3.33 \(\int \frac{\sinh ^4(c+d x)}{(a+b \text{sech}^2(c+d x))^2} \, dx\)

Optimal. Leaf size=194 \[ \frac{3 x \left (a^2+8 a b+8 b^2\right )}{8 a^4}-\frac{3 \sqrt{b} \sqrt{a+b} (a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{2 a^4 d}-\frac{3 b (3 a+4 b) \tanh (c+d x)}{8 a^3 d \left (a-b \tanh ^2(c+d x)+b\right )}-\frac{(5 a+6 b) \sinh (c+d x) \cosh (c+d x)}{8 a^2 d \left (a-b \tanh ^2(c+d x)+b\right )}+\frac{\sinh (c+d x) \cosh ^3(c+d x)}{4 a d \left (a-b \tanh ^2(c+d x)+b\right )} \]

[Out]

(3*(a^2 + 8*a*b + 8*b^2)*x)/(8*a^4) - (3*Sqrt[b]*Sqrt[a + b]*(a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a
+ b]])/(2*a^4*d) - ((5*a + 6*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*a^2*d*(a + b - b*Tanh[c + d*x]^2)) + (Cosh[c +
d*x]^3*Sinh[c + d*x])/(4*a*d*(a + b - b*Tanh[c + d*x]^2)) - (3*b*(3*a + 4*b)*Tanh[c + d*x])/(8*a^3*d*(a + b -
b*Tanh[c + d*x]^2))

________________________________________________________________________________________

Rubi [A]  time = 0.273275, antiderivative size = 194, 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, 470, 527, 522, 206, 208} \[ \frac{3 x \left (a^2+8 a b+8 b^2\right )}{8 a^4}-\frac{3 \sqrt{b} \sqrt{a+b} (a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{2 a^4 d}-\frac{3 b (3 a+4 b) \tanh (c+d x)}{8 a^3 d \left (a-b \tanh ^2(c+d x)+b\right )}-\frac{(5 a+6 b) \sinh (c+d x) \cosh (c+d x)}{8 a^2 d \left (a-b \tanh ^2(c+d x)+b\right )}+\frac{\sinh (c+d x) \cosh ^3(c+d x)}{4 a d \left (a-b \tanh ^2(c+d x)+b\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2,x]

[Out]

