3.338 \(\int \frac{\text{sech}^4(c+d x)}{(a+b \sinh ^2(c+d x))^2} \, dx\)
Optimal. Leaf size=143 \[ \frac{b^2 (6 a-b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} d (a-b)^{7/2}}-\frac{b^3 \tanh (c+d x)}{2 a d (a-b)^3 \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac{\tanh ^3(c+d x)}{3 d (a-b)^2}+\frac{(a-3 b) \tanh (c+d x)}{d (a-b)^3} \]
[Out]
((6*a - b)*b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^(7/2)*d) + ((a - 3*b)*Tanh[c +
d*x])/((a - b)^3*d) - Tanh[c + d*x]^3/(3*(a - b)^2*d) - (b^3*Tanh[c + d*x])/(2*a*(a - b)^3*d*(a - (a - b)*Tan
h[c + d*x]^2))
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Rubi [A] time = 0.207041, antiderivative size = 143, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used =
{3191, 390, 385, 208} \[ \frac{b^2 (6 a-b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} d (a-b)^{7/2}}-\frac{b^3 \tanh (c+d x)}{2 a d (a-b)^3 \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac{\tanh ^3(c+d x)}{3 d (a-b)^2}+\frac{(a-3 b) \tanh (c+d x)}{d (a-b)^3} \]
Antiderivative was successfully verified.
[In]
Int[Sech[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2,x]
[Out]
((6*a - b)*b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^(7/2)*d) + ((a - 3*b)*Tanh[c +
d*x])/((a - b)^3*d) - Tanh[c + d*x]^3/(3*(a - b)^2*d) - (b^3*Tanh[c + d*x])/(2*a*(a - b)^3*d*(a - (a - b)*Tan
h[c + d*x]^2))
Rule 3191
Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
Rule 390
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]
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 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}^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{\left (a-(a-b) x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a-3 b}{(a-b)^3}-\frac{x^2}{(a-b)^2}+\frac{(3 a-b) b^2-3 (a-b) b^2 x^2}{(a-b)^3 \left (a+(-a+b) x^2\right )^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{(a-3 b) \tanh (c+d x)}{(a-b)^3 d}-\frac{\tanh ^3(c+d x)}{3 (a-b)^2 d}+\frac{\operatorname{Subst}\left (\int \frac{(3 a-b) b^2-3 (a-b) b^2 x^2}{\left (a+(-a+b) x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{(a-b)^3 d}\\ &=\frac{(a-3 b) \tanh (c+d x)}{(a-b)^3 d}-\frac{\tanh ^3(c+d x)}{3 (a-b)^2 d}-\frac{b^3 \tanh (c+d x)}{2 a (a-b)^3 d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac{\left ((6 a-b) b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{2 a (a-b)^3 d}\\ &=\frac{(6 a-b) b^2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} (a-b)^{7/2} d}+\frac{(a-3 b) \tanh (c+d x)}{(a-b)^3 d}-\frac{\tanh ^3(c+d x)}{3 (a-b)^2 d}-\frac{b^3 \tanh (c+d x)}{2 a (a-b)^3 d \left (a-(a-b) \tanh ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 1.89396, size = 130, normalized size = 0.91 \[ \frac{\frac{3 b^2 (6 a-b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{3/2} (a-b)^{7/2}}+\frac{2 \tanh (c+d x) \left ((a-b) \text{sech}^2(c+d x)+2 (a-4 b)\right )-\frac{3 b^3 \sinh (2 (c+d x))}{a (2 a+b \cosh (2 (c+d x))-b)}}{(a-b)^3}}{6 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Sech[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2,x]
[Out]
((3*(6*a - b)*b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*(a - b)^(7/2)) + ((-3*b^3*Sinh[2*(c +
d*x)])/(a*(2*a - b + b*Cosh[2*(c + d*x)])) + 2*(2*(a - 4*b) + (a - b)*Sech[c + d*x]^2)*Tanh[c + d*x])/(a - b)
