3.164 \(\int \frac{1}{(a+b \text{sech}^2(c+d x))^3} \, dx\)

Optimal. Leaf size=146 \[ -\frac{\sqrt{b} \left (15 a^2+20 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^3 d (a+b)^{5/2}}-\frac{b (7 a+4 b) \tanh (c+d x)}{8 a^2 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}+\frac{x}{a^3}-\frac{b \tanh (c+d x)}{4 a d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

[Out]

x/a^3 - (Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(5/2)*
d) - (b*Tanh[c + d*x])/(4*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) - (b*(7*a + 4*b)*Tanh[c + d*x])/(8*a^2*(a
+ b)^2*d*(a + b - b*Tanh[c + d*x]^2))

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Rubi [A]  time = 0.181249, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4128, 414, 527, 522, 206, 208} \[ -\frac{\sqrt{b} \left (15 a^2+20 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^3 d (a+b)^{5/2}}-\frac{b (7 a+4 b) \tanh (c+d x)}{8 a^2 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}+\frac{x}{a^3}-\frac{b \tanh (c+d x)}{4 a d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sech[c + d*x]^2)^(-3),x]

[Out]

x/a^3 - (Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(5/2)*
d) - (b*Tanh[c + d*x])/(4*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) - (b*(7*a + 4*b)*Tanh[c + d*x])/(8*a^2*(a
+ b)^2*d*(a + b - b*Tanh[c + d*x]^2))

Rule 4128

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rule 414

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

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

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])

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}{\left (a+b \text{sech}^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{b \tanh (c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-4 a-b-3 b x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a (a+b) d}\\ &=-\frac{b \tanh (c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{b (7 a+4 b) \tanh (c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{8 a^2+9 a b+4 b^2+b (7 a+4 b) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a+b)^2 d}\\ &=-\frac{b \tanh (c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{b (7 a+4 b) \tanh (c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{a^3 d}-\frac{\left (b \left (15 a^2+20 a b+8 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^3 (a+b)^2 d}\\ &=\frac{x}{a^3}-\frac{\sqrt{b} \left (15 a^2+20 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^3 (a+b)^{5/2} d}-\frac{b \tanh (c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{b (7 a+4 b) \tanh (c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 6.11276, size = 301, normalized size = 2.06 \[ \frac{\text{sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\frac{b \text{sech}(2 c) \left (\left (9 a^2+28 a b+16 b^2\right ) \sinh (2 c)-3 a (3 a+2 b) \sinh (2 d x)\right ) (a \cosh (2 (c+d x))+a+2 b)}{d (a+b)^2}-\frac{b \left (15 a^2+20 a b+8 b^2\right ) (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b)^2 \tanh ^{-1}\left (\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right )}{d (a+b)^{5/2} \sqrt{b (\cosh (c)-\sinh (c))^4}}-\frac{4 b^2 \text{sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{d (a+b)}+8 x (a \cosh (2 (c+d x))+a+2 b)^2\right )}{64 a^3 \left (a+b \text{sech}^2(c+d x)\right )^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*(8*x*(a + 2*b + a*Cosh[2*(c + d*x)])^2 - (b*(15*a^2 + 20*a*b
+ 8*b^2)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[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]))/((a + b)^(5/2)*d*Sq
rt[b*(Cosh[c] - Sinh[c])^4]) - (4*b^2*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/((a + b)*d) + (b*(a + 2
*b + a*Cosh[2*(c + d*x)])*Sech[2*c]*((9*a^2 + 28*a*b + 16*b^2)*Sinh[2*c] - 3*a*(3*a + 2*b)*Sinh[2*d*x]))/((a +
b)^2*d)))/(64*a^3*(a + b*Sech[c + d*x]^2)^3)

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Maple [B]  time = 0.093, size = 1292, normalized size = 8.9 \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)^2)^3,x)

[Out]

1/d/a^3*ln(tanh(1/2*d*x+1/2*c)+1)-9/4/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)/a*tanh(1/2*d*x+1/2*c)^7-1/d*b^2/a^2/(tanh(1/2*d*x+1/2*c)^4*a+b*t
anh(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)^7-27
/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)^5-11/4/d*b^2/a/(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+1/d*b^3/a^2/(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)^5-27/4/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)^3-11/4/d*b^2/a/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tan
h(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+1/d*b^3/a^2/(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)^3-9/4/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)/a*tanh(1/2*d*x+1/2*c)-1/d*b^2/a^2/(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)-15/16/d*b^(1/2)/a/(a^2+2*a
*b+b^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))-5/4/d*b^(3
/2)/a^2/(a^2+2*a*b+b^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))-1/2/d*b^(5/2)/a^3/(a^2+2*a*b+b^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))+15/16/d*b^(1/2)/a/(a^2+2*a*b+b^2)/(a+b)^(1/2)*ln(-(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*ta
nh(1/2*d*x+1/2*c)*b^(1/2)-(a+b)^(1/2))+5/4/d*b^(3/2)/a^2/(a^2+2*a*b+b^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))+1/2/d*b^(5/2)/a^3/(a^2+2*a*b+b^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))-1/d/a^3*ln(tanh(1/2*d*x+1/2*c)-1)

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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)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.32923, size = 15505, normalized size = 106.2 \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)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(16*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^8 + 128*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)*sinh
(d*x + c)^7 + 16*(a^4 + 2*a^3*b + a^2*b^2)*d*x*sinh(d*x + c)^8 + 4*(9*a^3*b + 28*a^2*b^2 + 16*a*b^3 + 16*(a^4
+ 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c)^6 + 4*(112*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^2 +
9*a^3*b + 28*a^2*b^2 + 16*a*b^3 + 16*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*x)*sinh(d*x + c)^6 + 8*(112*(a^4
+ 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^3 + 3*(9*a^3*b + 28*a^2*b^2 + 16*a*b^3 + 16*(a^4 + 4*a^3*b + 5*a^2*b^2
+ 2*a*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + 4*(27*a^3*b + 90*a^2*b^2 + 120*a*b^3 + 48*b^4 + 8*(3*a^4 + 1
4*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c)^4 + 4*(280*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x
+ c)^4 + 27*a^3*b + 90*a^2*b^2 + 120*a*b^3 + 48*b^4 + 8*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*d*x
+ 15*(9*a^3*b + 28*a^2*b^2 + 16*a*b^3 + 16*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d
*x + c)^4 + 36*a^3*b + 24*a^2*b^2 + 16*(56*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^5 + 5*(9*a^3*b + 28*a^2
*b^2 + 16*a*b^3 + 16*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c)^3 + (27*a^3*b + 90*a^2*b^2 + 120
*a*b^3 + 48*b^4 + 8*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 1
6*(a^4 + 2*a^3*b + a^2*b^2)*d*x + 4*(27*a^3*b + 68*a^2*b^2 + 32*a*b^3 + 16*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^
3)*d*x)*cosh(d*x + c)^2 + 4*(112*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^6 + 15*(9*a^3*b + 28*a^2*b^2 + 16
*a*b^3 + 16*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c)^4 + 27*a^3*b + 68*a^2*b^2 + 32*a*b^3 + 16
*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*x + 6*(27*a^3*b + 90*a^2*b^2 + 120*a*b^3 + 48*b^4 + 8*(3*a^4 + 14*a^3
*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((15*a^4 + 20*a^3*b + 8*a^2*b^2)*c
osh(d*x + c)^8 + 8*(15*a^4 + 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (15*a^4 + 20*a^3*b + 8*a^2*
b^2)*sinh(d*x + c)^8 + 4*(15*a^4 + 50*a^3*b + 48*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^6 + 4*(15*a^4 + 50*a^3*b +
48*a^2*b^2 + 16*a*b^3 + 7*(15*a^4 + 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(15*a^4 + 20
*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^3 + 3*(15*a^4 + 50*a^3*b + 48*a^2*b^2 + 16*a*b^3)*cosh(d*x + c))*sinh(d*x +
c)^5 + 2*(45*a^4 + 180*a^3*b + 304*a^2*b^2 + 224*a*b^3 + 64*b^4)*cosh(d*x + c)^4 + 2*(35*(15*a^4 + 20*a^3*b +
8*a^2*b^2)*cosh(d*x + c)^4 + 45*a^4 + 180*a^3*b + 304*a^2*b^2 + 224*a*b^3 + 64*b^4 + 30*(15*a^4 + 50*a^3*b + 4
8*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 15*a^4 + 20*a^3*b + 8*a^2*b^2 + 8*(7*(15*a^4 + 20*a^3
*b + 8*a^2*b^2)*cosh(d*x + c)^5 + 10*(15*a^4 + 50*a^3*b + 48*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^3 + (45*a^4 + 1
80*a^3*b + 304*a^2*b^2 + 224*a*b^3 + 64*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(15*a^4 + 50*a^3*b + 48*a^2*b^
2 + 16*a*b^3)*cosh(d*x + c)^2 + 4*(7*(15*a^4 + 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^6 + 15*(15*a^4 + 50*a^3*b +
48*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^4 + 15*a^4 + 50*a^3*b + 48*a^2*b^2 + 16*a*b^3 + 3*(45*a^4 + 180*a^3*b +
304*a^2*b^2 + 224*a*b^3 + 64*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((15*a^4 + 20*a^3*b + 8*a^2*b^2)*cosh(d
*x + c)^7 + 3*(15*a^4 + 50*a^3*b + 48*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^5 + (45*a^4 + 180*a^3*b + 304*a^2*b^2
+ 224*a*b^3 + 64*b^4)*cosh(d*x + c)^3 + (15*a^4 + 50*a^3*b + 48*a^2*b^2 + 16*a*b^3)*cosh(d*x + c))*sinh(d*x +
c))*sqrt(b/(a + b))*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^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2
+ a*b)*cosh(d*x + c)*sinh(d*x + c) + (a^2 + a*b)*sinh(d*x + c)^2 + a^2 + 3*a*b + 2*b^2)*sqrt(b/(a + b)))/(a*c
osh(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*(16*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^7 + 3*(9*a^3*b + 28*a^2*b^2 + 16*a*b^3 + 16*(a^4 + 4*a^
3*b + 5*a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c)^5 + 2*(27*a^3*b + 90*a^2*b^2 + 120*a*b^3 + 48*b^4 + 8*(3*a^4 + 1
4*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c)^3 + (27*a^3*b + 68*a^2*b^2 + 32*a*b^3 + 16*(a^4 +
4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^8
+ 8*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^7 + 2*a^6*b + a^5*b^2)*d*sinh(d*x + c)^8 +
4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^6 + 4*(7*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^
2 + (a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d)*sinh(d*x + c)^6 + 2*(3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3
+ 8*a^3*b^4)*d*cosh(d*x + c)^4 + 8*(7*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^3 + 3*(a^7 + 4*a^6*b + 5*a^5*
b^2 + 2*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^4 + 30*(a^
7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^2 + (3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b
^4)*d)*sinh(d*x + c)^4 + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^2 + 8*(7*(a^7 + 2*a^6*b + a
^5*b^2)*d*cosh(d*x + c)^5 + 10*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^3 + (3*a^7 + 14*a^6*b +
27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh
(d*x + c)^6 + 15*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^4 + 3*(3*a^7 + 14*a^6*b + 27*a^5*b^2
+ 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^2 + (a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d)*sinh(d*x + c)^2 + (a^
7 + 2*a^6*b + a^5*b^2)*d + 8*((a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^7 + 3*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a
^4*b^3)*d*cosh(d*x + c)^5 + (3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^3 + (a^7
+ 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(8*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(
d*x + c)^8 + 64*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + 8*(a^4 + 2*a^3*b + a^2*b^2)*d*x*
sinh(d*x + c)^8 + 2*(9*a^3*b + 28*a^2*b^2 + 16*a*b^3 + 16*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x
+ c)^6 + 2*(112*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^2 + 9*a^3*b + 28*a^2*b^2 + 16*a*b^3 + 16*(a^4 + 4*
a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*x)*sinh(d*x + c)^6 + 4*(112*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^3 + 3*(
9*a^3*b + 28*a^2*b^2 + 16*a*b^3 + 16*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5
+ 2*(27*a^3*b + 90*a^2*b^2 + 120*a*b^3 + 48*b^4 + 8*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*d*x)*c
osh(d*x + c)^4 + 2*(280*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^4 + 27*a^3*b + 90*a^2*b^2 + 120*a*b^3 + 48
*b^4 + 8*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*d*x + 15*(9*a^3*b + 28*a^2*b^2 + 16*a*b^3 + 16*(a^
4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 18*a^3*b + 12*a^2*b^2 + 8*(56*(a^4
+ 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^5 + 5*(9*a^3*b + 28*a^2*b^2 + 16*a*b^3 + 16*(a^4 + 4*a^3*b + 5*a^2*b^2
+ 2*a*b^3)*d*x)*cosh(d*x + c)^3 + (27*a^3*b + 90*a^2*b^2 + 120*a*b^3 + 48*b^4 + 8*(3*a^4 + 14*a^3*b + 27*a^2*b
^2 + 24*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 8*(a^4 + 2*a^3*b + a^2*b^2)*d*x + 2*(27*a^3*b + 6
8*a^2*b^2 + 32*a*b^3 + 16*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c)^2 + 2*(112*(a^4 + 2*a^3*b +
a^2*b^2)*d*x*cosh(d*x + c)^6 + 15*(9*a^3*b + 28*a^2*b^2 + 16*a*b^3 + 16*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)
*d*x)*cosh(d*x + c)^4 + 27*a^3*b + 68*a^2*b^2 + 32*a*b^3 + 16*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*x + 6*(2
7*a^3*b + 90*a^2*b^2 + 120*a*b^3 + 48*b^4 + 8*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*d*x)*cosh(d*x
+ c)^2)*sinh(d*x + c)^2 - ((15*a^4 + 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^8 + 8*(15*a^4 + 20*a^3*b + 8*a^2*b^2
)*cosh(d*x + c)*sinh(d*x + c)^7 + (15*a^4 + 20*a^3*b + 8*a^2*b^2)*sinh(d*x + c)^8 + 4*(15*a^4 + 50*a^3*b + 48*
a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^6 + 4*(15*a^4 + 50*a^3*b + 48*a^2*b^2 + 16*a*b^3 + 7*(15*a^4 + 20*a^3*b + 8*
a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(15*a^4 + 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^3 + 3*(15*a^4 +
50*a^3*b + 48*a^2*b^2 + 16*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(45*a^4 + 180*a^3*b + 304*a^2*b^2 + 224*
a*b^3 + 64*b^4)*cosh(d*x + c)^4 + 2*(35*(15*a^4 + 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^4 + 45*a^4 + 180*a^3*b +
304*a^2*b^2 + 224*a*b^3 + 64*b^4 + 30*(15*a^4 + 50*a^3*b + 48*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^2)*sinh(d*x +
c)^4 + 15*a^4 + 20*a^3*b + 8*a^2*b^2 + 8*(7*(15*a^4 + 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^5 + 10*(15*a^4 + 50
