3.93 \(\int \frac{1}{(a+b \text{sech}(c+d x))^3} \, dx\)

Optimal. Leaf size=173 \[ -\frac{b \left (-5 a^2 b^2+6 a^4+2 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{b^2 \left (5 a^2-2 b^2\right ) \tanh (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \text{sech}(c+d x))}+\frac{b^2 \tanh (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \text{sech}(c+d x))^2}+\frac{x}{a^3} \]

[Out]

x/a^3 - (b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(Sqrt[a - b]*Tanh[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)
*(a + b)^(5/2)*d) + (b^2*Tanh[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sech[c + d*x])^2) + (b^2*(5*a^2 - 2*b^2)*Tan
h[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sech[c + d*x]))

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Rubi [A]  time = 0.308491, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3785, 4060, 3919, 3831, 2659, 208} \[ -\frac{b \left (-5 a^2 b^2+6 a^4+2 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{b^2 \left (5 a^2-2 b^2\right ) \tanh (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \text{sech}(c+d x))}+\frac{b^2 \tanh (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \text{sech}(c+d x))^2}+\frac{x}{a^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sech[c + d*x])^(-3),x]

[Out]

x/a^3 - (b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(Sqrt[a - b]*Tanh[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)
*(a + b)^(5/2)*d) + (b^2*Tanh[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sech[c + d*x])^2) + (b^2*(5*a^2 - 2*b^2)*Tan
h[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sech[c + d*x]))

Rule 3785

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(b^2*Cot[c + d*x]*(a + b*Csc[c + d*x])^(n +
1))/(a*d*(n + 1)*(a^2 - b^2)), x] + Dist[1/(a*(n + 1)*(a^2 - b^2)), Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^
2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 4060

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(a*f*(m + 1
)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*(a^2 - b^2)*(m +
1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

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

Mathematica [A]  time = 0.726719, size = 205, normalized size = 1.18 \[ \frac{\text{sech}^3(c+d x) (a \cosh (c+d x)+b) \left (\frac{3 a b^2 \left (2 a^2-b^2\right ) \sinh (c+d x) (a \cosh (c+d x)+b)}{(a-b)^2 (a+b)^2}+\frac{2 b \left (-5 a^2 b^2+6 a^4+2 b^4\right ) (a \cosh (c+d x)+b)^2 \tan ^{-1}\left (\frac{(b-a) \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac{a b^3 \sinh (c+d x)}{(b-a) (a+b)}+2 (c+d x) (a \cosh (c+d x)+b)^2\right )}{2 a^3 d (a+b \text{sech}(c+d x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b + a*Cosh[c + d*x])*Sech[c + d*x]^3*(2*(c + d*x)*(b + a*Cosh[c + d*x])^2 + (2*b*(6*a^4 - 5*a^2*b^2 + 2*b^4)
*ArcTan[((-a + b)*Tanh[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cosh[c + d*x])^2)/(a^2 - b^2)^(5/2) + (a*b^3*Sinh
[c + d*x])/((-a + b)*(a + b)) + (3*a*b^2*(2*a^2 - b^2)*(b + a*Cosh[c + d*x])*Sinh[c + d*x])/((a - b)^2*(a + b)
^2)))/(2*a^3*d*(a + b*Sech[c + d*x])^3)

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Maple [B]  time = 0.05, size = 660, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sech(d*x+c))^3,x)

[Out]

1/d/a^3*ln(tanh(1/2*d*x+1/2*c)+1)-1/d/a^3*ln(tanh(1/2*d*x+1/2*c)-1)+6/d*b^2/(tanh(1/2*d*x+1/2*c)^2*a-tanh(1/2*
d*x+1/2*c)^2*b+a+b)^2/(a-b)/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3+1/d*b^3/a/(tanh(1/2*d*x+1/2*c)^2*a-tanh(1/2*
d*x+1/2*c)^2*b+a+b)^2/(a-b)/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3-2/d*b^4/a^2/(tanh(1/2*d*x+1/2*c)^2*a-tanh(1/
2*d*x+1/2*c)^2*b+a+b)^2/(a-b)/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3+6/d*b^2/(tanh(1/2*d*x+1/2*c)^2*a-tanh(1/2*
d*x+1/2*c)^2*b+a+b)^2/(a+b)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)-1/d*b^3/a/(tanh(1/2*d*x+1/2*c)^2*a-tanh(1/2*d*
x+1/2*c)^2*b+a+b)^2/(a+b)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)-2/d*b^4/a^2/(tanh(1/2*d*x+1/2*c)^2*a-tanh(1/2*d*
x+1/2*c)^2*b+a+b)^2/(a+b)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)-6/d*b*a/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*
arctan((a-b)*tanh(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))+5/d*b^3/a/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctan
((a-b)*tanh(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))-2/d*b^5/a^3/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctan((a-
b)*tanh(1/2*d*x+1/2*c)/((a+b)*(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(1/(a+b*sech(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.17152, size = 8986, normalized size = 51.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sech(d*x+c))^3,x, algorithm="fricas")

[Out]

[-1/2*(12*a^6*b^2 - 18*a^4*b^4 + 6*a^2*b^6 - 2*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x*cosh(d*x + c)^4 - 2
*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x*sinh(d*x + c)^4 + 2*(7*a^5*b^3 - 11*a^3*b^5 + 4*a*b^7 - 4*(a^7*b
- 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*x)*cosh(d*x + c)^3 + 2*(7*a^5*b^3 - 11*a^3*b^5 + 4*a*b^7 - 4*(a^8 - 3*a^6*b
^2 + 3*a^4*b^4 - a^2*b^6)*d*x*cosh(d*x + c) - 4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*x)*sinh(d*x + c)^3 -
2*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x + 2*(6*a^6*b^2 + 3*a^4*b^4 - 15*a^2*b^6 + 6*b^8 - 2*(a^8 - a^6*
b^2 - 3*a^4*b^4 + 5*a^2*b^6 - 2*b^8)*d*x)*cosh(d*x + c)^2 + 2*(6*a^6*b^2 + 3*a^4*b^4 - 15*a^2*b^6 + 6*b^8 - 6*
(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x*cosh(d*x + c)^2 - 2*(a^8 - a^6*b^2 - 3*a^4*b^4 + 5*a^2*b^6 - 2*b^8
)*d*x + 3*(7*a^5*b^3 - 11*a^3*b^5 + 4*a*b^7 - 4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*x)*cosh(d*x + c))*si
nh(d*x + c)^2 + (6*a^6*b - 5*a^4*b^3 + 2*a^2*b^5 + (6*a^6*b - 5*a^4*b^3 + 2*a^2*b^5)*cosh(d*x + c)^4 + (6*a^6*
b - 5*a^4*b^3 + 2*a^2*b^5)*sinh(d*x + c)^4 + 4*(6*a^5*b^2 - 5*a^3*b^4 + 2*a*b^6)*cosh(d*x + c)^3 + 4*(6*a^5*b^
2 - 5*a^3*b^4 + 2*a*b^6 + (6*a^6*b - 5*a^4*b^3 + 2*a^2*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(6*a^6*b + 7*a^
4*b^3 - 8*a^2*b^5 + 4*b^7)*cosh(d*x + c)^2 + 2*(6*a^6*b + 7*a^4*b^3 - 8*a^2*b^5 + 4*b^7 + 3*(6*a^6*b - 5*a^4*b
^3 + 2*a^2*b^5)*cosh(d*x + c)^2 + 6*(6*a^5*b^2 - 5*a^3*b^4 + 2*a*b^6)*cosh(d*x + c))*sinh(d*x + c)^2 + 4*(6*a^
5*b^2 - 5*a^3*b^4 + 2*a*b^6)*cosh(d*x + c) + 4*(6*a^5*b^2 - 5*a^3*b^4 + 2*a*b^6 + (6*a^6*b - 5*a^4*b^3 + 2*a^2
*b^5)*cosh(d*x + c)^3 + 3*(6*a^5*b^2 - 5*a^3*b^4 + 2*a*b^6)*cosh(d*x + c)^2 + (6*a^6*b + 7*a^4*b^3 - 8*a^2*b^5
+ 4*b^7)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a^2 + b^2)*log((a^2*cosh(d*x + c)^2 + a^2*sinh(d*x + c)^2 + 2*a*
b*cosh(d*x + c) - a^2 + 2*b^2 + 2*(a^2*cosh(d*x + c) + a*b)*sinh(d*x + c) + 2*sqrt(-a^2 + b^2)*(a*cosh(d*x + c
) + a*sinh(d*x + c) + b))/(a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + 2*b*cosh(d*x + c) + 2*(a*cosh(d*x + c) + b)
*sinh(d*x + c) + a)) + 2*(17*a^5*b^3 - 25*a^3*b^5 + 8*a*b^7 - 4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*x)*c
osh(d*x + c) + 2*(17*a^5*b^3 - 25*a^3*b^5 + 8*a*b^7 - 4*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x*cosh(d*x +
c)^3 - 4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*x + 3*(7*a^5*b^3 - 11*a^3*b^5 + 4*a*b^7 - 4*(a^7*b - 3*a^5
*b^3 + 3*a^3*b^5 - a*b^7)*d*x)*cosh(d*x + c)^2 + 2*(6*a^6*b^2 + 3*a^4*b^4 - 15*a^2*b^6 + 6*b^8 - 2*(a^8 - a^6*
b^2 - 3*a^4*b^4 + 5*a^2*b^6 - 2*b^8)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b
^6)*d*cosh(d*x + c)^4 + (a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*d*sinh(d*x + c)^4 + 4*(a^10*b - 3*a^8*b^3 + 3
*a^6*b^5 - a^4*b^7)*d*cosh(d*x + c)^3 + 2*(a^11 - a^9*b^2 - 3*a^7*b^4 + 5*a^5*b^6 - 2*a^3*b^8)*d*cosh(d*x + c)
^2 + 4*((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*d*cosh(d*x + c) + (a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a^4*b^7)*
d)*sinh(d*x + c)^3 + 4*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a^4*b^7)*d*cosh(d*x + c) + 2*(3*(a^11 - 3*a^9*b^2 + 3
*a^7*b^4 - a^5*b^6)*d*cosh(d*x + c)^2 + 6*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a^4*b^7)*d*cosh(d*x + c) + (a^11 -
a^9*b^2 - 3*a^7*b^4 + 5*a^5*b^6 - 2*a^3*b^8)*d)*sinh(d*x + c)^2 + (a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*d
+ 4*((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*d*cosh(d*x + c)^3 + 3*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a^4*b^7)
*d*cosh(d*x + c)^2 + (a^11 - a^9*b^2 - 3*a^7*b^4 + 5*a^5*b^6 - 2*a^3*b^8)*d*cosh(d*x + c) + (a^10*b - 3*a^8*b^
3 + 3*a^6*b^5 - a^4*b^7)*d)*sinh(d*x + c)), -(6*a^6*b^2 - 9*a^4*b^4 + 3*a^2*b^6 - (a^8 - 3*a^6*b^2 + 3*a^4*b^4
- a^2*b^6)*d*x*cosh(d*x + c)^4 - (a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x*sinh(d*x + c)^4 + (7*a^5*b^3 - 1
1*a^3*b^5 + 4*a*b^7 - 4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*x)*cosh(d*x + c)^3 + (7*a^5*b^3 - 11*a^3*b^5
+ 4*a*b^7 - 4*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x*cosh(d*x + c) - 4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 -
a*b^7)*d*x)*sinh(d*x + c)^3 - (a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x + (6*a^6*b^2 + 3*a^4*b^4 - 15*a^2*b^
6 + 6*b^8 - 2*(a^8 - a^6*b^2 - 3*a^4*b^4 + 5*a^2*b^6 - 2*b^8)*d*x)*cosh(d*x + c)^2 + (6*a^6*b^2 + 3*a^4*b^4 -
15*a^2*b^6 + 6*b^8 - 6*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x*cosh(d*x + c)^2 - 2*(a^8 - a^6*b^2 - 3*a^4*
b^4 + 5*a^2*b^6 - 2*b^8)*d*x + 3*(7*a^5*b^3 - 11*a^3*b^5 + 4*a*b^7 - 4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)
*d*x)*cosh(d*x + c))*sinh(d*x + c)^2 - (6*a^6*b - 5*a^4*b^3 + 2*a^2*b^5 + (6*a^6*b - 5*a^4*b^3 + 2*a^2*b^5)*co
sh(d*x + c)^4 + (6*a^6*b - 5*a^4*b^3 + 2*a^2*b^5)*sinh(d*x + c)^4 + 4*(6*a^5*b^2 - 5*a^3*b^4 + 2*a*b^6)*cosh(d
*x + c)^3 + 4*(6*a^5*b^2 - 5*a^3*b^4 + 2*a*b^6 + (6*a^6*b - 5*a^4*b^3 + 2*a^2*b^5)*cosh(d*x + c))*sinh(d*x + c
)^3 + 2*(6*a^6*b + 7*a^4*b^3 - 8*a^2*b^5 + 4*b^7)*cosh(d*x + c)^2 + 2*(6*a^6*b + 7*a^4*b^3 - 8*a^2*b^5 + 4*b^7
+ 3*(6*a^6*b - 5*a^4*b^3 + 2*a^2*b^5)*cosh(d*x + c)^2 + 6*(6*a^5*b^2 - 5*a^3*b^4 + 2*a*b^6)*cosh(d*x + c))*si
nh(d*x + c)^2 + 4*(6*a^5*b^2 - 5*a^3*b^4 + 2*a*b^6)*cosh(d*x + c) + 4*(6*a^5*b^2 - 5*a^3*b^4 + 2*a*b^6 + (6*a^
6*b - 5*a^4*b^3 + 2*a^2*b^5)*cosh(d*x + c)^3 + 3*(6*a^5*b^2 - 5*a^3*b^4 + 2*a*b^6)*cosh(d*x + c)^2 + (6*a^6*b
+ 7*a^4*b^3 - 8*a^2*b^5 + 4*b^7)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*cosh(d*x + c) + a*si
nh(d*x + c) + b)/sqrt(a^2 - b^2)) + (17*a^5*b^3 - 25*a^3*b^5 + 8*a*b^7 - 4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*
b^7)*d*x)*cosh(d*x + c) + (17*a^5*b^3 - 25*a^3*b^5 + 8*a*b^7 - 4*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x*c
osh(d*x + c)^3 - 4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*x + 3*(7*a^5*b^3 - 11*a^3*b^5 + 4*a*b^7 - 4*(a^7*
b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*x)*cosh(d*x + c)^2 + 2*(6*a^6*b^2 + 3*a^4*b^4 - 15*a^2*b^6 + 6*b^8 - 2*(a
^8 - a^6*b^2 - 3*a^4*b^4 + 5*a^2*b^6 - 2*b^8)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^11 - 3*a^9*b^2 + 3*a^7*b^
4 - a^5*b^6)*d*cosh(d*x + c)^4 + (a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*d*sinh(d*x + c)^4 + 4*(a^10*b - 3*a^
8*b^3 + 3*a^6*b^5 - a^4*b^7)*d*cosh(d*x + c)^3 + 2*(a^11 - a^9*b^2 - 3*a^7*b^4 + 5*a^5*b^6 - 2*a^3*b^8)*d*cosh
(d*x + c)^2 + 4*((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*d*cosh(d*x + c) + (a^10*b - 3*a^8*b^3 + 3*a^6*b^5 -
a^4*b^7)*d)*sinh(d*x + c)^3 + 4*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a^4*b^7)*d*cosh(d*x + c) + 2*(3*(a^11 - 3*a^
9*b^2 + 3*a^7*b^4 - a^5*b^6)*d*cosh(d*x + c)^2 + 6*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a^4*b^7)*d*cosh(d*x + c)
+ (a^11 - a^9*b^2 - 3*a^7*b^4 + 5*a^5*b^6 - 2*a^3*b^8)*d)*sinh(d*x + c)^2 + (a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^
5*b^6)*d + 4*((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*d*cosh(d*x + c)^3 + 3*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 -
a^4*b^7)*d*cosh(d*x + c)^2 + (a^11 - a^9*b^2 - 3*a^7*b^4 + 5*a^5*b^6 - 2*a^3*b^8)*d*cosh(d*x + c) + (a^10*b -
3*a^8*b^3 + 3*a^6*b^5 - a^4*b^7)*d)*sinh(d*x + c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \operatorname{sech}{\left (c + d x \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sech(d*x+c))**3,x)

