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3.1.43 \(\int \frac {\sinh ^2(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [43]

Optimal. Leaf size=187 \[ -\frac {(a+6 b) x}{2 a^4}+\frac {\sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^4 (a+b)^{3/2} d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {3 b \tanh (c+d x)}{4 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {b (11 a+12 b) \tanh (c+d x)}{8 a^3 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )} \]

[Out]

-1/2*(a+6*b)*x/a^4+1/8*(15*a^2+40*a*b+24*b^2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))*b^(1/2)/a^4/(a+b)^(3/2)
/d+1/2*cosh(d*x+c)*sinh(d*x+c)/a/d/(a+b-b*tanh(d*x+c)^2)^2+3/4*b*tanh(d*x+c)/a^2/d/(a+b-b*tanh(d*x+c)^2)^2+1/8
*b*(11*a+12*b)*tanh(d*x+c)/a^3/(a+b)/d/(a+b-b*tanh(d*x+c)^2)

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Rubi [A]
time = 0.21, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4217, 482, 541, 536, 212, 214} \begin {gather*} -\frac {x (a+6 b)}{2 a^4}+\frac {b (11 a+12 b) \tanh (c+d x)}{8 a^3 d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}+\frac {3 b \tanh (c+d x)}{4 a^2 d \left (a-b \tanh ^2(c+d x)+b\right )^2}+\frac {\sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^4 d (a+b)^{3/2}}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 a d \left (a-b \tanh ^2(c+d x)+b\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((a + 6*b)*x)/a^4 + (Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*
a^4*(a + b)^(3/2)*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*a*d*(a + b - b*Tanh[c + d*x]^2)^2) + (3*b*Tanh[c + d*x
])/(4*a^2*d*(a + b - b*Tanh[c + d*x]^2)^2) + (b*(11*a + 12*b)*Tanh[c + d*x])/(8*a^3*(a + b)*d*(a + b - b*Tanh[
c + d*x]^2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
 /; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 482

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
 a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

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 541

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 4217

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_), x_Symbol] :> With[{ff = Fr
eeFactors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[x^m*(ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p/(1
 + ff^2*x^2)^(m/2 + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && Integer
Q[n/2]