(3*(a^2 + 8*a*b + 8*b^2)*x)/(8*a^4) - (3*Sqrt[b]*Sqrt[a + b]*(a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a
+ b]])/(2*a^4*d) - ((5*a + 6*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*a^2*d*(a + b - b*Tanh[c + d*x]^2)) + (Cosh[c +
d*x]^3*Sinh[c + d*x])/(4*a*d*(a + b - b*Tanh[c + d*x]^2)) - (3*b*(3*a + 4*b)*Tanh[c + d*x])/(8*a^3*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 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2
*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2
*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +
(a*d*(m - n + n*q + 1) + b*c*n*(p + 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] && GtQ[m - n + 1, n] && 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 ^4(c+d x)}{\left (a+b \text{sech}^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right )^3 \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{a+b+(4 a+5 b) x^2}{\left (1-x^2\right )^2 \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a d}\\ &=-\frac{(5 a+6 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac{\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-3 (a+b) (a+2 b)-3 b (5 a+6 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 d}\\ &=-\frac{(5 a+6 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac{\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac{3 b (3 a+4 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{6 (a+b)^2 (a+4 b)+6 b (a+b) (3 a+4 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) d}\\ &=-\frac{(5 a+6 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac{\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac{3 b (3 a+4 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac{(3 b (a+b) (a+2 b)) \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{2 a^4 d}+\frac{\left (3 \left (a^2+8 a b+8 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^4 d}\\ &=\frac{3 \left (a^2+8 a b+8 b^2\right ) x}{8 a^4}-\frac{3 \sqrt{b} \sqrt{a+b} (a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{2 a^4 d}-\frac{(5 a+6 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac{\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac{3 b (3 a+4 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 14.197, size = 1330, normalized size = 6.86 \[ -\frac{(\cosh (2 c+2 d x) a+a+2 b)^2 \left (16 x+\frac{\left (a^3-6 b a^2-24 b^2 a-16 b^3\right ) \tanh ^{-1}\left (\frac{\text{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}}\right ) (\cosh (2 c)-\sinh (2 c))}{b (a+b)^{3/2} d \sqrt{b (\cosh (c)-\sinh (c))^4}}+\frac{\left (a^2+8 b a+8 b^2\right ) \text{sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{b (a+b) d (\cosh (2 (c+d x)) a+a+2 b)}\right ) \text{sech}^4(c+d x)}{256 a^2 \left (b \text{sech}^2(c+d x)+a\right )^2}+\frac{3 (\cosh (2 c+2 d x) a+a+2 b)^2 \left (\frac{(a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 b^{3/2} (a+b)^{3/2} d}-\frac{a \sinh (2 (c+d x))}{8 b (a+b) d (\cosh (2 (c+d x)) a+a+2 b)}\right ) \text{sech}^4(c+d x)}{128 \left (b \text{sech}^2(c+d x)+a\right )^2}+\frac{(\cosh (2 c+2 d x) a+a+2 b)^2 \left (\frac{\left (a^5-30 b a^4-480 b^2 a^3-1600 b^3 a^2-1920 b^4 a-768 b^5\right ) \left (\frac{i \tan ^{-1}\left (\text{sech}(d x) \left (\frac{i \sinh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}-\frac{i \cosh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}\right ) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right ) \sinh (2 c)}{8 a^4 b \sqrt{a+b} d \sqrt{b \cosh (4 c)-b \sinh (4 c)}}-\frac{i \tan ^{-1}\left (\text{sech}(d x) \left (\frac{i \sinh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}-\frac{i \cosh (2 c)}{2 \sqrt{a+b} \sqrt{b \cosh (4 c)-b \sinh (4 c)}}\right ) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right ) \cosh (2 c)}{8 a^4 b \sqrt{a+b} d \sqrt{b \cosh (4 c)-b \sinh (4 c)}}\right )}{a+b}+\frac{\text{sech}(2 c) \left (\sinh (2 c) a^5-\sinh (2 d x) a^5+160 b d x \cosh (2 c) a^4+80 b d x \cosh (2 d x) a^4+80 b d x \cosh (4 c+2 d x) a^4+34 b \sinh (2 c) a^4-62 b \sinh (2 d x) a^4-30 b \sinh (4 c+2 d x) a^4-12 b \sinh (2 c+4 d x) a^4-12 b \sinh (6 c+4 d x) a^4+2 b \sinh (4 c+6 d x) a^4+2 b \sinh (8 c+6 d x) a^4+1248 b^2 d x \cosh (2 c) a^3+464 b^2 d x \cosh (2 d x) a^3+464 b^2 d x \cosh (4 c+2 d x) a^3+224 b^2 \sinh (2 c) a^3-318 b^2 \sinh (2 d x) a^3-158 b^2 \sinh (4 c+2 d x) a^3-36 b^2 \sinh (2 c+4 d x) a^3-36 b^2 \sinh (6 c+4 d x) a^3+2 b^2 \sinh (4 c+6 d x) a^3+2 b^2 \sinh (8 c+6 d x) a^3+3392 b^3 d x \cosh (2 c) a^2+768 b^3 d x \cosh (2 d x) a^2+768 b^3 d x \cosh (4 c+2 d x) a^2+576 b^3 \sinh (2 c) a^2-512 b^3 \sinh (2 d x) a^2-256 b^3 \sinh (4 c+2 d x) a^2-24 b^3 \sinh (2 c+4 d x) a^2-24 b^3 \sinh (6 c+4 d x) a^2+3840 b^4 d x \cosh (2 c) a+384 b^4 d x \cosh (2 d x) a+384 b^4 d x \cosh (4 c+2 d x) a+640 b^4 \sinh (2 c) a-256 b^4 \sinh (2 d x) a-128 b^4 \sinh (4 c+2 d x) a+1536 b^5 d x \cosh (2 c)+256 b^5 \sinh (2 c)\right )}{8 a^4 b (a+b) d (\cosh (2 c+2 d x) a+a+2 b)}\right ) \text{sech}^4(c+d x)}{128 \left (b \text{sech}^2(c+d x)+a\right )^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sinh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2,x]

