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

Optimal. Leaf size=123 \[ \frac{(4 a+b) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{8 a^{3/2} d (a+b)^{5/2}}+\frac{(4 a+b) \sinh (c+d x)}{8 a d (a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )}-\frac{b \sinh (c+d x)}{4 a d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2} \]

[Out]

((4*a + b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(8*a^(3/2)*(a + b)^(5/2)*d) - (b*Sinh[c + d*x])/(4*a*(
a + b)*d*(a + b + a*Sinh[c + d*x]^2)^2) + ((4*a + b)*Sinh[c + d*x])/(8*a*(a + b)^2*d*(a + b + a*Sinh[c + d*x]^
2))

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Rubi [A]  time = 0.10837, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {4147, 385, 199, 205} \[ \frac{(4 a+b) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{8 a^{3/2} d (a+b)^{5/2}}+\frac{(4 a+b) \sinh (c+d x)}{8 a d (a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )}-\frac{b \sinh (c+d x)}{4 a d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a + b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(8*a^(3/2)*(a + b)^(5/2)*d) - (b*Sinh[c + d*x])/(4*a*(
a + b)*d*(a + b + a*Sinh[c + d*x]^2)^2) + ((4*a + b)*Sinh[c + d*x])/(8*a*(a + b)^2*d*(a + b + a*Sinh[c + d*x]^
2))

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 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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.811625, size = 159, normalized size = 1.29 \[ -\frac{\text{sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b)^3 \left (\frac{8 \sinh (c+d x)}{\left (a \sinh ^2(c+d x)+a+b\right )^2}-(4 a+b) \left (\frac{5 (a+b) \sinh (c+d x)+3 a \sinh ^3(c+d x)}{(a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{\sqrt{a} (a+b)^{5/2}}\right )\right )}{192 a d \left (a+b \text{sech}^2(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*((8*Sinh[c + d*x])/(a + b + a*Sinh[c + d*x]^2)^2 - (4*a +
b)*((3*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(5/2)) + (5*(a + b)*Sinh[c + d*x] + 3*a*S
inh[c + d*x]^3)/((a + b)^2*(a + b + a*Sinh[c + d*x]^2)^2))))/(192*a*d*(a + b*Sech[c + d*x]^2)^3)

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Maple [B]  time = 0.079, size = 1038, normalized size = 8.4 \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)^3/(a+b*sech(d*x+c)^2)^3,x)

[Out]

-1/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+1/4/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/(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+5/4/d/(ta
nh(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)^5-3/4/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/(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-5/4/d/(ta
nh(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+3/4/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)^3+1/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)-1/4/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*t
anh(1/2*d*x+1/2*c)+1/2/d/(a^2+2*a*b+b^2)/(a+b)^(1/2)/a^(1/2)*arctan(1/2*(2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b
^(1/2))/a^(1/2))-1/2/d/(a^2+2*a*b+b^2)/(a+b)^(1/2)/a^(1/2)*arctan(1/2*(-2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^
(1/2))/a^(1/2))+1/8/d*b/a^(3/2)/(a^2+2*a*b+b^2)/(a+b)^(1/2)*arctan(1/2*(2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^
(1/2))/a^(1/2))-1/8/d/a^(3/2)/(a^2+2*a*b+b^2)/(a+b)^(1/2)*arctan(1/2*(-2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^(
1/2))/a^(1/2))*b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (4 \, a^{2} e^{\left (7 \, c\right )} + a b e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} +{\left (4 \, a^{2} e^{\left (5 \, c\right )} + 9 \, a b e^{\left (5 \, c\right )} - 4 \, b^{2} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} -{\left (4 \, a^{2} e^{\left (3 \, c\right )} + 9 \, a b e^{\left (3 \, c\right )} - 4 \, b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} -{\left (4 \, a^{2} e^{c} + a b e^{c}\right )} e^{\left (d x\right )}}{4 \,{\left (a^{5} d + 2 \, a^{4} b d + a^{3} b^{2} d +{\left (a^{5} d e^{\left (8 \, c\right )} + 2 \, a^{4} b d e^{\left (8 \, c\right )} + a^{3} b^{2} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 4 \,{\left (a^{5} d e^{\left (6 \, c\right )} + 4 \, a^{4} b d e^{\left (6 \, c\right )} + 5 \, a^{3} b^{2} d e^{\left (6 \, c\right )} + 2 \, a^{2} b^{3} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 2 \,{\left (3 \, a^{5} d e^{\left (4 \, c\right )} + 14 \, a^{4} b d e^{\left (4 \, c\right )} + 27 \, a^{3} b^{2} d e^{\left (4 \, c\right )} + 24 \, a^{2} b^{3} d e^{\left (4 \, c\right )} + 8 \, a b^{4} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \,{\left (a^{5} d e^{\left (2 \, c\right )} + 4 \, a^{4} b d e^{\left (2 \, c\right )} + 5 \, a^{3} b^{2} d e^{\left (2 \, c\right )} + 2 \, a^{2} b^{3} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}} + 8 \, \int \frac{{\left (4 \, a e^{\left (3 \, c\right )} + b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (4 \, a e^{c} + b e^{c}\right )} e^{\left (d x\right )}}{32 \,{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2} +{\left (a^{4} e^{\left (4 \, c\right )} + 2 \, a^{3} b e^{\left (4 \, c\right )} + a^{2} b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{4} e^{\left (2 \, c\right )} + 4 \, a^{3} b e^{\left (2 \, c\right )} + 5 \, a^{2} b^{2} e^{\left (2 \, c\right )} + 2 \, a b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \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)^3,x, algorithm="maxima")