Rubi steps

\begin {align*} \int \frac {\sinh ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {x^2}{\left (1-x^2\right )^2 \left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {a+b+5 b x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{2 a d}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {3 b \tanh (c+d x)}{4 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {-2 (a+b) (2 a+3 b)-18 b (a+b) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a+b) d}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {3 b \tanh (c+d x)}{4 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {b (11 a+12 b) \tanh (c+d x)}{8 a^3 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {2 (a+b) \left (4 a^2+17 a b+12 b^2\right )+2 b (a+b) (11 a+12 b) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{16 a^3 (a+b)^2 d}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {3 b \tanh (c+d x)}{4 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {b (11 a+12 b) \tanh (c+d x)}{8 a^3 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {(a+6 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 a^4 d}+\frac {\left (b \left (15 a^2+40 a b+24 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^4 (a+b) d}\\ &=-\frac {(a+6 b) x}{2 a^4}+\frac {\sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^4 (a+b)^{3/2} d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {3 b \tanh (c+d x)}{4 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {b (11 a+12 b) \tanh (c+d x)}{8 a^3 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2544\) vs. \(2(187)=374\).
time = 15.67, size = 2544, normalized size = 13.60 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-5*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*(((3*a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x]
)/Sqrt[a + b]])/(a + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cosh[2*(c + d*x)])*Sinh[2*
(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(8192*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3) - ((a +
 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-3*a*(a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(
a + b)^(5/2) + (Sqrt[b]*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a*b + 4*b^2)*Cosh[2*(c + d*x)])*S
inh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(2048*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3) +
 ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-2*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*
b^4 + 256*b^5)*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])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (
Sech[2*c]*(256*b^2*(a + b)^2*(3*a^2 + 8*a*b + 8*b^2)*d*x*Cosh[2*c] + 512*a*b^2*(a + b)^2*(a + 2*b)*d*x*Cosh[2*
d*x] + 128*a^4*b^2*d*x*Cosh[2*(c + 2*d*x)] + 256*a^3*b^3*d*x*Cosh[2*(c + 2*d*x)] + 128*a^2*b^4*d*x*Cosh[2*(c +
 2*d*x)] + 512*a^4*b^2*d*x*Cosh[4*c + 2*d*x] + 2048*a^3*b^3*d*x*Cosh[4*c + 2*d*x] + 2560*a^2*b^4*d*x*Cosh[4*c
+ 2*d*x] + 1024*a*b^5*d*x*Cosh[4*c + 2*d*x] + 128*a^4*b^2*d*x*Cosh[6*c + 4*d*x] + 256*a^3*b^3*d*x*Cosh[6*c + 4
*d*x] + 128*a^2*b^4*d*x*Cosh[6*c + 4*d*x] - 9*a^6*Sinh[2*c] + 12*a^5*b*Sinh[2*c] + 684*a^4*b^2*Sinh[2*c] + 288
0*a^3*b^3*Sinh[2*c] + 5280*a^2*b^4*Sinh[2*c] + 4608*a*b^5*Sinh[2*c] + 1536*b^6*Sinh[2*c] + 9*a^6*Sinh[2*d*x] -