[Out]

-((a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(16*x + ((a^3 - 6*a^2*b - 24*a*b^2 - 16*b^3)*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]))/(b*(a + b)^(3/2)*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + ((a^2 + 8*a*b + 8*b^2)*
Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/(b*(a + b)*d*(a + 2*b + a*Cosh[2*(c + d*x)]))))/(256*a^2*(a +
b*Sech[c + d*x]^2)^2) + (3*(a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(((a + 2*b)*ArcTanh[(Sqrt[b]*Tan
h[c + d*x])/Sqrt[a + b]])/(8*b^(3/2)*(a + b)^(3/2)*d) - (a*Sinh[2*(c + d*x)])/(8*b*(a + b)*d*(a + 2*b + a*Cosh
[2*(c + d*x)]))))/(128*(a + b*Sech[c + d*x]^2)^2) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(((a^5
- 30*a^4*b - 480*a^3*b^2 - 1600*a^2*b^3 - 1920*a*b^4 - 768*b^5)*(((-I/8)*ArcTan[Sech[d*x]*(((-I/2)*Cosh[2*c])/
(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]
))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Cosh[2*c])/(a^4*b*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*
Sinh[4*c]]) + ((I/8)*ArcTan[Sech[d*x]*(((-I/2)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/
2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*
x])]*Sinh[2*c])/(a^4*b*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/(a + b) + (Sech[2*c]*(160*a^4*b*d*x*Co
sh[2*c] + 1248*a^3*b^2*d*x*Cosh[2*c] + 3392*a^2*b^3*d*x*Cosh[2*c] + 3840*a*b^4*d*x*Cosh[2*c] + 1536*b^5*d*x*Co
sh[2*c] + 80*a^4*b*d*x*Cosh[2*d*x] + 464*a^3*b^2*d*x*Cosh[2*d*x] + 768*a^2*b^3*d*x*Cosh[2*d*x] + 384*a*b^4*d*x
*Cosh[2*d*x] + 80*a^4*b*d*x*Cosh[4*c + 2*d*x] + 464*a^3*b^2*d*x*Cosh[4*c + 2*d*x] + 768*a^2*b^3*d*x*Cosh[4*c +
2*d*x] + 384*a*b^4*d*x*Cosh[4*c + 2*d*x] + a^5*Sinh[2*c] + 34*a^4*b*Sinh[2*c] + 224*a^3*b^2*Sinh[2*c] + 576*a
^2*b^3*Sinh[2*c] + 640*a*b^4*Sinh[2*c] + 256*b^5*Sinh[2*c] - a^5*Sinh[2*d*x] - 62*a^4*b*Sinh[2*d*x] - 318*a^3*
b^2*Sinh[2*d*x] - 512*a^2*b^3*Sinh[2*d*x] - 256*a*b^4*Sinh[2*d*x] - 30*a^4*b*Sinh[4*c + 2*d*x] - 158*a^3*b^2*S
inh[4*c + 2*d*x] - 256*a^2*b^3*Sinh[4*c + 2*d*x] - 128*a*b^4*Sinh[4*c + 2*d*x] - 12*a^4*b*Sinh[2*c + 4*d*x] -
36*a^3*b^2*Sinh[2*c + 4*d*x] - 24*a^2*b^3*Sinh[2*c + 4*d*x] - 12*a^4*b*Sinh[6*c + 4*d*x] - 36*a^3*b^2*Sinh[6*c
+ 4*d*x] - 24*a^2*b^3*Sinh[6*c + 4*d*x] + 2*a^4*b*Sinh[4*c + 6*d*x] + 2*a^3*b^2*Sinh[4*c + 6*d*x] + 2*a^4*b*S
inh[8*c + 6*d*x] + 2*a^3*b^2*Sinh[8*c + 6*d*x]))/(8*a^4*b*(a + b)*d*(a + 2*b + a*Cosh[2*c + 2*d*x]))))/(128*(a
+ b*Sech[c + d*x]^2)^2)

________________________________________________________________________________________

Maple [B]  time = 0.109, size = 1025, normalized size = 5.3 \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)^4/(a+b*sech(d*x+c)^2)^2,x)

[Out]

-1/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)*tanh(1/2*d*x+1/2*c)-3/4/d*b^(1/2)/a^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))+3/4/d*b^(1/2)/a^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))+3/8/d/a^2*ln(tanh(1/2*d*x+1/2*c)+1)-3/8/d/a^2*ln(tanh(1/2*d*x+1/2*c)-1)+1/4/d/a^2/(
tanh(1/2*d*x+1/2*c)-1)^4+1/2/d/a^2/(tanh(1/2*d*x+1/2*c)-1)^3-1/8/d/a^2/(tanh(1/2*d*x+1/2*c)-1)^2-3/8/d/a^2/(ta
nh(1/2*d*x+1/2*c)-1)-1/4/d/a^2/(tanh(1/2*d*x+1/2*c)+1)^4+1/2/d/a^2/(tanh(1/2*d*x+1/2*c)+1)^3+1/8/d/a^2/(tanh(1
/2*d*x+1/2*c)+1)^2-3/8/d/a^2/(tanh(1/2*d*x+1/2*c)+1)-1/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)*tanh(1/2*d*x+1/2*c)^3-1/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)*tanh(1/2*d*x+1/2*c)-1
/d/a^3/(tanh(1/2*d*x+1/2*c)-1)*b-3/d/a^3*ln(tanh(1/2*d*x+1/2*c)-1)*b-3/d/a^4*ln(tanh(1/2*d*x+1/2*c)-1)*b^2-1/d
/a^3/(tanh(1/2*d*x+1/2*c)+1)*b+3/d/a^3*ln(tanh(1/2*d*x+1/2*c)+1)*b+3/d/a^4*ln(tanh(1/2*d*x+1/2*c)+1)*b^2-1/d/a
^3/(tanh(1/2*d*x+1/2*c)-1)^2*b+1/d/a^3/(tanh(1/2*d*x+1/2*c)+1)^2*b+9/4/d*b^(3/2)/a^3/(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))-3/2/d*b^(5/2)/a^4/(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))+3/2/d*b^(5/2)/a^4/(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/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)*tanh(1/2*d*x+1/2*c)^3-9/4/d*b^(3/2)
/a^3/(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))

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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)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.92577, size = 12945, normalized size = 66.73 \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)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[1/64*(a^3*cosh(d*x + c)^12 + 12*a^3*cosh(d*x + c)*sinh(d*x + c)^11 + a^3*sinh(d*x + c)^12 - 6*(a^3 + 2*a^2*b)
*cosh(d*x + c)^10 + 6*(11*a^3*cosh(d*x + c)^2 - a^3 - 2*a^2*b)*sinh(d*x + c)^10 + 20*(11*a^3*cosh(d*x + c)^3 -
3*(a^3 + 2*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^9 - (15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^
2)*d*x)*cosh(d*x + c)^8 + (495*a^3*cosh(d*x + c)^4 - 15*a^3 - 64*a^2*b - 64*a*b^2 + 24*(a^3 + 8*a^2*b + 8*a*b^
2)*d*x - 270*(a^3 + 2*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(99*a^3*cosh(d*x + c)^5 - 90*(a^3 + 2*a^2*b)
*cosh(d*x + c)^3 - (15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c))*sinh(d*x +
c)^7 + 16*(4*a^2*b + 12*a*b^2 + 8*b^3 + 3*(a^3 + 10*a^2*b + 24*a*b^2 + 16*b^3)*d*x)*cosh(d*x + c)^6 + 4*(231*
a^3*cosh(d*x + c)^6 - 315*(a^3 + 2*a^2*b)*cosh(d*x + c)^4 + 16*a^2*b + 48*a*b^2 + 32*b^3 + 12*(a^3 + 10*a^2*b
+ 24*a*b^2 + 16*b^3)*d*x - 7*(15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^2
)*sinh(d*x + c)^6 + 8*(99*a^3*cosh(d*x + c)^7 - 189*(a^3 + 2*a^2*b)*cosh(d*x + c)^5 - 7*(15*a^3 + 64*a^2*b + 6
4*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^3 + 12*(4*a^2*b + 12*a*b^2 + 8*b^3 + 3*(a^3 + 10*a^2
*b + 24*a*b^2 + 16*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + (15*a^3 + 128*a^2*b + 128*a*b^2 + 24*(a^3 + 8*a^
2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^4 + (495*a^3*cosh(d*x + c)^8 - 1260*(a^3 + 2*a^2*b)*cosh(d*x + c)^6 - 70*(15
*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^4 + 15*a^3 + 128*a^2*b + 128*a*b^
2 + 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x + 240*(4*a^2*b + 12*a*b^2 + 8*b^3 + 3*(a^3 + 10*a^2*b + 24*a*b^2 + 16*b^3
)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(55*a^3*cosh(d*x + c)^9 - 180*(a^3 + 2*a^2*b)*cosh(d*x + c)^7 - 14
*(15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^5 + 80*(4*a^2*b + 12*a*b^2 +
8*b^3 + 3*(a^3 + 10*a^2*b + 24*a*b^2 + 16*b^3)*d*x)*cosh(d*x + c)^3 + (15*a^3 + 128*a^2*b + 128*a*b^2 + 24*(a^
3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - a^3 + 6*(a^3 + 2*a^2*b)*cosh(d*x + c)^2 + 2*(33*a
^3*cosh(d*x + c)^10 - 135*(a^3 + 2*a^2*b)*cosh(d*x + c)^8 - 14*(15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2
*b + 8*a*b^2)*d*x)*cosh(d*x + c)^6 + 120*(4*a^2*b + 12*a*b^2 + 8*b^3 + 3*(a^3 + 10*a^2*b + 24*a*b^2 + 16*b^3)*
d*x)*cosh(d*x + c)^4 + 3*a^3 + 6*a^2*b + 3*(15*a^3 + 128*a^2*b + 128*a*b^2 + 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)
*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 48*((a^2 + 2*a*b)*cosh(d*x + c)^8 + 8*(a^2 + 2*a*b)*cosh(d*x + c)*sinh(d*x
+ c)^7 + (a^2 + 2*a*b)*sinh(d*x + c)^8 + 2*(a^2 + 4*a*b + 4*b^2)*cosh(d*x + c)^6 + 2*(14*(a^2 + 2*a*b)*cosh(d
*x + c)^2 + a^2 + 4*a*b + 4*b^2)*sinh(d*x + c)^6 + 4*(14*(a^2 + 2*a*b)*cosh(d*x + c)^3 + 3*(a^2 + 4*a*b + 4*b^
2)*cosh(d*x + c))*sinh(d*x + c)^5 + (a^2 + 2*a*b)*cosh(d*x + c)^4 + (70*(a^2 + 2*a*b)*cosh(d*x + c)^4 + 30*(a^
2 + 4*a*b + 4*b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^4 + 4*(14*(a^2 + 2*a*b)*cosh(d*x + c)^5 + 10*(
a^2 + 4*a*b + 4*b^2)*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(14*(a^2 + 2*a*b)*cosh
(d*x + c)^6 + 15*(a^2 + 4*a*b + 4*b^2)*cosh(d*x + c)^4 + 3*(a^2 + 2*a*b)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*
(2*(a^2 + 2*a*b)*cosh(d*x + c)^7 + 3*(a^2 + 4*a*b + 4*b^2)*cosh(d*x + c)^5 + (a^2 + 2*a*b)*cosh(d*x + c)^3)*si
nh(d*x + c))*sqrt(a*b + b^2)*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*cosh(d*x + c)^2 + 2*a*cos
h(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a + 2*b)*sqrt(a*b + b^2))/(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)) + 4*(3*a^3*cosh(d*x + c)^11 -
15*(a^3 + 2*a^2*b)*cosh(d*x + c)^9 - 2*(15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(
d*x + c)^7 + 24*(4*a^2*b + 12*a*b^2 + 8*b^3 + 3*(a^3 + 10*a^2*b + 24*a*b^2 + 16*b^3)*d*x)*cosh(d*x + c)^5 + (1
5*a^3 + 128*a^2*b + 128*a*b^2 + 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^3 + 3*(a^3 + 2*a^2*b)*cosh(d*x
+ c))*sinh(d*x + c))/(a^5*d*cosh(d*x + c)^8 + 8*a^5*d*cosh(d*x + c)*sinh(d*x + c)^7 + a^5*d*sinh(d*x + c)^8 +
a^5*d*cosh(d*x + c)^4 + 2*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^6 + 2*(14*a^5*d*cosh(d*x + c)^2 + (a^5 + 2*a^4*b)*d
)*sinh(d*x + c)^6 + 4*(14*a^5*d*cosh(d*x + c)^3 + 3*(a^5 + 2*a^4*b)*d*cosh(d*x + c))*sinh(d*x + c)^5 + (70*a^5
*d*cosh(d*x + c)^4 + a^5*d + 30*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(14*a^5*d*cosh(d*x + c)
^5 + a^5*d*cosh(d*x + c) + 10*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^3)*sinh(d*x + c)^3 + 2*(14*a^5*d*cosh(d*x + c)^6
+ 3*a^5*d*cosh(d*x + c)^2 + 15*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^4)*sinh(d*x + c)^2 + 4*(2*a^5*d*cosh(d*x + c)^
7 + a^5*d*cosh(d*x + c)^3 + 3*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^5)*sinh(d*x + c)), 1/64*(a^3*cosh(d*x + c)^12 +
12*a^3*cosh(d*x + c)*sinh(d*x + c)^11 + a^3*sinh(d*x + c)^12 - 6*(a^3 + 2*a^2*b)*cosh(d*x + c)^10 + 6*(11*a^3*
cosh(d*x + c)^2 - a^3 - 2*a^2*b)*sinh(d*x + c)^10 + 20*(11*a^3*cosh(d*x + c)^3 - 3*(a^3 + 2*a^2*b)*cosh(d*x +
c))*sinh(d*x + c)^9 - (15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^8 + (495
*a^3*cosh(d*x + c)^4 - 15*a^3 - 64*a^2*b - 64*a*b^2 + 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x - 270*(a^3 + 2*a^2*b)*c
osh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(99*a^3*cosh(d*x + c)^5 - 90*(a^3 + 2*a^2*b)*cosh(d*x + c)^3 - (15*a^3 + 6
4*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^7 + 16*(4*a^2*b + 12*a*b^2
+ 8*b^3 + 3*(a^3 + 10*a^2*b + 24*a*b^2 + 16*b^3)*d*x)*cosh(d*x + c)^6 + 4*(231*a^3*cosh(d*x + c)^6 - 315*(a^3
+ 2*a^2*b)*cosh(d*x + c)^4 + 16*a^2*b + 48*a*b^2 + 32*b^3 + 12*(a^3 + 10*a^2*b + 24*a*b^2 + 16*b^3)*d*x - 7*(
15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(99*a^3*
cosh(d*x + c)^7 - 189*(a^3 + 2*a^2*b)*cosh(d*x + c)^5 - 7*(15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b +
8*a*b^2)*d*x)*cosh(d*x + c)^3 + 12*(4*a^2*b + 12*a*b^2 + 8*b^3 + 3*(a^3 + 10*a^2*b + 24*a*b^2 + 16*b^3)*d*x)*c
osh(d*x + c))*sinh(d*x + c)^5 + (15*a^3 + 128*a^2*b + 128*a*b^2 + 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x +