[Out]

1/4*((4*a^2*e^(7*c) + a*b*e^(7*c))*e^(7*d*x) + (4*a^2*e^(5*c) + 9*a*b*e^(5*c) - 4*b^2*e^(5*c))*e^(5*d*x) - (4*
a^2*e^(3*c) + 9*a*b*e^(3*c) - 4*b^2*e^(3*c))*e^(3*d*x) - (4*a^2*e^c + a*b*e^c)*e^(d*x))/(a^5*d + 2*a^4*b*d + a
^3*b^2*d + (a^5*d*e^(8*c) + 2*a^4*b*d*e^(8*c) + a^3*b^2*d*e^(8*c))*e^(8*d*x) + 4*(a^5*d*e^(6*c) + 4*a^4*b*d*e^
(6*c) + 5*a^3*b^2*d*e^(6*c) + 2*a^2*b^3*d*e^(6*c))*e^(6*d*x) + 2*(3*a^5*d*e^(4*c) + 14*a^4*b*d*e^(4*c) + 27*a^
3*b^2*d*e^(4*c) + 24*a^2*b^3*d*e^(4*c) + 8*a*b^4*d*e^(4*c))*e^(4*d*x) + 4*(a^5*d*e^(2*c) + 4*a^4*b*d*e^(2*c) +
5*a^3*b^2*d*e^(2*c) + 2*a^2*b^3*d*e^(2*c))*e^(2*d*x)) + 8*integrate(1/32*((4*a*e^(3*c) + b*e^(3*c))*e^(3*d*x)
+ (4*a*e^c + b*e^c)*e^(d*x))/(a^4 + 2*a^3*b + a^2*b^2 + (a^4*e^(4*c) + 2*a^3*b*e^(4*c) + a^2*b^2*e^(4*c))*e^(
4*d*x) + 2*(a^4*e^(2*c) + 4*a^3*b*e^(2*c) + 5*a^2*b^2*e^(2*c) + 2*a*b^3*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 2.93106, size = 14071, normalized size = 114.4 \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)^3,x, algorithm="fricas")

[Out]