 14*a^5*b*Sinh[2*d*x] - 608*a^4*b^2*Sinh[2*d*x] - 2112*a^3*b^3*Sinh[2*d*x] - 2560*a^2*b^4*Sinh[2*d*x] - 1024*a
*b^5*Sinh[2*d*x] + 3*a^6*Sinh[2*(c + 2*d*x)] - 12*a^5*b*Sinh[2*(c + 2*d*x)] - 204*a^4*b^2*Sinh[2*(c + 2*d*x)]
- 384*a^3*b^3*Sinh[2*(c + 2*d*x)] - 192*a^2*b^4*Sinh[2*(c + 2*d*x)] - 3*a^6*Sinh[4*c + 2*d*x] + 10*a^5*b*Sinh[
4*c + 2*d*x] + 304*a^4*b^2*Sinh[4*c + 2*d*x] + 1056*a^3*b^3*Sinh[4*c + 2*d*x] + 1280*a^2*b^4*Sinh[4*c + 2*d*x]
 + 512*a*b^5*Sinh[4*c + 2*d*x]))/(a + 2*b + a*Cosh[2*(c + d*x)])^2))/(4096*a^3*b^2*(a + b)^2*d*(a + b*Sech[c +
 d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((6*(a^6 - 8*a^5*b + 120*a^4*b^2 + 1280*a^3*b
^3 + 3200*a^2*b^4 + 3072*a*b^5 + 1024*b^6)*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])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b
*(Cosh[c] - Sinh[c])^4]) + (Sech[2*c]*(-1536*b^2*(a + b)^2*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3)*d*x*Cosh[2*c
] - 3072*a*b^2*(a^2 + 3*a*b + 2*b^2)^2*d*x*Cosh[2*d*x] - 768*a^5*b^2*d*x*Cosh[2*(c + 2*d*x)] - 3072*a^4*b^3*d*
x*Cosh[2*(c + 2*d*x)] - 3840*a^3*b^4*d*x*Cosh[2*(c + 2*d*x)] - 1536*a^2*b^5*d*x*Cosh[2*(c + 2*d*x)] - 3072*a^5
*b^2*d*x*Cosh[4*c + 2*d*x] - 18432*a^4*b^3*d*x*Cosh[4*c + 2*d*x] - 39936*a^3*b^4*d*x*Cosh[4*c + 2*d*x] - 36864
*a^2*b^5*d*x*Cosh[4*c + 2*d*x] - 12288*a*b^6*d*x*Cosh[4*c + 2*d*x] - 768*a^5*b^2*d*x*Cosh[6*c + 4*d*x] - 3072*
a^4*b^3*d*x*Cosh[6*c + 4*d*x] - 3840*a^3*b^4*d*x*Cosh[6*c + 4*d*x] - 1536*a^2*b^5*d*x*Cosh[6*c + 4*d*x] + 9*a^
7*Sinh[2*c] - 54*a^6*b*Sinh[2*c] - 2392*a^5*b^2*Sinh[2*c] - 13968*a^4*b^3*Sinh[2*c] - 36480*a^3*b^4*Sinh[2*c]
- 50432*a^2*b^5*Sinh[2*c] - 35840*a*b^6*Sinh[2*c] - 10240*b^7*Sinh[2*c] - 9*a^7*Sinh[2*d*x] + 56*a^6*b*Sinh[2*
d*x] + 2552*a^5*b^2*Sinh[2*d*x] + 13184*a^4*b^3*Sinh[2*d*x] + 27072*a^3*b^4*Sinh[2*d*x] + 24576*a^2*b^5*Sinh[2
*d*x] + 8192*a*b^6*Sinh[2*d*x] - 3*a^7*Sinh[2*(c + 2*d*x)] + 26*a^6*b*Sinh[2*(c + 2*d*x)] + 992*a^5*b^2*Sinh[2
*(c + 2*d*x)] + 3648*a^4*b^3*Sinh[2*(c + 2*d*x)] + 4480*a^3*b^4*Sinh[2*(c + 2*d*x)] + 1792*a^2*b^5*Sinh[2*(c +
 2*d*x)] + 3*a^7*Sinh[4*c + 2*d*x] - 24*a^6*b*Sinh[4*c + 2*d*x] - 600*a^5*b^2*Sinh[4*c + 2*d*x] - 3200*a^4*b^3
*Sinh[4*c + 2*d*x] - 6720*a^3*b^4*Sinh[4*c + 2*d*x] - 6144*a^2*b^5*Sinh[4*c + 2*d*x] - 2048*a*b^6*Sinh[4*c + 2
*d*x] + 256*a^5*b^2*Sinh[6*c + 4*d*x] + 1024*a^4*b^3*Sinh[6*c + 4*d*x] + 1280*a^3*b^4*Sinh[6*c + 4*d*x] + 512*
a^2*b^5*Sinh[6*c + 4*d*x] + 64*a^5*b^2*Sinh[4*c + 6*d*x] + 128*a^4*b^3*Sinh[4*c + 6*d*x] + 64*a^3*b^4*Sinh[4*c
 + 6*d*x] + 64*a^5*b^2*Sinh[8*c + 6*d*x] + 128*a^4*b^3*Sinh[8*c + 6*d*x] + 64*a^3*b^4*Sinh[8*c + 6*d*x]))/(a +
 2*b + a*Cosh[2*(c + d*x)])^2))/(16384*a^4*b^2*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c
 + 2*d*x])^3*Sech[c + d*x]^6*((6*a^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])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh
[c] - Sinh[c])^4]) + (a*Sech[2*c]*((-9*a^4 - 16*a^3*b + 48*a^2*b^2 + 128*a*b^3 + 64*b^4)*Sinh[2*d*x] + a*(-3*a
^3 + 2*a^2*b + 24*a*b^2 + 16*b^3)*Sinh[2*(c + 2*d*x)] + (3*a^4 - 64*a^2*b^2 - 128*a*b^3 - 64*b^4)*Sinh[4*c + 2
*d*x]) + (9*a^5 + 18*a^4*b - 64*a^3*b^2 - 256*a^2*b^3 - 320*a*b^4 - 128*b^5)*Tanh[2*c])/(a^2*(a + 2*b + a*Cosh
[2*(c + d*x)])^2)))/(4096*b^2*(a + b)^2*d*(a + ...