^3)/(6*d)
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Maple [B] time = 0.092, size = 998, normalized size = 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)^4/(a+b*sinh(d*x+c)^2)^2,x)
[Out]
-1/d*b^3/(a-b)^3/(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^3/(a-b)^3/(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*t
anh(1/2*d*x+1/2*c)+3/d*b^2/(a-b)^3/((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/d*b^3/(a-b)^3/(-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/d*b^2/(a-b)^3/((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/d*b^3/(a-b)^3/(-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^3/(a-b)
^3/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^4/(a-b)^3/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^3/(a-b)^3/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^4/(a-b)^3/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)^3/(tanh(1/2*d*x+1/2*c)^
2+1)^3*tanh(1/2*d*x+1/2*c)^5*a-6/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^3*tanh(1/2*d*x+1/2*c)^5*b+4/3/d/(a-b)^3/(
tanh(1/2*d*x+1/2*c)^2+1)^3*tanh(1/2*d*x+1/2*c)^3*a-28/3/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^3*tanh(1/2*d*x+1/2
*c)^3*b+2/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^3*tanh(1/2*d*x+1/2*c)*a-6/d/(a-b)^3/(tanh(1/2*d*x+1/2*c)^2+1)^3*
tanh(1/2*d*x+1/2*c)*b
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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)^4/(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.66689, size = 17920, normalized size = 125.31 \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)^4/(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")
[Out]
[1/12*(12*(6*a^3*b^2 - 7*a^2*b^3 + a*b^4)*cosh(d*x + c)^8 + 96*(6*a^3*b^2 - 7*a^2*b^3 + a*b^4)*cosh(d*x + c)*s
inh(d*x + c)^7 + 12*(6*a^3*b^2 - 7*a^2*b^3 + a*b^4)*sinh(d*x + c)^8 + 24*(6*a^4*b - a^3*b^2 - 6*a^2*b^3 + a*b^
4)*cosh(d*x + c)^6 + 24*(6*a^4*b - a^3*b^2 - 6*a^2*b^3 + a*b^4 + 14*(6*a^3*b^2 - 7*a^2*b^3 + a*b^4)*cosh(d*x +
c)^2)*sinh(d*x + c)^6 + 48*(14*(6*a^3*b^2 - 7*a^2*b^3 + a*b^4)*cosh(d*x + c)^3 + 3*(6*a^4*b - a^3*b^2 - 6*a^2
*b^3 + a*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 - 16*a^4*b + 80*a^3*b^2 - 52*a^2*b^3 - 12*a*b^4 - 8*(24*a^5 - 106
*a^4*b + 95*a^3*b^2 - 13*a^2*b^3)*cosh(d*x + c)^4 - 8*(24*a^5 - 106*a^4*b + 95*a^3*b^2 - 13*a^2*b^3 - 105*(6*a
^3*b^2 - 7*a^2*b^3 + a*b^4)*cosh(d*x + c)^4 - 45*(6*a^4*b - a^3*b^2 - 6*a^2*b^3 + a*b^4)*cosh(d*x + c)^2)*sinh
(d*x + c)^4 + 32*(21*(6*a^3*b^2 - 7*a^2*b^3 + a*b^4)*cosh(d*x + c)^5 + 15*(6*a^4*b - a^3*b^2 - 6*a^2*b^3 + a*b
^4)*cosh(d*x + c)^3 - (24*a^5 - 106*a^4*b + 95*a^3*b^2 - 13*a^2*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 8*(8*a^5
- 38*a^4*b + 25*a^3*b^2 + 2*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^2 + 8*(42*(6*a^3*b^2 - 7*a^2*b^3 + a*b^4)*cosh(d
*x + c)^6 - 8*a^5 + 38*a^4*b - 25*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 + 45*(6*a^4*b - a^3*b^2 - 6*a^2*b^3 + a*b^4)*c
osh(d*x + c)^4 - 6*(24*a^5 - 106*a^4*b + 95*a^3*b^2 - 13*a^2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 3*((6*a*b
^3 - b^4)*cosh(d*x + c)^10 + 10*(6*a*b^3 - b^4)*cosh(d*x + c)*sinh(d*x + c)^9 + (6*a*b^3 - b^4)*sinh(d*x + c)^
10 + (24*a^2*b^2 + 2*a*b^3 - b^4)*cosh(d*x + c)^8 + (24*a^2*b^2 + 2*a*b^3 - b^4 + 45*(6*a*b^3 - b^4)*cosh(d*x
+ c)^2)*sinh(d*x + c)^8 + 8*(15*(6*a*b^3 - b^4)*cosh(d*x + c)^3 + (24*a^2*b^2 + 2*a*b^3 - b^4)*cosh(d*x + c))*
sinh(d*x + c)^7 + 2*(36*a^2*b^2 - 12*a*b^3 + b^4)*cosh(d*x + c)^6 + 2*(105*(6*a*b^3 - b^4)*cosh(d*x + c)^4 + 3
6*a^2*b^2 - 12*a*b^3 + b^4 + 14*(24*a^2*b^2 + 2*a*b^3 - b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(6*a*b^3
- b^4)*cosh(d*x + c)^5 + 14*(24*a^2*b^2 + 2*a*b^3 - b^4)*cosh(d*x + c)^3 + 3*(36*a^2*b^2 - 12*a*b^3 + b^4)*co
sh(d*x + c))*sinh(d*x + c)^5 + 2*(36*a^2*b^2 - 12*a*b^3 + b^4)*cosh(d*x + c)^4 + 2*(105*(6*a*b^3 - b^4)*cosh(d
*x + c)^6 + 35*(24*a^2*b^2 + 2*a*b^3 - b^4)*cosh(d*x + c)^4 + 36*a^2*b^2 - 12*a*b^3 + b^4 + 15*(36*a^2*b^2 - 1
2*a*b^3 + b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 6*a*b^3 - b^4 + 8*(15*(6*a*b^3 - b^4)*cosh(d*x + c)^7 + 7*(2
4*a^2*b^2 + 2*a*b^3 - b^4)*cosh(d*x + c)^5 + 5*(36*a^2*b^2 - 12*a*b^3 + b^4)*cosh(d*x + c)^3 + (36*a^2*b^2 - 1
2*a*b^3 + b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + (24*a^2*b^2 + 2*a*b^3 - b^4)*cosh(d*x + c)^2 + (45*(6*a*b^3 -
b^4)*cosh(d*x + c)^8 + 28*(24*a^2*b^2 + 2*a*b^3 - b^4)*cosh(d*x + c)^6 + 30*(36*a^2*b^2 - 12*a*b^3 + b^4)*cosh
(d*x + c)^4 + 24*a^2*b^2 + 2*a*b^3 - b^4 + 12*(36*a^2*b^2 - 12*a*b^3 + b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 +
2*(5*(6*a*b^3 - b^4)*cosh(d*x + c)^9 + 4*(24*a^2*b^2 + 2*a*b^3 - b^4)*cosh(d*x + c)^7 + 6*(36*a^2*b^2 - 12*a*
b^3 + b^4)*cosh(d*x + c)^5 + 4*(36*a^2*b^2 - 12*a*b^3 + b^4)*cosh(d*x + c)^3 + (24*a^2*b^2 + 2*a*b^3 - b^4)*co
sh(d*x + c))*sinh(d*x + c))*sqrt(a^2 - a*b)*log((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b
^2*sinh(d*x + c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2
+ 8*a^2 - 8*a*b + b^2 + 4*(b^2*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c) - 4*(b*cosh(d*x +
c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(a^2 - 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)) + 16*(6*(6*a^3*
b^2 - 7*a^2*b^3 + a*b^4)*cosh(d*x + c)^7 + 9*(6*a^4*b - a^3*b^2 - 6*a^2*b^3 + a*b^4)*cosh(d*x + c)^5 - 2*(24*a
^5 - 106*a^4*b + 95*a^3*b^2 - 13*a^2*b^3)*cosh(d*x + c)^3 - (8*a^5 - 38*a^4*b + 25*a^3*b^2 + 2*a^2*b^3 + 3*a*b
^4)*cosh(d*x + c))*sinh(d*x + c))/((a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^10 +
10*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^6*b - 4*a^5*b^2
+ 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*sinh(d*x + c)^10 + (4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)
*d*cosh(d*x + c)^8 + (45*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^2 + (4*a^7 - 15
*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*d)*sinh(d*x + c)^8 + 2*(6*a^7 - 25*a^6*b + 40*a^5*b^2 - 30*a^4*b^3
+ 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^6 + 8*(15*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*cos
h(d*x + c)^3 + (4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(10
5*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^4 + 14*(4*a^7 - 15*a^6*b + 20*a^5*b^2
- 10*a^4*b^3 + a^2*b^5)*d*cosh(d*x + c)^2 + (6*a^7 - 25*a^6*b + 40*a^5*b^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*b^5
)*d)*sinh(d*x + c)^6 + 2*(6*a^7 - 25*a^6*b + 40*a^5*b^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^4