[1/16*(4*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)^7 + 28*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)*sinh(d*x +
c)^6 + 4*(4*a^4 + 5*a^3*b + a^2*b^2)*sinh(d*x + c)^7 + 4*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x +
c)^5 + 4*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 21*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x +
c)^5 + 20*(7*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)^3 + (4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x +
c))*sinh(d*x + c)^4 - 4*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^3 + 4*(35*(4*a^4 + 5*a^3*b + a
^2*b^2)*cosh(d*x + c)^4 - 4*a^4 - 13*a^3*b - 5*a^2*b^2 + 4*a*b^3 + 10*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)
*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(21*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)^5 + 10*(4*a^4 + 13*a^3*b +
5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^3 - 3*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c))*sinh(d*x + c
)^2 - ((4*a^3 + a^2*b)*cosh(d*x + c)^8 + 8*(4*a^3 + a^2*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (4*a^3 + a^2*b)*sin
h(d*x + c)^8 + 4*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^6 + 4*(4*a^3 + 9*a^2*b + 2*a*b^2 + 7*(4*a^3 + a^2*b
)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(4*a^3 + a^2*b)*cosh(d*x + c)^3 + 3*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh
(d*x + c))*sinh(d*x + c)^5 + 2*(12*a^3 + 35*a^2*b + 40*a*b^2 + 8*b^3)*cosh(d*x + c)^4 + 2*(35*(4*a^3 + a^2*b)*
cosh(d*x + c)^4 + 12*a^3 + 35*a^2*b + 40*a*b^2 + 8*b^3 + 30*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^2)*sinh(
d*x + c)^4 + 8*(7*(4*a^3 + a^2*b)*cosh(d*x + c)^5 + 10*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^3 + (12*a^3 +
35*a^2*b + 40*a*b^2 + 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*a^3 + a^2*b + 4*(4*a^3 + 9*a^2*b + 2*a*b^2)*c
osh(d*x + c)^2 + 4*(7*(4*a^3 + a^2*b)*cosh(d*x + c)^6 + 15*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^4 + 4*a^3
+ 9*a^2*b + 2*a*b^2 + 3*(12*a^3 + 35*a^2*b + 40*a*b^2 + 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((4*a^3 +
a^2*b)*cosh(d*x + c)^7 + 3*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^5 + (12*a^3 + 35*a^2*b + 40*a*b^2 + 8*b^
3)*cosh(d*x + c)^3 + (4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a^2 - a*b)*log((a*cosh(d*
x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(3*a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh
(d*x + c)^2 - 3*a - 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (3*a + 2*b)*cosh(d*x + c))*sinh(d*x + c) - 4
*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1)*sinh(d*x + c)
- cosh(d*x + c))*sqrt(-a^2 - a*b) + a)/(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)) - 4*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c) + 4*(7*(4*a^4 + 5*
a^3*b + a^2*b^2)*cosh(d*x + c)^6 + 5*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^4 - 4*a^4 - 5*a^3*
b - a^2*b^2 - 3*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^7 + 3*a^6*b + 3*a
^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^8 + 8*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7
+ (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*sinh(d*x + c)^8 + 4*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b
^4)*d*cosh(d*x + c)^6 + 4*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^2 + (a^7 + 5*a^6*b + 9*a^5*
b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d)*sinh(d*x + c)^6 + 2*(3*a^7 + 17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 +
8*a^2*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^3 + 3*(a^7 + 5*a^6*
b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^7 + 3*a^6*b + 3*a^5*b^2 + a
^4*b^3)*d*cosh(d*x + c)^4 + 30*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^2 + (3*a^7
+ 17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d)*sinh(d*x + c)^4 + 4*(a^7 + 5*a^6*b + 9*a^5*b
^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^2 + 8*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^5 +
10*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^3 + (3*a^7 + 17*a^6*b + 41*a^5*b^2 + 5
1*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b
^3)*d*cosh(d*x + c)^6 + 15*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^4 + 3*(3*a^7 +
17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c)^2 + (a^7 + 5*a^6*b + 9*a^5*b^2 +
7*a^4*b^3 + 2*a^3*b^4)*d)*sinh(d*x + c)^2 + (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d + 8*((a^7 + 3*a^6*b + 3*a^
5*b^2 + a^4*b^3)*d*cosh(d*x + c)^7 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^5 +
(3*a^7 + 17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c)^3 + (a^7 + 5*a^6*b + 9*
a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(2*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x
+ c)^7 + 14*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(4*a^4 + 5*a^3*b + a^2*b^2)*sinh(d*x
+ c)^7 + 2*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^5 + 2*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b
^3 + 21*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(7*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(
d*x + c)^3 + (4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 2*(4*a^4 + 13*a^3*b + 5
*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^3 + 2*(35*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)^4 - 4*a^4 - 13*a^3*b - 5