*a^3*b + 48*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^3 + (45*a^4 + 180*a^3*b + 304*a^2*b^2 + 224*a*b^3 + 64*b^4)*cosh
(d*x + c))*sinh(d*x + c)^3 + 4*(15*a^4 + 50*a^3*b + 48*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^2 + 4*(7*(15*a^4 + 20
*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^6 + 15*(15*a^4 + 50*a^3*b + 48*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^4 + 15*a^4
+ 50*a^3*b + 48*a^2*b^2 + 16*a*b^3 + 3*(45*a^4 + 180*a^3*b + 304*a^2*b^2 + 224*a*b^3 + 64*b^4)*cosh(d*x + c)^2
)*sinh(d*x + c)^2 + 8*((15*a^4 + 20*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^7 + 3*(15*a^4 + 50*a^3*b + 48*a^2*b^2 + 1
6*a*b^3)*cosh(d*x + c)^5 + (45*a^4 + 180*a^3*b + 304*a^2*b^2 + 224*a*b^3 + 64*b^4)*cosh(d*x + c)^3 + (15*a^4 +
50*a^3*b + 48*a^2*b^2 + 16*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/(a + b))*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(-b/(a + b))/b) + 4*(16*(a^4 + 2*a^3*b
+ a^2*b^2)*d*x*cosh(d*x + c)^7 + 3*(9*a^3*b + 28*a^2*b^2 + 16*a*b^3 + 16*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)
*d*x)*cosh(d*x + c)^5 + 2*(27*a^3*b + 90*a^2*b^2 + 120*a*b^3 + 48*b^4 + 8*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*
a*b^3 + 8*b^4)*d*x)*cosh(d*x + c)^3 + (27*a^3*b + 68*a^2*b^2 + 32*a*b^3 + 16*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*
b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^8 + 8*(a^7 + 2*a^6*b + a^5*
b^2)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^7 + 2*a^6*b + a^5*b^2)*d*sinh(d*x + c)^8 + 4*(a^7 + 4*a^6*b + 5*a^5*
b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^6 + 4*(7*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^2 + (a^7 + 4*a^6*b + 5*a^5
*b^2 + 2*a^4*b^3)*d)*sinh(d*x + c)^6 + 2*(3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x +
c)^4 + 8*(7*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^3 + 3*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*
x + c))*sinh(d*x + c)^5 + 2*(35*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^4 + 30*(a^7 + 4*a^6*b + 5*a^5*b^2 +
2*a^4*b^3)*d*cosh(d*x + c)^2 + (3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4)*d)*sinh(d*x + c)^4 + 4
*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^2 + 8*(7*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^5
+ 10*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^3 + (3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 +
8*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^6 + 15*(a^7 + 4*
a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^4 + 3*(3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4)*
d*cosh(d*x + c)^2 + (a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d)*sinh(d*x + c)^2 + (a^7 + 2*a^6*b + a^5*b^2)*d +
8*((a^7 + 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^7 + 3*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^5
+ (3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^3 + (a^7 + 4*a^6*b + 5*a^5*b^2 + 2*
a^4*b^3)*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(1/(a+b*sech(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.1767, size = 450, normalized size = 3.08 \begin{align*} -\frac{{\left (15 \, a^{2} b + 20 \, a b^{2} + 8 \, b^{3}\right )} \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^{5} d + 2 \, a^{4} b d + a^{3} b^{2} d\right )} \sqrt{-a b - b^{2}}} + \frac{9 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 28 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 27 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 90 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 120 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 68 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 32 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{3} b + 6 \, a^{2} b^{2}}{4 \,{\left (a^{5} d + 2 \, a^{4} b d + a^{3} 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}} + \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)^2)^3,x, algorithm="giac")

[Out]

-1/8*(15*a^2*b + 20*a*b^2 + 8*b^3)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^5*d + 2*a^4*
b*d + a^3*b^2*d)*sqrt(-a*b - b^2)) + 1/4*(9*a^3*b*e^(6*d*x + 6*c) + 28*a^2*b^2*e^(6*d*x + 6*c) + 16*a*b^3*e^(6
*d*x + 6*c) + 27*a^3*b*e^(4*d*x + 4*c) + 90*a^2*b^2*e^(4*d*x + 4*c) + 120*a*b^3*e^(4*d*x + 4*c) + 48*b^4*e^(4*
d*x + 4*c) + 27*a^3*b*e^(2*d*x + 2*c) + 68*a^2*b^2*e^(2*d*x + 2*c) + 32*a*b^3*e^(2*d*x + 2*c) + 9*a^3*b + 6*a^
2*b^2)/((a^5*d + 2*a^4*b*d + a^3*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)
+ (d*x + c)/(a^3*d)