3.4.46 \(\int \frac {\text {sech}^2(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\) [346]
Optimal. Leaf size=172 \[ -\frac {3 b \left (8 a^2-4 a b+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a-b)^{7/2} d}+\frac {\tanh (c+d x)}{(a-b)^3 d}-\frac {b^3 \tanh (c+d x)}{4 a (a-b)^3 d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {3 (4 a-b) b^2 \tanh (c+d x)}{8 a^2 (a-b)^3 d \left (a-(a-b) \tanh ^2(c+d x)\right )} \]
[Out]
-3/8*b*(8*a^2-4*a*b+b^2)*arctanh((a-b)^(1/2)*tanh(d*x+c)/a^(1/2))/a^(5/2)/(a-b)^(7/2)/d+tanh(d*x+c)/(a-b)^3/d-
1/4*b^3*tanh(d*x+c)/a/(a-b)^3/d/(a-(a-b)*tanh(d*x+c)^2)^2+3/8*(4*a-b)*b^2*tanh(d*x+c)/a^2/(a-b)^3/d/(a-(a-b)*t
anh(d*x+c)^2)
________________________________________________________________________________________
Rubi [A]
time = 0.20, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3270, 398,
1171, 393, 214} \begin {gather*} \frac {3 b^2 (4 a-b) \tanh (c+d x)}{8 a^2 d (a-b)^3 \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac {3 b \left (8 a^2-4 a b+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} d (a-b)^{7/2}}-\frac {b^3 \tanh (c+d x)}{4 a d (a-b)^3 \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {\tanh (c+d x)}{d (a-b)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sech[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^3,x]
[Out]
(-3*b*(8*a^2 - 4*a*b + b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^(7/2)*d) + Tanh[c
+ d*x]/((a - b)^3*d) - (b^3*Tanh[c + d*x])/(4*a*(a - b)^3*d*(a - (a - b)*Tanh[c + d*x]^2)^2) + (3*(4*a - b)*b
^2*Tanh[c + d*x])/(8*a^2*(a - b)^3*d*(a - (a - b)*Tanh[c + d*x]^2))
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 393
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 398
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 1171
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Rule 3270
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]
Rubi steps
\begin {align*} \int \frac {\text {sech}^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{\left (a-(a-b) x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{(a-b)^3}-\frac {b \left (3 a^2-3 a b+b^2\right )-3 (a-b) (2 a-b) b x^2+3 (a-b)^2 b x^4}{(a-b)^3 \left (a+(-a+b) x^2\right )^3}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\tanh (c+d x)}{(a-b)^3 d}-\frac {\text {Subst}\left (\int \frac {b \left (3 a^2-3 a b+b^2\right )-3 (a-b) (2 a-b) b x^2+3 (a-b)^2 b x^4}{\left (a+(-a+b) x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{(a-b)^3 d}\\ &=\frac {\tanh (c+d x)}{(a-b)^3 d}-\frac {b^3 \tanh (c+d x)}{4 a (a-b)^3 d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {-3 (2 a-b)^2 b+12 a (a-b) b x^2}{\left (a+(-a+b) x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a (a-b)^3 d}\\ &=\frac {\tanh (c+d x)}{(a-b)^3 d}-\frac {b^3 \tanh (c+d x)}{4 a (a-b)^3 d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {3 (4 a-b) b^2 \tanh (c+d x)}{8 a^2 (a-b)^3 d \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac {\left (3 b \left (8 a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a-b)^3 d}\\ &=-\frac {3 b \left (8 a^2-4 a b+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a-b)^{7/2} d}+\frac {\tanh (c+d x)}{(a-b)^3 d}-\frac {b^3 \tanh (c+d x)}{4 a (a-b)^3 d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {3 (4 a-b) b^2 \tanh (c+d x)}{8 a^2 (a-b)^3 d \left (a-(a-b) \tanh ^2(c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.01, size = 165, normalized size = 0.96 \begin {gather*} \frac {-\frac {3 b \left (8 a^2-4 a b+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{5/2} (a-b)^{7/2}}+\frac {4 b^2 \sinh (2 (c+d x))}{a (a-b)^2 (2 a-b+b \cosh (2 (c+d x)))^2}+\frac {(10 a-3 b) b^2 \sinh (2 (c+d x))}{a^2 (a-b)^3 (2 a-b+b \cosh (2 (c+d x)))}+\frac {8 \tanh (c+d x)}{(a-b)^3}}{8 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sech[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^3,x]
[Out]
((-3*b*(8*a^2 - 4*a*b + b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(5/2)*(a - b)^(7/2)) + (4*b^2*Si
nh[2*(c + d*x)])/(a*(a - b)^2*(2*a - b + b*Cosh[2*(c + d*x)])^2) + ((10*a - 3*b)*b^2*Sinh[2*(c + d*x)])/(a^2*(
a - b)^3*(2*a - b + b*Cosh[2*(c + d*x)])) + (8*Tanh[c + d*x])/(a - b)^3)/(8*d)
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(393\) vs.