+ 4*(63*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^5 + 14*(4*a^7 - 15*a^6*b + 20*a
^5*b^2 - 10*a^4*b^3 + a^2*b^5)*d*cosh(d*x + c)^3 + 3*(6*a^7 - 25*a^6*b + 40*a^5*b^2 - 30*a^4*b^3 + 10*a^3*b^4
- a^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*c
osh(d*x + c)^6 + 35*(4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*d*cosh(d*x + c)^4 + 15*(6*a^7 - 25*
a^6*b + 40*a^5*b^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^2 + (6*a^7 - 25*a^6*b + 40*a^5*b^2 - 3
0*a^4*b^3 + 10*a^3*b^4 - a^2*b^5)*d)*sinh(d*x + c)^4 + (4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*
d*cosh(d*x + c)^2 + 8*(15*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^7 + 7*(4*a^7 -
15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*d*cosh(d*x + c)^5 + 5*(6*a^7 - 25*a^6*b + 40*a^5*b^2 - 30*a^4*b
^3 + 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^3 + (6*a^7 - 25*a^6*b + 40*a^5*b^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*
b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*cosh(d*x +
c)^8 + 28*(4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*d*cosh(d*x + c)^6 + 30*(6*a^7 - 25*a^6*b + 4
0*a^5*b^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^4 + 12*(6*a^7 - 25*a^6*b + 40*a^5*b^2 - 30*a^4*
b^3 + 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^2 + (4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*d)*sinh
(d*x + c)^2 + (a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d + 2*(5*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 -
4*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^9 + 4*(4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*d*cosh(d*x +
c)^7 + 6*(6*a^7 - 25*a^6*b + 40*a^5*b^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^5 + 4*(6*a^7 - 2
5*a^6*b + 40*a^5*b^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^3 + (4*a^7 - 15*a^6*b + 20*a^5*b^2 -
10*a^4*b^3 + a^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)), 1/6*(6*(6*a^3*b^2 - 7*a^2*b^3 + a*b^4)*cosh(d*x + c)^8
+ 48*(6*a^3*b^2 - 7*a^2*b^3 + a*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + 6*(6*a^3*b^2 - 7*a^2*b^3 + a*b^4)*sinh(d
*x + c)^8 + 12*(6*a^4*b - a^3*b^2 - 6*a^2*b^3 + a*b^4)*cosh(d*x + c)^6 + 12*(6*a^4*b - a^3*b^2 - 6*a^2*b^3 + a
*b^4 + 14*(6*a^3*b^2 - 7*a^2*b^3 + a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 24*(14*(6*a^3*b^2 - 7*a^2*b^3 + a
*b^4)*cosh(d*x + c)^3 + 3*(6*a^4*b - a^3*b^2 - 6*a^2*b^3 + a*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 - 8*a^4*b + 4
0*a^3*b^2 - 26*a^2*b^3 - 6*a*b^4 - 4*(24*a^5 - 106*a^4*b + 95*a^3*b^2 - 13*a^2*b^3)*cosh(d*x + c)^4 - 4*(24*a^
5 - 106*a^4*b + 95*a^3*b^2 - 13*a^2*b^3 - 105*(6*a^3*b^2 - 7*a^2*b^3 + a*b^4)*cosh(d*x + c)^4 - 45*(6*a^4*b -
a^3*b^2 - 6*a^2*b^3 + a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(21*(6*a^3*b^2 - 7*a^2*b^3 + a*b^4)*cosh(d*
x + c)^5 + 15*(6*a^4*b - a^3*b^2 - 6*a^2*b^3 + a*b^4)*cosh(d*x + c)^3 - (24*a^5 - 106*a^4*b + 95*a^3*b^2 - 13*
a^2*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(8*a^5 - 38*a^4*b + 25*a^3*b^2 + 2*a^2*b^3 + 3*a*b^4)*cosh(d*x + c
)^2 + 4*(42*(6*a^3*b^2 - 7*a^2*b^3 + a*b^4)*cosh(d*x + c)^6 - 8*a^5 + 38*a^4*b - 25*a^3*b^2 - 2*a^2*b^3 - 3*a*
b^4 + 45*(6*a^4*b - a^3*b^2 - 6*a^2*b^3 + a*b^4)*cosh(d*x + c)^4 - 6*(24*a^5 - 106*a^4*b + 95*a^3*b^2 - 13*a^2