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(420\) vs. \(2(169)=338\).
time = 2.35, size = 421, normalized size = 2.25 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2/a^3/(tanh(1/2*d*x+1/2*c)+1)^2+1/2/a^3/(tanh(1/2*d*x+1/2*c)+1)+1/2/a^4*(-a-6*b)*ln(tanh(1/2*d*x+1/2*c
)+1)-2*b/a^4*(((-9/8*a^2-a*b)*tanh(1/2*d*x+1/2*c)^7-1/8*a*(27*a^2+23*a*b-8*b^2)/(a+b)*tanh(1/2*d*x+1/2*c)^5-1/
8*a*(27*a^2+23*a*b-8*b^2)/(a+b)*tanh(1/2*d*x+1/2*c)^3+(-9/8*a^2-a*b)*tanh(1/2*d*x+1/2*c))/(a*tanh(1/2*d*x+1/2*
c)^4+b*tanh(1/2*d*x+1/2*c)^4+2*a*tanh(1/2*d*x+1/2*c)^2-2*b*tanh(1/2*d*x+1/2*c)^2+a+b)^2+1/8*(15*a^2+40*a*b+24*
b^2)/(a+b)*(-1/4/b^(1/2)/(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))+1/4/b^(1/2)/(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/a^3/(tanh(1/2*d*x+1/2*c)-1)^2+1/2/a^3/(tanh(1/2*d*x+1/2*c)-1)+1/2*(a+6*b)/a^4*ln(tanh(1/2*d*x+1/2*c)-1)
)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1373 vs. \(2 (178) = 356\).
time = 0.58, size = 1373, normalized size = 7.34 \begin {gather*} \frac {3 \, {\left (5 \, a^{3} b + 30 \, a^{2} b^{2} + 40 \, a b^{3} + 16 \, b^{4}\right )} \log \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{64 \, {\left (a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {3 \, {\left (5 \, a^{3} b + 30 \, a^{2} b^{2} + 40 \, a b^{3} + 16 \, b^{4}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{64 \, {\left (a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {{\left (15 \, a^{2} b + 20 \, a b^{2} + 8 \, b^{3}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{32 \, {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {9 \, a^{4} b + 32 \, a^{3} b^{2} + 20 \, a^{2} b^{3} + 3 \, {\left (3 \, a^{4} b + 34 \, a^{3} b^{2} + 64 \, a^{2} b^{3} + 32 \, a b^{4}\right )} e^{\left (6 \, d x + 6 \, c\right )} + {\left (27 \, a^{4} b + 264 \, a^{3} b^{2} + 740 \, a^{2} b^{3} + 832 \, a b^{4} + 320 \, b^{5}\right )} e^{\left (4 \, d x + 4 \, c\right )} + {\left (27 \, a^{4} b + 194 \, a^{3} b^{2} + 336 \, a^{2} b^{3} + 160 \, a b^{4}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{16 \, {\left (a^{8} + 2 \, a^{7} b + a^{6} b^{2} + {\left (a^{8} + 2 \, a^{7} b + a^{6} b^{2}\right )} e^{\left (8 \, d x + 8 \, c\right )} + 4 \, {\left (a^{8} + 4 \, a^{7} b + 5 \, a^{6} b^{2} + 2 \, a^{5} b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )} + 2 \, {\left (3 \, a^{8} + 14 \, a^{7} b + 27 \, a^{6} b^{2} + 24 \, a^{5} b^{3} + 8 \, a^{4} b^{4}\right )} e^{\left (4 \, d x + 4 \, c\right )} + 4 \, {\left (a^{8} + 4 \, a^{7} b + 5 \, a^{6} b^{2} + 2 \, a^{5} b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} d} + \frac {9 \, a^{4} b + 32 \, a^{3} b^{2} + 20 \, a^{2} b^{3} + {\left (27 \, a^{4} b + 194 \, a^{3} b^{2} + 336 \, a^{2} b^{3} + 160 \, a b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (27 \, a^{4} b + 264 \, a^{3} b^{2} + 740 \, a^{2} b^{3} + 832 \, a b^{4} + 320 \, b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, {\left (3 \, a^{4} b + 34 \, a^{3} b^{2} + 64 \, a^{2} b^{3} + 32 \, a b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{16 \, {\left (a^{8} + 2 \, a^{7} b + a^{6} b^{2} + 4 \, {\left (a^{8} + 4 \, a^{7} b + 5 \, a^{6} b^{2} + 2 \, a^{5} b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{8} + 14 \, a^{7} b + 27 \, a^{6} b^{2} + 24 \, a^{5} b^{3} + 8 \, a^{4} b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{8} + 4 \, a^{7} b + 5 \, a^{6} b^{2} + 2 \, a^{5} b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{8} + 2 \, a^{7} b + a^{6} b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} + \frac {9 \, a^{3} b + 6 \, a^{2} b^{2} + {\left (27 \, a^{3} b + 68 \, a^{2} b^{2} + 32 \, a b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (9 \, a^{3} b + 30 \, a^{2} b^{2} + 40 \, a b^{3} + 16 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (9 \, a^{3} b + 28 \, a^{2} b^{2} + 16 \, a b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{8 \, {\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2} + 4 \, {\left (a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} + 2 \, a^{4} b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{7} + 14 \, a^{6} b + 27 \, a^{5} b^{2} + 24 \, a^{4} b^{3} + 8 \, a^{3} b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} + 2 \, a^{4} b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} - \frac {d x + c}{2 \, a^{3} d} + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{8 \, a^{3} d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, a^{3} d} - \frac {3 \, b \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a + 2 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{4 \, a^{4} d} + \frac {3 \, b \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{4 \, a^{4} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

3/64*(5*a^3*b + 30*a^2*b^2 + 40*a*b^3 + 16*b^4)*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2*
d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^6 + 2*a^5*b + a^4*b^2)*sqrt((a + b)*b)*d) - 3/64*(5*a^3*b + 30*
a^2*b^2 + 40*a*b^3 + 16*b^4)*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a +
2*b + 2*sqrt((a + b)*b)))/((a^6 + 2*a^5*b + a^4*b^2)*sqrt((a + b)*b)*d) - 1/32*(15*a^2*b + 20*a*b^2 + 8*b^3)*l
og((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^
5 + 2*a^4*b + a^3*b^2)*sqrt((a + b)*b)*d) - 1/16*(9*a^4*b + 32*a^3*b^2 + 20*a^2*b^3 + 3*(3*a^4*b + 34*a^3*b^2
+ 64*a^2*b^3 + 32*a*b^4)*e^(6*d*x + 6*c) + (27*a^4*b + 264*a^3*b^2 + 740*a^2*b^3 + 832*a*b^4 + 320*b^5)*e^(4*d
*x + 4*c) + (27*a^4*b + 194*a^3*b^2 + 336*a^2*b^3 + 160*a*b^4)*e^(2*d*x + 2*c))/((a^8 + 2*a^7*b + a^6*b^2 + (a
^8 + 2*a^7*b + a^6*b^2)*e^(8*d*x + 8*c) + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*e^(6*d*x + 6*c) + 2*(3*a^8
 + 14*a^7*b + 27*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*e^(4*d*x + 4*c) + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)
*e^(2*d*x + 2*c))*d) + 1/16*(9*a^4*b + 32*a^3*b^2 + 20*a^2*b^3 + (27*a^4*b + 194*a^3*b^2 + 336*a^2*b^3 + 160*a
*b^4)*e^(-2*d*x - 2*c) + (27*a^4*b + 264*a^3*b^2 + 740*a^2*b^3 + 832*a*b^4 + 320*b^5)*e^(-4*d*x - 4*c) + 3*(3*
a^4*b + 34*a^3*b^2 + 64*a^2*b^3 + 32*a*b^4)*e^(-6*d*x - 6*c))/((a^8 + 2*a^7*b + a^6*b^2 + 4*(a^8 + 4*a^7*b + 5
*a^6*b^2 + 2*a^5*b^3)*e^(-2*d*x - 2*c) + 2*(3*a^8 + 14*a^7*b + 27*a^6*b^2 + 24*a^5*b^3 + 8*a^4*b^4)*e^(-4*d*x
- 4*c) + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 + 2*a^5*b^3)*e^(-6*d*x - 6*c) + (a^8 + 2*a^7*b + a^6*b^2)*e^(-8*d*x - 8*
c))*d) + 1/8*(9*a^3*b + 6*a^2*b^2 + (27*a^3*b + 68*a^2*b^2 + 32*a*b^3)*e^(-2*d*x - 2*c) + 3*(9*a^3*b + 30*a^2*
b^2 + 40*a*b^3 + 16*b^4)*e^(-4*d*x - 4*c) + (9*a^3*b + 28*a^2*b^2 + 16*a*b^3)*e^(-6*d*x - 6*c))/((a^7 + 2*a^6*
b + a^5*b^2 + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*e^(-2*d*x - 2*c) + 2*(3*a^7 + 14*a^6*b + 27*a^5*b^2 +
24*a^4*b^3 + 8*a^3*b^4)*e^(-4*d*x - 4*c) + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*e^(-6*d*x - 6*c) + (a^7 +
 2*a^6*b + a^5*b^2)*e^(-8*d*x - 8*c))*d) - 1/2*(d*x + c)/(a^3*d) + 1/8*e^(2*d*x + 2*c)/(a^3*d) - 1/8*e^(-2*d*x
 - 2*c)/(a^3*d) - 3/4*b*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/(a^4*d) + 3/4*b*log(2*(a + 2*
b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(a^4*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4724 vs. \(2 (178) = 356\).