\(2(158)=316\).
time = 1.69, size = 394, normalized size = 2.29 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sech(d*x+c)^2/(a+b*sinh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
[Out]
1/d*(2*b/(a-b)^3*((1/8*b*(12*a-5*b)/a*tanh(1/2*d*x+1/2*c)^7-3/8*(4*a^2-15*a*b+4*b^2)/a^2*b*tanh(1/2*d*x+1/2*c)
^5-3/8*(4*a^2-15*a*b+4*b^2)/a^2*b*tanh(1/2*d*x+1/2*c)^3+1/8*b*(12*a-5*b)/a*tanh(1/2*d*x+1/2*c))/(a*tanh(1/2*d*
x+1/2*c)^4-2*a*tanh(1/2*d*x+1/2*c)^2+4*b*tanh(1/2*d*x+1/2*c)^2+a)^2+3/8/a*(8*a^2-4*a*b+b^2)*(-1/2*((-b*(a-b))^
(1/2)-b)/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*((-b*(a-b))^(1/2)+b)/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/(a-b)^3*tanh(1/2*d*x+1/2*c)/(tanh(1/2*d*x+1/2
*c)^2+1))
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^2/(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4593 vs.
\(2 (160) = 320\).
time = 0.49, size = 9442, normalized size = 54.90 \begin {gather*} \text {too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^2/(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")
[Out]
[-1/16*(12*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(d*x + c)^8 + 96*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b
^4 - a*b^5)*cosh(d*x + c)*sinh(d*x + c)^7 + 12*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*sinh(d*x + c)^8 +
24*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^3 - 8*a^2*b^4 + a*b^5)*cosh(d*x + c)^6 + 24*(24*a^5*b - 44*a^4*b^2 + 27*a
^3*b^3 - 8*a^2*b^4 + a*b^5 + 14*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^6
+ 32*a^4*b^2 + 8*a^3*b^3 - 52*a^2*b^4 + 12*a*b^5 + 48*(14*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(d*
x + c)^3 + 3*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^3 - 8*a^2*b^4 + a*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 + 8*(64*a
^6 - 88*a^5*b + 28*a^4*b^2 - 3*a^3*b^3 - a^2*b^4)*cosh(d*x + c)^4 + 8*(64*a^6 - 88*a^5*b + 28*a^4*b^2 - 3*a^3*
b^3 - a^2*b^4 + 105*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(d*x + c)^4 + 45*(24*a^5*b - 44*a^4*b^2 +
27*a^3*b^3 - 8*a^2*b^4 + a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 32*(21*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4
- a*b^5)*cosh(d*x + c)^5 + 15*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^3 - 8*a^2*b^4 + a*b^5)*cosh(d*x + c)^3 + (64*
a^6 - 88*a^5*b + 28*a^4*b^2 - 3*a^3*b^3 - a^2*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 8*(32*a^5*b - 16*a^4*b^2 -
37*a^3*b^3 + 24*a^2*b^4 - 3*a*b^5)*cosh(d*x + c)^2 + 8*(42*(8*a^4*b^2 - 12*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cosh(
d*x + c)^6 + 32*a^5*b - 16*a^4*b^2 - 37*a^3*b^3 + 24*a^2*b^4 - 3*a*b^5 + 45*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^
3 - 8*a^2*b^4 + a*b^5)*cosh(d*x + c)^4 + 6*(64*a^6 - 88*a^5*b + 28*a^4*b^2 - 3*a^3*b^3 - a^2*b^4)*cosh(d*x + c
)^2)*sinh(d*x + c)^2 + 3*((8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^10 + 10*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d
*x + c)*sinh(d*x + c)^9 + (8*a^2*b^3 - 4*a*b^4 + b^5)*sinh(d*x + c)^10 + (64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 -
3*b^5)*cosh(d*x + c)^8 + (64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5 + 45*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*
x + c)^2)*sinh(d*x + c)^8 + 8*(15*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^3 + (64*a^3*b^2 - 56*a^2*b^3 + 20*
a*b^4 - 3*b^5)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*
x + c)^6 + 2*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5 + 105*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x +
c)^4 + 14*(64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(8*a^2*b^3 - 4
*a*b^4 + b^5)*cosh(d*x + c)^5 + 14*(64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x + c)^3 + 3*(64*a^4*b
- 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c))*sinh(d*x + c)^5 + 8*a^2*b^3 - 4*a*b^4 + b^5 + 2*(64*
a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^4 + 2*(105*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x
+ c)^6 + 64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5 + 35*(64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)
*cosh(d*x + c)^4 + 15*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 +
8*(15*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^7 + 7*(64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x +
c)^5 + 5*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^3 + (64*a^4*b - 64*a^3*b^2 + 32*a^
2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + (64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x
+ c)^2 + (45*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^8 + 28*(64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cos
h(d*x + c)^6 + 64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5 + 30*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 +
b^5)*cosh(d*x + c)^4 + 12*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^2)*sinh(d*x + c)
^2 + 2*(5*(8*a^2*b^3 - 4*a*b^4 + b^5)*cosh(d*x + c)^9 + 4*(64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*
x + c)^7 + 6*(64*a^4*b - 64*a^3*b^2 + 32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^5 + 4*(64*a^4*b - 64*a^3*b^2 +
32*a^2*b^3 - 8*a*b^4 + b^5)*cosh(d*x + c)^3 + (64*a^3*b^2 - 56*a^2*b^3 + 20*a*b^4 - 3*b^5)*cosh(d*x + c))*sin
h(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*(8*a^4*b^2 - 12*a^3*b^3
+ 5*a^2*b^4 - a*b^5)*cosh(d*x + c)^7 + 9*(24*a^5*b - 44*a^4*b^2 + 27*a^3*b^3 - 8*a^2*b^4 + a*b^5)*cosh(d*x +
c)^5 + 2*(64*a^6 - 88*a^5*b + 28*a^4*b^2 - 3*a^3*b^3 - a^2*b^4)*cosh(d*x + c)^3 + (32*a^5*b - 16*a^4*b^2 - 37*
a^3*b^3 + 24*a^2*b^4 - 3*a*b^5)*cosh(d*x + c))*sinh(d*x + c))/((a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 +
a^3*b^6)*d*cosh(d*x + c)^10 + 10*(a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^4*b^5 + a^3*b^6)*d*cosh(d*x + c)*sinh(
d*x + c)^9 + (a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 -...
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)**2/(a+b*sinh(d*x+c)**2)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 367 vs.
\(2 (160) = 320\).
time = 1.33, size = 367, normalized size = 2.13 \begin {gather*} -\frac {\frac {3 \, {\left (8 \, a^{2} b - 4 \, 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 )}{{\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )} \sqrt {-a^{2} + a b}} + \frac {2 \, {\left (16 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 12 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 3 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 80 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} - 104 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 54 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 9 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 64 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 52 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 10 \, a b^{3} - 3 \, b^{4}\right )}}{{\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\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 )}^{2}} + \frac {16}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^2/(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")
[Out]
-1/8*(3*(8*a^2*b - 4*a*b^2 + b^3)*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/((a^5 - 3*a^4*b +
3*a^3*b^2 - a^2*b^3)*sqrt(-a^2 + a*b)) + 2*(16*a^2*b^2*e^(6*d*x + 6*c) - 12*a*b^3*e^(6*d*x + 6*c) + 3*b^4*e^(
6*d*x + 6*c) + 80*a^3*b*e^(4*d*x + 4*c) - 104*a^2*b^2*e^(4*d*x + 4*c) + 54*a*b^3*e^(4*d*x + 4*c) - 9*b^4*e^(4*
d*x + 4*c) + 64*a^2*b^2*e^(2*d*x + 2*c) - 52*a*b^3*e^(2*d*x + 2*c) + 9*b^4*e^(2*d*x + 2*c) + 10*a*b^3 - 3*b^4)
/((a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*(b*e^(4*d*x + 4*c) + 4*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + b)^2)
+ 16/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*(e^(2*d*x + 2*c) + 1)))/d
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(cosh(c + d*x)^2*(a + b*sinh(c + d*x)^2)^3),x)
[Out]
int(1/(cosh(c + d*x)^2*(a + b*sinh(c + d*x)^2)^3), x)
________________________________________________________________________________________