*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 3*((6*a*b^3 - b^4)*cosh(d*x + c)^10 + 10*(6*a*b^3 - b^4)*cosh(d*x + c
)*sinh(d*x + c)^9 + (6*a*b^3 - b^4)*sinh(d*x + c)^10 + (24*a^2*b^2 + 2*a*b^3 - b^4)*cosh(d*x + c)^8 + (24*a^2*
b^2 + 2*a*b^3 - b^4 + 45*(6*a*b^3 - b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(6*a*b^3 - b^4)*cosh(d*x + c
)^3 + (24*a^2*b^2 + 2*a*b^3 - b^4)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(36*a^2*b^2 - 12*a*b^3 + b^4)*cosh(d*x +
c)^6 + 2*(105*(6*a*b^3 - b^4)*cosh(d*x + c)^4 + 36*a^2*b^2 - 12*a*b^3 + b^4 + 14*(24*a^2*b^2 + 2*a*b^3 - b^4)
*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(6*a*b^3 - b^4)*cosh(d*x + c)^5 + 14*(24*a^2*b^2 + 2*a*b^3 - b^4)*co
sh(d*x + c)^3 + 3*(36*a^2*b^2 - 12*a*b^3 + b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(36*a^2*b^2 - 12*a*b^3 + b^
4)*cosh(d*x + c)^4 + 2*(105*(6*a*b^3 - b^4)*cosh(d*x + c)^6 + 35*(24*a^2*b^2 + 2*a*b^3 - b^4)*cosh(d*x + c)^4
+ 36*a^2*b^2 - 12*a*b^3 + b^4 + 15*(36*a^2*b^2 - 12*a*b^3 + b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 6*a*b^3 -
b^4 + 8*(15*(6*a*b^3 - b^4)*cosh(d*x + c)^7 + 7*(24*a^2*b^2 + 2*a*b^3 - b^4)*cosh(d*x + c)^5 + 5*(36*a^2*b^2 -
12*a*b^3 + b^4)*cosh(d*x + c)^3 + (36*a^2*b^2 - 12*a*b^3 + b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + (24*a^2*b^2
+ 2*a*b^3 - b^4)*cosh(d*x + c)^2 + (45*(6*a*b^3 - b^4)*cosh(d*x + c)^8 + 28*(24*a^2*b^2 + 2*a*b^3 - b^4)*cosh(
d*x + c)^6 + 30*(36*a^2*b^2 - 12*a*b^3 + b^4)*cosh(d*x + c)^4 + 24*a^2*b^2 + 2*a*b^3 - b^4 + 12*(36*a^2*b^2 -
12*a*b^3 + b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(6*a*b^3 - b^4)*cosh(d*x + c)^9 + 4*(24*a^2*b^2 + 2*a*
b^3 - b^4)*cosh(d*x + c)^7 + 6*(36*a^2*b^2 - 12*a*b^3 + b^4)*cosh(d*x + c)^5 + 4*(36*a^2*b^2 - 12*a*b^3 + b^4)
*cosh(d*x + c)^3 + (24*a^2*b^2 + 2*a*b^3 - b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a^2 + a*b)*arctan(-1/2*(b*
cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(-a^2 + a*b)/(a^2 - a*b))
+ 8*(6*(6*a^3*b^2 - 7*a^2*b^3 + a*b^4)*cosh(d*x + c)^7 + 9*(6*a^4*b - a^3*b^2 - 6*a^2*b^3 + a*b^4)*cosh(d*x +
c)^5 - 2*(24*a^5 - 106*a^4*b + 95*a^3*b^2 - 13*a^2*b^3)*cosh(d*x + c)^3 - (8*a^5 - 38*a^4*b + 25*a^3*b^2 + 2*
a^2*b^3 + 3*a*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*cosh
(d*x + c)^10 + 10*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^6
*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*sinh(d*x + c)^10 + (4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4
*b^3 + a^2*b^5)*d*cosh(d*x + c)^8 + (45*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^
2 + (4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*d)*sinh(d*x + c)^8 + 2*(6*a^7 - 25*a^6*b + 40*a^5*b
^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^6 + 8*(15*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 +
a^2*b^5)*d*cosh(d*x + c)^3 + (4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*d*cosh(d*x + c))*sinh(d*x
+ c)^7 + 2*(105*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^4 + 14*(4*a^7 - 15*a^6*
b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*d*cosh(d*x + c)^2 + (6*a^7 - 25*a^6*b + 40*a^5*b^2 - 30*a^4*b^3 + 10*a^
3*b^4 - a^2*b^5)*d)*sinh(d*x + c)^6 + 2*(6*a^7 - 25*a^6*b + 40*a^5*b^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*b^5)*d*
cosh(d*x + c)^4 + 4*(63*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^5 + 14*(4*a^7 -
15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*d*cosh(d*x + c)^3 + 3*(6*a^7 - 25*a^6*b + 40*a^5*b^2 - 30*a^4*b^
3 + 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4
+ a^2*b^5)*d*cosh(d*x + c)^6 + 35*(4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*d*cosh(d*x + c)^4 +
15*(6*a^7 - 25*a^6*b + 40*a^5*b^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^2 + (6*a^7 - 25*a^6*b +
40*a^5*b^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*b^5)*d)*sinh(d*x + c)^4 + (4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*
b^3 + a^2*b^5)*d*cosh(d*x + c)^2 + 8*(15*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)
^7 + 7*(4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*d*cosh(d*x + c)^5 + 5*(6*a^7 - 25*a^6*b + 40*a^5
*b^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^3 + (6*a^7 - 25*a^6*b + 40*a^5*b^2 - 30*a^4*b^3 + 10
*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^
5)*d*cosh(d*x + c)^8 + 28*(4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*d*cosh(d*x + c)^6 + 30*(6*a^7
- 25*a^6*b + 40*a^5*b^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^4 + 12*(6*a^7 - 25*a^6*b + 40*a^
5*b^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^2 + (4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a
^2*b^5)*d)*sinh(d*x + c)^2 + (a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d + 2*(5*(a^6*b - 4*a^5*b^2
+ 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^9 + 4*(4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^
5)*d*cosh(d*x + c)^7 + 6*(6*a^7 - 25*a^6*b + 40*a^5*b^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^5
+ 4*(6*a^7 - 25*a^6*b + 40*a^5*b^2 - 30*a^4*b^3 + 10*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^3 + (4*a^7 - 15*a^6*b
+ 20*a^5*b^2 - 10*a^4*b^3 + a^2*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)**4/(a+b*sinh(d*x+c)**2)**2,x)
[Out]
Timed out
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Giac [B] time = 1.38008, size = 377, normalized size = 2.64 \begin{align*} \frac{{\left (6 \, a b^{2} - b^{3}\right )} \arctan \left (\frac{b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt{-a^{2} + a b}}\right )}{2 \,{\left (a^{4} d - 3 \, a^{3} b d + 3 \, a^{2} b^{2} d - a b^{3} d\right )} \sqrt{-a^{2} + a b}} + \frac{2 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - b^{3} e^{\left (2 \, d x + 2 \, c\right )} + b^{3}}{{\left (a^{4} d - 3 \, a^{3} b d + 3 \, a^{2} b^{2} d - a b^{3} d\right )}{\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}} + \frac{4 \,{\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 9 \, b e^{\left (2 \, d x + 2 \, c\right )} - a + 4 \, b\right )}}{3 \,{\left (a^{3} d - 3 \, a^{2} b d + 3 \, a b^{2} d - b^{3} d\right )}{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")
[Out]
1/2*(6*a*b^2 - b^3)*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/((a^4*d - 3*a^3*b*d + 3*a^2*b^2
*d - a*b^3*d)*sqrt(-a^2 + a*b)) + (2*a*b^2*e^(2*d*x + 2*c) - b^3*e^(2*d*x + 2*c) + b^3)/((a^4*d - 3*a^3*b*d +
3*a^2*b^2*d - a*b^3*d)*(b*e^(4*d*x + 4*c) + 4*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + b)) + 4/3*(3*b*e^(4*d*
x + 4*c) - 3*a*e^(2*d*x + 2*c) + 9*b*e^(2*d*x + 2*c) - a + 4*b)/((a^3*d - 3*a^2*b*d + 3*a*b^2*d - b^3*d)*(e^(2
*d*x + 2*c) + 1)^3)