time = 0.49, size = 9730, normalized size = 52.03 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(2*(a^4 + a^3*b)*cosh(d*x + c)^12 + 24*(a^4 + a^3*b)*cosh(d*x + c)*sinh(d*x + c)^11 + 2*(a^4 + a^3*b)*si
nh(d*x + c)^12 + 8*(a^4 + 3*a^3*b + 2*a^2*b^2 - (a^4 + 7*a^3*b + 6*a^2*b^2)*d*x)*cosh(d*x + c)^10 + 4*(2*a^4 +
 6*a^3*b + 4*a^2*b^2 - 2*(a^4 + 7*a^3*b + 6*a^2*b^2)*d*x + 33*(a^4 + a^3*b)*cosh(d*x + c)^2)*sinh(d*x + c)^10
+ 40*(11*(a^4 + a^3*b)*cosh(d*x + c)^3 + 2*(a^4 + 3*a^3*b + 2*a^2*b^2 - (a^4 + 7*a^3*b + 6*a^2*b^2)*d*x)*cosh(
d*x + c))*sinh(d*x + c)^9 + 2*(5*a^4 + 3*a^3*b - 32*a^2*b^2 - 32*a*b^3 - 16*(a^4 + 9*a^3*b + 20*a^2*b^2 + 12*a
*b^3)*d*x)*cosh(d*x + c)^8 + 2*(495*(a^4 + a^3*b)*cosh(d*x + c)^4 + 5*a^4 + 3*a^3*b - 32*a^2*b^2 - 32*a*b^3 -
16*(a^4 + 9*a^3*b + 20*a^2*b^2 + 12*a*b^3)*d*x + 180*(a^4 + 3*a^3*b + 2*a^2*b^2 - (a^4 + 7*a^3*b + 6*a^2*b^2)*
d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 16*(99*(a^4 + a^3*b)*cosh(d*x + c)^5 + 60*(a^4 + 3*a^3*b + 2*a^2*b^2 -
 (a^4 + 7*a^3*b + 6*a^2*b^2)*d*x)*cosh(d*x + c)^3 + (5*a^4 + 3*a^3*b - 32*a^2*b^2 - 32*a*b^3 - 16*(a^4 + 9*a^3
*b + 20*a^2*b^2 + 12*a*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^7 - 4*(27*a^3*b + 102*a^2*b^2 + 152*a*b^3 + 80*b
^4 + 4*(3*a^4 + 29*a^3*b + 82*a^2*b^2 + 104*a*b^3 + 48*b^4)*d*x)*cosh(d*x + c)^6 + 4*(462*(a^4 + a^3*b)*cosh(d
*x + c)^6 + 420*(a^4 + 3*a^3*b + 2*a^2*b^2 - (a^4 + 7*a^3*b + 6*a^2*b^2)*d*x)*cosh(d*x + c)^4 - 27*a^3*b - 102
*a^2*b^2 - 152*a*b^3 - 80*b^4 - 4*(3*a^4 + 29*a^3*b + 82*a^2*b^2 + 104*a*b^3 + 48*b^4)*d*x + 14*(5*a^4 + 3*a^3
*b - 32*a^2*b^2 - 32*a*b^3 - 16*(a^4 + 9*a^3*b + 20*a^2*b^2 + 12*a*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^6
+ 8*(198*(a^4 + a^3*b)*cosh(d*x + c)^7 + 252*(a^4 + 3*a^3*b + 2*a^2*b^2 - (a^4 + 7*a^3*b + 6*a^2*b^2)*d*x)*cos
h(d*x + c)^5 + 14*(5*a^4 + 3*a^3*b - 32*a^2*b^2 - 32*a*b^3 - 16*(a^4 + 9*a^3*b + 20*a^2*b^2 + 12*a*b^3)*d*x)*c
osh(d*x + c)^3 - 3*(27*a^3*b + 102*a^2*b^2 + 152*a*b^3 + 80*b^4 + 4*(3*a^4 + 29*a^3*b + 82*a^2*b^2 + 104*a*b^3
 + 48*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(5*a^4 + 75*a^3*b + 192*a^2*b^2 + 128*a*b^3 + 16*(a^4 + 9*a
^3*b + 20*a^2*b^2 + 12*a*b^3)*d*x)*cosh(d*x + c)^4 + 2*(495*(a^4 + a^3*b)*cosh(d*x + c)^8 + 840*(a^4 + 3*a^3*b
 + 2*a^2*b^2 - (a^4 + 7*a^3*b + 6*a^2*b^2)*d*x)*cosh(d*x + c)^6 + 70*(5*a^4 + 3*a^3*b - 32*a^2*b^2 - 32*a*b^3
- 16*(a^4 + 9*a^3*b + 20*a^2*b^2 + 12*a*b^3)*d*x)*cosh(d*x + c)^4 - 5*a^4 - 75*a^3*b - 192*a^2*b^2 - 128*a*b^3
 - 16*(a^4 + 9*a^3*b + 20*a^2*b^2 + 12*a*b^3)*d*x - 30*(27*a^3*b + 102*a^2*b^2 + 152*a*b^3 + 80*b^4 + 4*(3*a^4
 + 29*a^3*b + 82*a^2*b^2 + 104*a*b^3 + 48*b^4)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 2*a^4 - 2*a^3*b + 8*(55
*(a^4 + a^3*b)*cosh(d*x + c)^9 + 120*(a^4 + 3*a^3*b + 2*a^2*b^2 - (a^4 + 7*a^3*b + 6*a^2*b^2)*d*x)*cosh(d*x +
c)^7 + 14*(5*a^4 + 3*a^3*b - 32*a^2*b^2 - 32*a*b^3 - 16*(a^4 + 9*a^3*b + 20*a^2*b^2 + 12*a*b^3)*d*x)*cosh(d*x
+ c)^5 - 10*(27*a^3*b + 102*a^2*b^2 + 152*a*b^3 + 80*b^4 + 4*(3*a^4 + 29*a^3*b + 82*a^2*b^2 + 104*a*b^3 + 48*b
^4)*d*x)*cosh(d*x + c)^3 - (5*a^4 + 75*a^3*b + 192*a^2*b^2 + 128*a*b^3 + 16*(a^4 + 9*a^3*b + 20*a^2*b^2 + 12*a
*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(2*a^4 + 15*a^3*b + 14*a^2*b^2 + 2*(a^4 + 7*a^3*b + 6*a^2*b^2)*d
*x)*cosh(d*x + c)^2 + 4*(33*(a^4 + a^3*b)*cosh(d*x + c)^10 + 90*(a^4 + 3*a^3*b + 2*a^2*b^2 - (a^4 + 7*a^3*b +
6*a^2*b^2)*d*x)*cosh(d*x + c)^8 + 14*(5*a^4 + 3*a^3*b - 32*a^2*b^2 - 32*a*b^3 - 16*(a^4 + 9*a^3*b + 20*a^2*b^2
 + 12*a*b^3)*d*x)*cosh(d*x + c)^6 - 15*(27*a^3*b + 102*a^2*b^2 + 152*a*b^3 + 80*b^4 + 4*(3*a^4 + 29*a^3*b + 82
*a^2*b^2 + 104*a*b^3 + 48*b^4)*d*x)*cosh(d*x + c)^4 - 2*a^4 - 15*a^3*b - 14*a^2*b^2 - 2*(a^4 + 7*a^3*b + 6*a^2
*b^2)*d*x - 3*(5*a^4 + 75*a^3*b + 192*a^2*b^2 + 128*a*b^3 + 16*(a^4 + 9*a^3*b + 20*a^2*b^2 + 12*a*b^3)*d*x)*co
sh(d*x + c)^2)*sinh(d*x + c)^2 + ((15*a^4 + 40*a^3*b + 24*a^2*b^2)*cosh(d*x + c)^10 + 10*(15*a^4 + 40*a^3*b +
24*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^9 + (15*a^4 + 40*a^3*b + 24*a^2*b^2)*sinh(d*x + c)^10 + 4*(15*a^4 + 70
*a^3*b + 104*a^2*b^2 + 48*a*b^3)*cosh(d*x + c)^8 + (60*a^4 + 280*a^3*b + 416*a^2*b^2 + 192*a*b^3 + 45*(15*a^4
+ 40*a^3*b + 24*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(15*a^4 + 40*a^3*b + 24*a^2*b^2)*cosh(d*x +
c)^3 + 4*(15*a^4 + 70*a^3*b + 104*a^2*b^2 + 48*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(45*a^4 + 240*a^3*b +
 512*a^2*b^2 + 512*a*b^3 + 192*b^4)*cosh(d*x + c)^6 + 2*(105*(15*a^4 + 40*a^3*b + 24*a^2*b^2)*cosh(d*x + c)^4
+ 45*a^4 + 240*a^3*b + 512*a^2*b^2 + 512*a*b^3 + 192*b^4 + 56*(15*a^4 + 70*a^3*b + 104*a^2*b^2 + 48*a*b^3)*cos
h(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(15*a^4 + 40*a^3*b + 24*a^2*b^2)*cosh(d*x + c)^5 + 56*(15*a^4 + 70*a^3*b
 + 104*a^2*b^2 + 48*a*b^3)*cosh(d*x + c)^3 + 3*(45*a^4 + 240*a^3*b + 512*a^2*b^2 + 512*a*b^3 + 192*b^4)*cosh(d
*x + c))*sinh(d*x + c)^5 + 4*(15*a^4 + 70*a^3*b + 104*a^2*b^2 + 48*a*b^3)*cosh(d*x + c)^4 + 2*(105*(15*a^4 + 4
0*a^3*b + 24*a^2*b^2)*cosh(d*x + c)^6 + 140*(15*a^4 + 70*a^3*b + 104*a^2*b^2 + 48*a*b^3)*cosh(d*x + c)^4 + 30*
a^4 + 140*a^3*b + 208*a^2*b^2 + 96*a*b^3 + 15*(45*a^4 + 240*a^3*b + 512*a^2*b^2 + 512*a*b^3 + 192*b^4)*cosh(d*
x + c)^2)*sinh(d*x + c)^4 + 8*(15*(15*a^4 + 40*...

