3.335 \(\int \frac{\text{sech}(c+d x)}{(a+b \sinh ^2(c+d x))^2} \, dx\)
Optimal. Leaf size=106 \[ -\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} d (a-b)^2}-\frac{b \sinh (c+d x)}{2 a d (a-b) \left (a+b \sinh ^2(c+d x)\right )}+\frac{\tan ^{-1}(\sinh (c+d x))}{d (a-b)^2} \]
[Out]
ArcTan[Sinh[c + d*x]]/((a - b)^2*d) - ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(
a - b)^2*d) - (b*Sinh[c + d*x])/(2*a*(a - b)*d*(a + b*Sinh[c + d*x]^2))
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Rubi [A] time = 0.107907, antiderivative size = 106, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used =
{3190, 414, 522, 203, 205} \[ -\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} d (a-b)^2}-\frac{b \sinh (c+d x)}{2 a d (a-b) \left (a+b \sinh ^2(c+d x)\right )}+\frac{\tan ^{-1}(\sinh (c+d x))}{d (a-b)^2} \]
Antiderivative was successfully verified.
[In]
Int[Sech[c + d*x]/(a + b*Sinh[c + d*x]^2)^2,x]
[Out]
ArcTan[Sinh[c + d*x]]/((a - b)^2*d) - ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(
a - b)^2*d) - (b*Sinh[c + d*x])/(2*a*(a - b)*d*(a + b*Sinh[c + d*x]^2))
Rule 3190
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
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 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 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])
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}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac{b \sinh (c+d x)}{2 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{2 a-b-b x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\sinh (c+d x)\right )}{2 a (a-b) d}\\ &=-\frac{b \sinh (c+d x)}{2 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{(a-b)^2 d}-\frac{((3 a-b) b) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sinh (c+d x)\right )}{2 a (a-b)^2 d}\\ &=\frac{\tan ^{-1}(\sinh (c+d x))}{(a-b)^2 d}-\frac{(3 a-b) \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} (a-b)^2 d}-\frac{b \sinh (c+d x)}{2 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.377087, size = 174, normalized size = 1.64 \[ \frac{\cosh (2 (c+d x)) \left (4 a^{3/2} b \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )-b^{3/2} (b-3 a) \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right )\right )+(2 a-b) \left (4 a^{3/2} \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )-\sqrt{b} (b-3 a) \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right )\right )-2 \sqrt{a} b (a-b) \sinh (c+d x)}{2 a^{3/2} d (a-b)^2 (2 a+b \cosh (2 (c+d x))-b)} \]
Antiderivative was successfully verified.
[In]
Integrate[Sech[c + d*x]/(a + b*Sinh[c + d*x]^2)^2,x]
[Out]
((2*a - b)*(-(Sqrt[b]*(-3*a + b)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]]) + 4*a^(3/2)*ArcTan[Tanh[(c + d*x)/2]
]) + (-(b^(3/2)*(-3*a + b)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]]) + 4*a^(3/2)*b*ArcTan[Tanh[(c + d*x)/2]])*C
osh[2*(c + d*x)] - 2*Sqrt[a]*(a - b)*b*Sinh[c + d*x])/(2*a^(3/2)*(a - b)^2*d*(2*a - b + b*Cosh[2*(c + d*x)]))
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Maple [B] time = 0.082, size = 1080, normalized size = 10.2 \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)/(a+b*sinh(d*x+c)^2)^2,x)
[Out]
1/d*b/(a-b)^2/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*tanh(1/2*d*x+1/2
*c)^3-1/d*b^2/(a-b)^2/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)/a*tanh(1
/2*d*x+1/2*c)^3-1/d*b/(a-b)^2/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*
tanh(1/2*d*x+1/2*c)+1/d*b^2/(a-b)^2/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2
*b+a)/a*tanh(1/2*d*x+1/2*c)+3/2/d*b/(a-b)^2*a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*
tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+3/2/d*b/(a-b)^2/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)
*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))-2/d*b^2/(a-b)^2/(-b*(a-b))^(1/2)/((2*(-b*
(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+3/2/d*b/(a-b)
^2*a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a
+2*b)*a)^(1/2))-3/2/d*b/(a-b)^2/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b
))^(1/2)-a+2*b)*a)^(1/2))-2/d*b^2/(a-b)^2/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(
1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))-1/2/d*b^2/(a-b)^2/a/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*
arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+1/2/d*b^3/(a-b)^2/a/(-b*(a-b))^(1/2)/((2*(
-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+1/2/d*b^2/
(a-b)^2/a/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/
2))+1/2/d*b^3/(a-b)^2/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2
*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))+2/d/(a-b)^2*arctan(tanh(1/2*d*x+1/2*c))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b e^{\left (3 \, d x + 3 \, c\right )} - b e^{\left (d x + c\right )}}{a^{2} b d - a b^{2} d +{\left (a^{2} b d e^{\left (4 \, c\right )} - a b^{2} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (2 \, a^{3} d e^{\left (2 \, c\right )} - 3 \, a^{2} b d e^{\left (2 \, c\right )} + a b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + \frac{2 \, \arctan \left (e^{\left (d x + c\right )}\right )}{a^{2} d - 2 \, a b d + b^{2} d} - 2 \, \int \frac{{\left (3 \, a b e^{\left (3 \, c\right )} - b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (3 \, a b e^{c} - b^{2} e^{c}\right )} e^{\left (d x\right )}}{2 \,{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3} +{\left (a^{3} b e^{\left (4 \, c\right )} - 2 \, a^{2} b^{2} e^{\left (4 \, c\right )} + a b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (2 \, a^{4} e^{\left (2 \, c\right )} - 5 \, a^{3} b e^{\left (2 \, c\right )} + 4 \, a^{2} b^{2} e^{\left (2 \, c\right )} - 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)/(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")
[Out]
-(b*e^(3*d*x + 3*c) - b*e^(d*x + c))/(a^2*b*d - a*b^2*d + (a^2*b*d*e^(4*c) - a*b^2*d*e^(4*c))*e^(4*d*x) + 2*(2
*a^3*d*e^(2*c) - 3*a^2*b*d*e^(2*c) + a*b^2*d*e^(2*c))*e^(2*d*x)) + 2*arctan(e^(d*x + c))/(a^2*d - 2*a*b*d + b^
2*d) - 2*integrate(1/2*((3*a*b*e^(3*c) - b^2*e^(3*c))*e^(3*d*x) + (3*a*b*e^c - b^2*e^c)*e^(d*x))/(a^3*b - 2*a^
2*b^2 + a*b^3 + (a^3*b*e^(4*c) - 2*a^2*b^2*e^(4*c) + a*b^3*e^(4*c))*e^(4*d*x) + 2*(2*a^4*e^(2*c) - 5*a^3*b*e^(
2*c) + 4*a^2*b^2*e^(2*c) - a*b^3*e^(2*c))*e^(2*d*x)), x)
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Fricas [B] time = 2.00276, size = 5192, normalized size = 48.98 \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)/(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")
[Out]
[-1/4*(4*(a*b - b^2)*cosh(d*x + c)^3 + 12*(a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^2 + 4*(a*b - b^2)*sinh(d*x +
c)^3 + ((3*a*b - b^2)*cosh(d*x + c)^4 + 4*(3*a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a*b - b^2)*sinh(d*
x + c)^4 + 2*(6*a^2 - 5*a*b + b^2)*cosh(d*x + c)^2 + 2*(3*(3*a*b - b^2)*cosh(d*x + c)^2 + 6*a^2 - 5*a*b + b^2)
*sinh(d*x + c)^2 + 3*a*b - b^2 + 4*((3*a*b - b^2)*cosh(d*x + c)^3 + (6*a^2 - 5*a*b + b^2)*cosh(d*x + c))*sinh(
d*x + c))*sqrt(-b/a)*log((b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 - 2*(2*a +
b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a + b)*cos