*a^2*b^2 + 4*a*b^3 + 10*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(21*(4*a
^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)^5 + 10*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^3 - 3*(4*a
^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + ((4*a^3 + a^2*b)*cosh(d*x + c)^8 + 8*(4*
a^3 + a^2*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (4*a^3 + a^2*b)*sinh(d*x + c)^8 + 4*(4*a^3 + 9*a^2*b + 2*a*b^2)*c
osh(d*x + c)^6 + 4*(4*a^3 + 9*a^2*b + 2*a*b^2 + 7*(4*a^3 + a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(4*a
^3 + a^2*b)*cosh(d*x + c)^3 + 3*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(12*a^3 + 35*a^
2*b + 40*a*b^2 + 8*b^3)*cosh(d*x + c)^4 + 2*(35*(4*a^3 + a^2*b)*cosh(d*x + c)^4 + 12*a^3 + 35*a^2*b + 40*a*b^2
+ 8*b^3 + 30*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(4*a^3 + a^2*b)*cosh(d*x + c
)^5 + 10*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^3 + (12*a^3 + 35*a^2*b + 40*a*b^2 + 8*b^3)*cosh(d*x + c))*s
inh(d*x + c)^3 + 4*a^3 + a^2*b + 4*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^2 + 4*(7*(4*a^3 + a^2*b)*cosh(d*x
+ c)^6 + 15*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^4 + 4*a^3 + 9*a^2*b + 2*a*b^2 + 3*(12*a^3 + 35*a^2*b +
40*a*b^2 + 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((4*a^3 + a^2*b)*cosh(d*x + c)^7 + 3*(4*a^3 + 9*a^2*b +
2*a*b^2)*cosh(d*x + c)^5 + (12*a^3 + 35*a^2*b + 40*a*b^2 + 8*b^3)*cosh(d*x + c)^3 + (4*a^3 + 9*a^2*b + 2*a*b^
2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 + a*b)*arctan(1/2*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x +
c)^2 + a*sinh(d*x + c)^3 + (3*a + 4*b)*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + 3*a + 4*b)*sinh(d*x + c))/sqrt(a
^2 + a*b)) + ((4*a^3 + a^2*b)*cosh(d*x + c)^8 + 8*(4*a^3 + a^2*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (4*a^3 + a^2
*b)*sinh(d*x + c)^8 + 4*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^6 + 4*(4*a^3 + 9*a^2*b + 2*a*b^2 + 7*(4*a^3
+ a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(4*a^3 + a^2*b)*cosh(d*x + c)^3 + 3*(4*a^3 + 9*a^2*b + 2*a*b^
2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(12*a^3 + 35*a^2*b + 40*a*b^2 + 8*b^3)*cosh(d*x + c)^4 + 2*(35*(4*a^3 +
a^2*b)*cosh(d*x + c)^4 + 12*a^3 + 35*a^2*b + 40*a*b^2 + 8*b^3 + 30*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^2
)*sinh(d*x + c)^4 + 8*(7*(4*a^3 + a^2*b)*cosh(d*x + c)^5 + 10*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^3 + (1
2*a^3 + 35*a^2*b + 40*a*b^2 + 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*a^3 + a^2*b + 4*(4*a^3 + 9*a^2*b + 2*a
*b^2)*cosh(d*x + c)^2 + 4*(7*(4*a^3 + a^2*b)*cosh(d*x + c)^6 + 15*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^4
+ 4*a^3 + 9*a^2*b + 2*a*b^2 + 3*(12*a^3 + 35*a^2*b + 40*a*b^2 + 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((
4*a^3 + a^2*b)*cosh(d*x + c)^7 + 3*(4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c)^5 + (12*a^3 + 35*a^2*b + 40*a*b^2
+ 8*b^3)*cosh(d*x + c)^3 + (4*a^3 + 9*a^2*b + 2*a*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 + a*b)*arctan(1
/2*sqrt(a^2 + a*b)*(cosh(d*x + c) + sinh(d*x + c))/(a + b)) - 2*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c) + 2*
(7*(4*a^4 + 5*a^3*b + a^2*b^2)*cosh(d*x + c)^6 + 5*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^4 -
4*a^4 - 5*a^3*b - a^2*b^2 - 3*(4*a^4 + 13*a^3*b + 5*a^2*b^2 - 4*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^7 +
3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^8 + 8*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)*si
nh(d*x + c)^7 + (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*sinh(d*x + c)^8 + 4*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4
*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^6 + 4*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^2 + (a^7 + 5*
a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d)*sinh(d*x + c)^6 + 2*(3*a^7 + 17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3
+ 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^3 + 3
*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^7 + 3*a^6*b +
3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^4 + 30*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x +
c)^2 + (3*a^7 + 17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d)*sinh(d*x + c)^4 + 4*(a^7 + 5*a
^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^2 + 8*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cos
h(d*x + c)^5 + 10*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^3 + (3*a^7 + 17*a^6*b +
41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7 + 3*a^6*b + 3*a
^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^6 + 15*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^4
+ 3*(3*a^7 + 17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c)^2 + (a^7 + 5*a^6*b
+ 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d)*sinh(d*x + c)^2 + (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d + 8*((a^7 +
3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^7 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cos
h(d*x + c)^5 + (3*a^7 + 17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c)^3 + (a^7
+ 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*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)**3/(a+b*sech(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError