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

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

[Out]

-((a + 6*b)*x)/(2*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))

________________________________________________________________________________________

Rubi [A]  time = 0.291762, antiderivative size = 187, normalized size of antiderivative = 1., 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 = {4132, 471, 527, 522, 206, 208} \[ \frac{\sqrt{b} \left (15 a^2+40 a b+24 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^4 d (a+b)^{3/2}}+\frac{b (11 a+12 b) \tanh (c+d x)}{8 a^3 d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}+\frac{3 b \tanh (c+d x)}{4 a^2 d \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac{x (a+6 b)}{2 a^4}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 a d \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + 6*b)*x)/(2*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 4132

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]

Rule 471

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 527

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 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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 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{\sinh ^2(c+d x)}{\left (a+b \text{sech}^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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) \operatorname{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 ) \operatorname{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*}

Mathematica [B]  time = 18.7883, size = 2544, normalized size = 13.6 \[ \text{Result too large to show} \]

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 + b*Sech[c + d*x]^2)^3)

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Maple [B]  time = 0.112, size = 1320, normalized size = 7.1 \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

-1/2/d/a^3/(tanh(1/2*d*x+1/2*c)+1)^2+1/2/d/a^3/(tanh(1/2*d*x+1/2*c)+1)-1/2/d/a^3*ln(tanh(1/2*d*x+1/2*c)+1)-3/d
/a^4*ln(tanh(1/2*d*x+1/2*c)+1)*b+9/4/d*b/a^2/(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*tanh(1/2*d*x+1/2*c)^7+2/d*b^2/a^3/(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*tanh(1/2*d*x+1/2*c)^7+27/4/d*b/a/(t
anh(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)^5+23/4/d*b^2/a^2/(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)^5-2/d*b^3/a^3/(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)^5+27/4/d*
b/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)*tanh(1/2*d*x+1/2*c)^3+23/4/d*b^2/a^2/(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)^3-2/d*b^3/a^3/(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)^3+9
/4/d*b/a^2/(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*tanh(1/2*d*x+1/2*c)+2/d*b^2/a^3/(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*tanh(1/2*d*x+1/2*c)+15/16/d*b^(1/2)/a^2/(a+b)^(3/2)*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))+5/2/d*b^(3/2)/a^3/(a+b)^(3/2)*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))+3/2/d*b^(5/2)/a^4/(a+b)^(3/2)*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))-15/16/d*b^(1/2)/a^2/(a+b)^(3/2)*ln((a+b)^(1/2)*ta
nh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))-5/2/d*b^(3/2)/a^3/(a+b)^(3/2)*ln((a+b)^(1/2)*ta
nh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))-3/2/d*b^(5/2)/a^4/(a+b)^(3/2)*ln((a+b)^(1/2)*ta
nh(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/d/a^3/(tanh(1/2*d*x+1/2*c)-1)^2+1/2/d/a^3/(
tanh(1/2*d*x+1/2*c)-1)+1/2/d/a^3*ln(tanh(1/2*d*x+1/2*c)-1)+3/d/a^4*ln(tanh(1/2*d*x+1/2*c)-1)*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(sinh(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 4.00341, size = 23448, normalized size = 125.39 \begin{align*} \text{result too large to display} \end{align*}