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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(sinh(d*x+c)**2/(a+b*sech(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (178) = 356\).
time = 1.65, size = 370, normalized size = 1.98 \begin {gather*} \frac {\frac {{\left (15 \, a^{2} b + 40 \, a b^{2} + 24 \, 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 )}{{\left (a^{5} + a^{4} b\right )} \sqrt {-a b - b^{2}}} - \frac {2 \, {\left (9 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 32 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 24 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 27 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 102 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 152 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 80 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 80 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 56 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{3} b + 10 \, a^{2} b^{2}\right )}}{{\left (a^{5} + a^{4} b\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 {4 \, {\left (d x + c\right )} {\left (a + 6 \, b\right )}}{a^{4}} + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{a^{3}} + \frac {{\left (2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b e^{\left (2 \, d x + 2 \, c\right )} - a\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{a^{4}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/8*((15*a^2*b + 40*a*b^2 + 24*b^3)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^5 + a^4*b)*
sqrt(-a*b - b^2)) - 2*(9*a^3*b*e^(6*d*x + 6*c) + 32*a^2*b^2*e^(6*d*x + 6*c) + 24*a*b^3*e^(6*d*x + 6*c) + 27*a^
3*b*e^(4*d*x + 4*c) + 102*a^2*b^2*e^(4*d*x + 4*c) + 152*a*b^3*e^(4*d*x + 4*c) + 80*b^4*e^(4*d*x + 4*c) + 27*a^
3*b*e^(2*d*x + 2*c) + 80*a^2*b^2*e^(2*d*x + 2*c) + 56*a*b^3*e^(2*d*x + 2*c) + 9*a^3*b + 10*a^2*b^2)/((a^5 + a^
4*b)*(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) - 4*(d*x + c)*(a + 6*b)/a^4 + e^(2
*d*x + 2*c)/a^3 + (2*a*e^(2*d*x + 2*c) + 12*b*e^(2*d*x + 2*c) - a)*e^(-2*d*x - 2*c)/a^4)/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {sinh}\left (c+d\,x\right )}^2}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^2/(a + b/cosh(c + d*x)^2)^3,x)

[Out]

int((cosh(c + d*x)^6*sinh(c + d*x)^2)/(b + a*cosh(c + d*x)^2)^3, x)

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