[Out]

Integral((a + b*sech(c + d*x))**(-3), x)

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Giac [A]  time = 1.1785, size = 362, normalized size = 2.09 \begin{align*} -\frac{{\left (6 \, a^{4} b - 5 \, a^{2} b^{3} + 2 \, b^{5}\right )} \arctan \left (\frac{a e^{\left (d x + c\right )} + b}{\sqrt{a^{2} - b^{2}}}\right )}{{\left (a^{7} d - 2 \, a^{5} b^{2} d + a^{3} b^{4} d\right )} \sqrt{a^{2} - b^{2}}} - \frac{7 \, a^{3} b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 4 \, a b^{5} e^{\left (3 \, d x + 3 \, c\right )} + 6 \, a^{4} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{2} b^{4} e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b^{6} e^{\left (2 \, d x + 2 \, c\right )} + 17 \, a^{3} b^{3} e^{\left (d x + c\right )} - 8 \, a b^{5} e^{\left (d x + c\right )} + 6 \, a^{4} b^{2} - 3 \, a^{2} b^{4}}{{\left (a^{7} d - 2 \, a^{5} b^{2} d + a^{3} b^{4} d\right )}{\left (a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b e^{\left (d x + c\right )} + a\right )}^{2}} + \frac{d x + c}{a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sech(d*x+c))^3,x, algorithm="giac")

[Out]

-(6*a^4*b - 5*a^2*b^3 + 2*b^5)*arctan((a*e^(d*x + c) + b)/sqrt(a^2 - b^2))/((a^7*d - 2*a^5*b^2*d + a^3*b^4*d)*
sqrt(a^2 - b^2)) - (7*a^3*b^3*e^(3*d*x + 3*c) - 4*a*b^5*e^(3*d*x + 3*c) + 6*a^4*b^2*e^(2*d*x + 2*c) + 9*a^2*b^
4*e^(2*d*x + 2*c) - 6*b^6*e^(2*d*x + 2*c) + 17*a^3*b^3*e^(d*x + c) - 8*a*b^5*e^(d*x + c) + 6*a^4*b^2 - 3*a^2*b
^4)/((a^7*d - 2*a^5*b^2*d + a^3*b^4*d)*(a*e^(2*d*x + 2*c) + 2*b*e^(d*x + c) + a)^2) + (d*x + c)/(a^3*d)