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*a^3*b + 24*a^2*b^2)*cosh(d*x + c)^7 + 28*(15*a^4 + 70*a^3*b + 1
04*a^2*b^2 + 48*a*b^3)*cosh(d*x + c)^5 + 5*(45*a^4 + 240*a^3*b + 512*a^2*b^2 + 512*a*b^3 + 192*b^4)*cosh(d*x +
c)^3 + 2*(15*a^4 + 70*a^3*b + 104*a^2*b^2 + 48*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + (15*a^4 + 40*a^3*b + 2
4*a^2*b^2)*cosh(d*x + c)^2 + (45*(15*a^4 + 40*a^3*b + 24*a^2*b^2)*cosh(d*x + c)^8 + 112*(15*a^4 + 70*a^3*b + 1
04*a^2*b^2 + 48*a*b^3)*cosh(d*x + c)^6 + 30*(45*a^4 + 240*a^3*b + 512*a^2*b^2 + 512*a*b^3 + 192*b^4)*cosh(d*x
+ c)^4 + 15*a^4 + 40*a^3*b + 24*a^2*b^2 + 24*(15*a^4 + 70*a^3*b + 104*a^2*b^2 + 48*a*b^3)*cosh(d*x + c)^2)*sin
h(d*x + c)^2 + 2*(5*(15*a^4 + 40*a^3*b + 24*a^2*b^2)*cosh(d*x + c)^9 + 16*(15*a^4 + 70*a^3*b + 104*a^2*b^2 + 4
8*a*b^3)*cosh(d*x + c)^7 + 6*(45*a^4 + 240*a^3*b + 512*a^2*b^2 + 512*a*b^3 + 192*b^4)*cosh(d*x + c)^5 + 8*(15*
a^4 + 70*a^3*b + 104*a^2*b^2 + 48*a*b^3)*cosh(d*x + c)^3 + (15*a^4 + 40*a^3*b + 24*a^2*b^2)*cosh(d*x + c))*sin
h(d*x + c))*sqrt(b/(a + b))*log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)
^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b +
8*b^2 + 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c) - 4*((a^2 + a*b)*cosh(d*x + c)^2
+ 2*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c) + (a^2 + a*b)*sinh(d*x + c)^2 + a^2 + 3*a*b + 2*b^2)*sqrt(b/(a + b
)))/(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)) + 8*(3*(a^4 + a^3*b)*cosh(d*x + c)^11 + 10*(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)^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)^7 - 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 + 1
04*a*b^3 + 48*b^4)*d*x)*cosh(d*x + c)^5 - (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)^3 - (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))*sinh(d*x + c))/((a^7 + a^6*b)*d*cosh(d*x + c)^10 + 10*(a^7 + a^6*b)*d*cosh(d*x + c)*sinh(d*x
+ c)^9 + (a^7 + a^6*b)*d*sinh(d*x + c)^10 + 4*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c)^8 + (45*(a^7 + a^6*b
)*d*cosh(d*x + c)^2 + 4*(a^7 + 3*a^6*b + 2*a^5*b^2)*d)*sinh(d*x + c)^8 + 2*(3*a^7 + 11*a^6*b + 16*a^5*b^2 + 8*
a^4*b^3)*d*cosh(d*x + c)^6 + 8*(15*(a^7 + a^6*b)*d*cosh(d*x + c)^3 + 4*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x
+ c))*sinh(d*x + c)^7 + 2*(105*(a^7 + a^6*b)*d*cosh(d*x + c)^4 + 56*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c
)^2 + (3*a^7 + 11*a^6*b + 16*a^5*b^2 + 8*a^4*b^3)*d)*sinh(d*x + c)^6 + 4*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*
x + c)^4 + 4*(63*(a^7 + a^6*b)*d*cosh(d*x + c)^5 + 56*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c)^3 + 3*(3*a^7
+ 11*a^6*b + 16*a^5*b^2 + 8*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^7 + a^6*b)*d*cosh(d*x + c)^
6 + 140*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c)^4 + 15*(3*a^7 + 11*a^6*b + 16*a^5*b^2 + 8*a^4*b^3)*d*cosh(
d*x + c)^2 + 2*(a^7 + 3*a^6*b + 2*a^5*b^2)*d)*sinh(d*x + c)^4 + (a^7 + a^6*b)*d*cosh(d*x + c)^2 + 8*(15*(a^7 +
a^6*b)*d*cosh(d*x + c)^7 + 28*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c)^5 + 5*(3*a^7 + 11*a^6*b + 16*a^5*b^
2 + 8*a^4*b^3)*d*cosh(d*x + c)^3 + 2*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^7 +
a^6*b)*d*cosh(d*x + c)^8 + 112*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c)^6 + 30*(3*a^7 + 11*a^6*b + 16*a^5*
b^2 + 8*a^4*b^3)*d*cosh(d*x + c)^4 + 24*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c)^2 + (a^7 + a^6*b)*d)*sinh(
d*x + c)^2 + 2*(5*(a^7 + a^6*b)*d*cosh(d*x + c)^9 + 16*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c)^7 + 6*(3*a^
7 + 11*a^6*b + 16*a^5*b^2 + 8*a^4*b^3)*d*cosh(d*x + c)^5 + 8*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c)^3 + (
a^7 + a^6*b)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*((a^4 + a^3*b)*cosh(d*x + c)^12 + 12*(a^4 + a^3*b)*cosh(d*x
+ c)*sinh(d*x + c)^11 + (a^4 + a^3*b)*sinh(d*x + c)^12 + 4*(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 + 2*(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 + 20*(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 + (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 + (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 + 8*(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 - 2*(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 + 2*(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 + 4*(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)*cosh(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)*cosh(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 - (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 + (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 - a^4 - a^3*b + 4*(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 - 2*(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 + 2*(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)*cosh(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 + 2
4*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)*cosh(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 + 40*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 + 5
12*a*b^3 + 192*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(15*(15*a^4 + 40*a^3*b + 24*a^2*b^2)*cosh(d*x + c)^7
+ 28*(15*a^4 + 70*a^3*b + 104*a^2*b^2 + 48*a*b^3)*cosh(d*x + c)^5 + 5*(45*a^4 + 240*a^3*b + 512*a^2*b^2 + 512*
a*b^3 + 192*b^4)*cosh(d*x + c)^3 + 2*(15*a^4 + 70*a^3*b + 104*a^2*b^2 + 48*a*b^3)*cosh(d*x + c))*sinh(d*x + c)
^3 + (15*a^4 + 40*a^3*b + 24*a^2*b^2)*cosh(d*x + c)^2 + (45*(15*a^4 + 40*a^3*b + 24*a^2*b^2)*cosh(d*x + c)^8 +
112*(15*a^4 + 70*a^3*b + 104*a^2*b^2 + 48*a*b^3)*cosh(d*x + c)^6 + 30*(45*a^4 + 240*a^3*b + 512*a^2*b^2 + 512
*a*b^3 + 192*b^4)*cosh(d*x + c)^4 + 15*a^4 + 40*a^3*b + 24*a^2*b^2 + 24*(15*a^4 + 70*a^3*b + 104*a^2*b^2 + 48*
a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(15*a^4 + 40*a^3*b + 24*a^2*b^2)*cosh(d*x + c)^9 + 16*(15*a^4 +
70*a^3*b + 104*a^2*b^2 + 48*a*b^3)*cosh(d*x + c)^7 + 6*(45*a^4 + 240*a^3*b + 512*a^2*b^2 + 512*a*b^3 + 192*b^
4)*cosh(d*x + c)^5 + 8*(15*a^4 + 70*a^3*b + 104*a^2*b^2 + 48*a*b^3)*cosh(d*x + c)^3 + (15*a^4 + 40*a^3*b + 24*
a^2*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/(a + b))*arctan(1/2*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh
(d*x + c) + a*sinh(d*x + c)^2 + a + 2*b)*sqrt(-b/(a + b))/b) + 4*(3*(a^4 + a^3*b)*cosh(d*x + c)^11 + 10*(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)^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)^7 - 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)^5 - (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)^3 - (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))*sinh(d*x + c))/((a^7 + a^6*b)*d*cosh(d*x +
c)^10 + 10*(a^7 + a^6*b)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^7 + a^6*b)*d*sinh(d*x + c)^10 + 4*(a^7 + 3*a^6*b
+ 2*a^5*b^2)*d*cosh(d*x + c)^8 + (45*(a^7 + a^6*b)*d*cosh(d*x + c)^2 + 4*(a^7 + 3*a^6*b + 2*a^5*b^2)*d)*sinh(
d*x + c)^8 + 2*(3*a^7 + 11*a^6*b + 16*a^5*b^2 + 8*a^4*b^3)*d*cosh(d*x + c)^6 + 8*(15*(a^7 + a^6*b)*d*cosh(d*x
+ c)^3 + 4*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^7 + a^6*b)*d*cosh(d*x + c)
^4 + 56*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c)^2 + (3*a^7 + 11*a^6*b + 16*a^5*b^2 + 8*a^4*b^3)*d)*sinh(d*
x + c)^6 + 4*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c)^4 + 4*(63*(a^7 + a^6*b)*d*cosh(d*x + c)^5 + 56*(a^7 +
3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c)^3 + 3*(3*a^7 + 11*a^6*b + 16*a^5*b^2 + 8*a^4*b^3)*d*cosh(d*x + c))*sinh(
d*x + c)^5 + 2*(105*(a^7 + a^6*b)*d*cosh(d*x + c)^6 + 140*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c)^4 + 15*(
3*a^7 + 11*a^6*b + 16*a^5*b^2 + 8*a^4*b^3)*d*cosh(d*x + c)^2 + 2*(a^7 + 3*a^6*b + 2*a^5*b^2)*d)*sinh(d*x + c)^
4 + (a^7 + a^6*b)*d*cosh(d*x + c)^2 + 8*(15*(a^7 + a^6*b)*d*cosh(d*x + c)^7 + 28*(a^7 + 3*a^6*b + 2*a^5*b^2)*d
*cosh(d*x + c)^5 + 5*(3*a^7 + 11*a^6*b + 16*a^5*b^2 + 8*a^4*b^3)*d*cosh(d*x + c)^3 + 2*(a^7 + 3*a^6*b + 2*a^5*
b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^7 + a^6*b)*d*cosh(d*x + c)^8 + 112*(a^7 + 3*a^6*b + 2*a^5*b^2)*
d*cosh(d*x + c)^6 + 30*(3*a^7 + 11*a^6*b + 16*a^5*b^2 + 8*a^4*b^3)*d*cosh(d*x + c)^4 + 24*(a^7 + 3*a^6*b + 2*a
^5*b^2)*d*cosh(d*x + c)^2 + (a^7 + a^6*b)*d)*sinh(d*x + c)^2 + 2*(5*(a^7 + a^6*b)*d*cosh(d*x + c)^9 + 16*(a^7
+ 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c)^7 + 6*(3*a^7 + 11*a^6*b + 16*a^5*b^2 + 8*a^4*b^3)*d*cosh(d*x + c)^5 + 8
*(a^7 + 3*a^6*b + 2*a^5*b^2)*d*cosh(d*x + c)^3 + (a^7 + a^6*b)*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(sinh(d*x+c)**2/(a+b*sech(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.18836, size = 517, normalized size = 2.76 \begin{align*} \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 )}{8 \,{\left (a^{5} d + a^{4} b d\right )} \sqrt{-a b - b^{2}}} - \frac{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}}{4 \,{\left (a^{5} d + a^{4} b d\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{{\left (d x + c\right )}{\left (a + 6 \, b\right )}}{2 \, a^{4} d} + \frac{e^{\left (2 \, d x + 2 \, c\right )}}{8 \, a^{3} d} + \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 )}}{8 \, a^{4} d} \end{align*}

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*d + a^4*b*
d)*sqrt(-a*b - b^2)) - 1/4*(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
*d + a^4*b*d)*(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) - 1/2*(d*x + c)*(a + 6*b)
/(a^4*d) + 1/8*e^(2*d*x + 2*c)/(a^3*d) + 1/8*(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)