a^3*b^5 + 8*a^2*b^6)*d*cosh(d*x + c))*sinh(d*x + c)
^3 + 4*(7*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*d*cosh(d*x + c)^6 + 15*(a^7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*
a^4*b^4 + 2*a^3*b^5)*d*cosh(d*x + c)^4 + 3*(3*a^7*b + 17*a^6*b^2 + 41*a^5*b^3 + 51*a^4*b^4 + 32*a^3*b^5 + 8*a^
2*b^6)*d*cosh(d*x + c)^2 + (a^7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)*d)*sinh(d*x + c)^2 + (a^7*b
+ 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*d + 8*((a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*d*cosh(d*x + c)^7 + 3*(a^
7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^5)*d*cosh(d*x + c)^5 + (3*a^7*b + 17*a^6*b^2 + 41*a^5*b^3 +
51*a^4*b^4 + 32*a^3*b^5 + 8*a^2*b^6)*d*cosh(d*x + c)^3 + (a^7*b + 5*a^6*b^2 + 9*a^5*b^3 + 7*a^4*b^4 + 2*a^3*b^
5)*d*cosh(d*x + c))*sinh(d*x + c))]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)**2/(a+b*sech(d*x+c)**2)**3,x)
[Out]
Timed out
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Giac [B] time = 1.35344, size = 386, normalized size = 3.57 \begin{align*} \frac{3 \, \arctan \left (\frac{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt{-a b - b^{2}}}\right )}{8 \,{\left (a^{2} d + 2 \, a b d + b^{2} d\right )} \sqrt{-a b - b^{2}}} - \frac{5 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 8 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 15 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 46 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 56 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 16 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 32 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, a^{3} + 2 \, a^{2} b}{4 \,{\left (a^{4} d + 2 \, a^{3} b d + a^{2} b^{2} d\right )}{\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
[Out]
3/8*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^2*d + 2*a*b*d + b^2*d)*sqrt(-a*b - b^2)) -
1/4*(5*a^3*e^(6*d*x + 6*c) + 16*a^2*b*e^(6*d*x + 6*c) + 8*a*b^2*e^(6*d*x + 6*c) + 15*a^3*e^(4*d*x + 4*c) + 46*
a^2*b*e^(4*d*x + 4*c) + 56*a*b^2*e^(4*d*x + 4*c) + 16*b^3*e^(4*d*x + 4*c) + 15*a^3*e^(2*d*x + 2*c) + 32*a^2*b*
e^(2*d*x + 2*c) + 8*a*b^2*e^(2*d*x + 2*c) + 5*a^3 + 2*a^2*b)/((a^4*d + 2*a^3*b*d + a^2*b^2*d)*(a*e^(4*d*x + 4*
c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)^2)