h(d*x + c))*sinh(d*x + c) + 4*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 - a*c
osh(d*x + c) + (3*a*cosh(d*x + c)^2 - a)*sinh(d*x + c))*sqrt(-b/a) + b)/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)
*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*
x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) - 8*(a*b*cosh(d*x + c)^4 + 4*a*
b*cosh(d*x + c)*sinh(d*x + c)^3 + a*b*sinh(d*x + c)^4 + 2*(2*a^2 - a*b)*cosh(d*x + c)^2 + 2*(3*a*b*cosh(d*x +
c)^2 + 2*a^2 - a*b)*sinh(d*x + c)^2 + a*b + 4*(a*b*cosh(d*x + c)^3 + (2*a^2 - a*b)*cosh(d*x + c))*sinh(d*x + c
))*arctan(cosh(d*x + c) + sinh(d*x + c)) - 4*(a*b - b^2)*cosh(d*x + c) + 4*(3*(a*b - b^2)*cosh(d*x + c)^2 - a*
b + b^2)*sinh(d*x + c))/((a^3*b - 2*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^4 + 4*(a^3*b - 2*a^2*b^2 + a*b^3)*d*cosh(
d*x + c)*sinh(d*x + c)^3 + (a^3*b - 2*a^2*b^2 + a*b^3)*d*sinh(d*x + c)^4 + 2*(2*a^4 - 5*a^3*b + 4*a^2*b^2 - a*
b^3)*d*cosh(d*x + c)^2 + 2*(3*(a^3*b - 2*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^2 + (2*a^4 - 5*a^3*b + 4*a^2*b^2 - a
*b^3)*d)*sinh(d*x + c)^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*d + 4*((a^3*b - 2*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^3 +
(2*a^4 - 5*a^3*b + 4*a^2*b^2 - a*b^3)*d*cosh(d*x + c))*sinh(d*x + c)), -1/2*(2*(a*b - b^2)*cosh(d*x + c)^3 + 6
*(a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^2 + 2*(a*b - b^2)*sinh(d*x + c)^3 + ((3*a*b - b^2)*cosh(d*x + c)^4 +
4*(3*a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a*b - b^2)*sinh(d*x + c)^4 + 2*(6*a^2 - 5*a*b + b^2)*cosh(d
*x + c)^2 + 2*(3*(3*a*b - b^2)*cosh(d*x + c)^2 + 6*a^2 - 5*a*b + b^2)*sinh(d*x + c)^2 + 3*a*b - b^2 + 4*((3*a*
b - b^2)*cosh(d*x + c)^3 + (6*a^2 - 5*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/a)*arctan(1/2*sqrt(b/a)*
(cosh(d*x + c) + sinh(d*x + c))) + ((3*a*b - b^2)*cosh(d*x + c)^4 + 4*(3*a*b - b^2)*cosh(d*x + c)*sinh(d*x + c
)^3 + (3*a*b - b^2)*sinh(d*x + c)^4 + 2*(6*a^2 - 5*a*b + b^2)*cosh(d*x + c)^2 + 2*(3*(3*a*b - b^2)*cosh(d*x +
c)^2 + 6*a^2 - 5*a*b + b^2)*sinh(d*x + c)^2 + 3*a*b - b^2 + 4*((3*a*b - b^2)*cosh(d*x + c)^3 + (6*a^2 - 5*a*b
+ b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/a)*arctan(1/2*(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)*sinh(d*x + c)
^2 + b*sinh(d*x + c)^3 + (4*a - b)*cosh(d*x + c) + (3*b*cosh(d*x + c)^2 + 4*a - b)*sinh(d*x + c))*sqrt(b/a)/b)
- 4*(a*b*cosh(d*x + c)^4 + 4*a*b*cosh(d*x + c)*sinh(d*x + c)^3 + a*b*sinh(d*x + c)^4 + 2*(2*a^2 - a*b)*cosh(d
*x + c)^2 + 2*(3*a*b*cosh(d*x + c)^2 + 2*a^2 - a*b)*sinh(d*x + c)^2 + a*b + 4*(a*b*cosh(d*x + c)^3 + (2*a^2 -
a*b)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - 2*(a*b - b^2)*cosh(d*x + c) + 2*(3*
(a*b - b^2)*cosh(d*x + c)^2 - a*b + b^2)*sinh(d*x + c))/((a^3*b - 2*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^4 + 4*(a^
3*b - 2*a^2*b^2 + a*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*b - 2*a^2*b^2 + a*b^3)*d*sinh(d*x + c)^4 + 2*(
2*a^4 - 5*a^3*b + 4*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^2 + 2*(3*(a^3*b - 2*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^2 +
(2*a^4 - 5*a^3*b + 4*a^2*b^2 - a*b^3)*d)*sinh(d*x + c)^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*d + 4*((a^3*b - 2*a^2*b
^2 + a*b^3)*d*cosh(d*x + c)^3 + (2*a^4 - 5*a^3*b + 4*a^2*b^2 - a*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(sech(d*x+c)/(a+b*sinh(d*x+c)**2)**2,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)/(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")
[Out]
Exception raised: TypeError