c)^4 + (495*a^3*cosh(d*x + c)^8 - 1260*(a^3 + 2*a^2*b)*cosh(d*x + c)^6 - 70*(15*a^3 + 64*a^2*b + 64*a*b^2 - 2
4*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^4 + 15*a^3 + 128*a^2*b + 128*a*b^2 + 24*(a^3 + 8*a^2*b + 8*a*b^
2)*d*x + 240*(4*a^2*b + 12*a*b^2 + 8*b^3 + 3*(a^3 + 10*a^2*b + 24*a*b^2 + 16*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d
*x + c)^4 + 4*(55*a^3*cosh(d*x + c)^9 - 180*(a^3 + 2*a^2*b)*cosh(d*x + c)^7 - 14*(15*a^3 + 64*a^2*b + 64*a*b^2
- 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^5 + 80*(4*a^2*b + 12*a*b^2 + 8*b^3 + 3*(a^3 + 10*a^2*b + 24
*a*b^2 + 16*b^3)*d*x)*cosh(d*x + c)^3 + (15*a^3 + 128*a^2*b + 128*a*b^2 + 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*co
sh(d*x + c))*sinh(d*x + c)^3 - a^3 + 6*(a^3 + 2*a^2*b)*cosh(d*x + c)^2 + 2*(33*a^3*cosh(d*x + c)^10 - 135*(a^3
+ 2*a^2*b)*cosh(d*x + c)^8 - 14*(15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x +
c)^6 + 120*(4*a^2*b + 12*a*b^2 + 8*b^3 + 3*(a^3 + 10*a^2*b + 24*a*b^2 + 16*b^3)*d*x)*cosh(d*x + c)^4 + 3*a^3 +
6*a^2*b + 3*(15*a^3 + 128*a^2*b + 128*a*b^2 + 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c
)^2 - 96*((a^2 + 2*a*b)*cosh(d*x + c)^8 + 8*(a^2 + 2*a*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 + 2*a*b)*sinh(d
*x + c)^8 + 2*(a^2 + 4*a*b + 4*b^2)*cosh(d*x + c)^6 + 2*(14*(a^2 + 2*a*b)*cosh(d*x + c)^2 + a^2 + 4*a*b + 4*b^
2)*sinh(d*x + c)^6 + 4*(14*(a^2 + 2*a*b)*cosh(d*x + c)^3 + 3*(a^2 + 4*a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c
)^5 + (a^2 + 2*a*b)*cosh(d*x + c)^4 + (70*(a^2 + 2*a*b)*cosh(d*x + c)^4 + 30*(a^2 + 4*a*b + 4*b^2)*cosh(d*x +
c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^4 + 4*(14*(a^2 + 2*a*b)*cosh(d*x + c)^5 + 10*(a^2 + 4*a*b + 4*b^2)*cosh(d*x
+ c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(14*(a^2 + 2*a*b)*cosh(d*x + c)^6 + 15*(a^2 + 4*a*b
+ 4*b^2)*cosh(d*x + c)^4 + 3*(a^2 + 2*a*b)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(2*(a^2 + 2*a*b)*cosh(d*x + c)
^7 + 3*(a^2 + 4*a*b + 4*b^2)*cosh(d*x + c)^5 + (a^2 + 2*a*b)*cosh(d*x + c)^3)*sinh(d*x + c))*sqrt(-a*b - b^2)*
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(-a*b - b^2
)/(a*b + b^2)) + 4*(3*a^3*cosh(d*x + c)^11 - 15*(a^3 + 2*a^2*b)*cosh(d*x + c)^9 - 2*(15*a^3 + 64*a^2*b + 64*a*
b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^7 + 24*(4*a^2*b + 12*a*b^2 + 8*b^3 + 3*(a^3 + 10*a^2*b +
24*a*b^2 + 16*b^3)*d*x)*cosh(d*x + c)^5 + (15*a^3 + 128*a^2*b + 128*a*b^2 + 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)
*cosh(d*x + c)^3 + 3*(a^3 + 2*a^2*b)*cosh(d*x + c))*sinh(d*x + c))/(a^5*d*cosh(d*x + c)^8 + 8*a^5*d*cosh(d*x +
c)*sinh(d*x + c)^7 + a^5*d*sinh(d*x + c)^8 + a^5*d*cosh(d*x + c)^4 + 2*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^6 + 2*
(14*a^5*d*cosh(d*x + c)^2 + (a^5 + 2*a^4*b)*d)*sinh(d*x + c)^6 + 4*(14*a^5*d*cosh(d*x + c)^3 + 3*(a^5 + 2*a^4*
b)*d*cosh(d*x + c))*sinh(d*x + c)^5 + (70*a^5*d*cosh(d*x + c)^4 + a^5*d + 30*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^2
)*sinh(d*x + c)^4 + 4*(14*a^5*d*cosh(d*x + c)^5 + a^5*d*cosh(d*x + c) + 10*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^3)*
sinh(d*x + c)^3 + 2*(14*a^5*d*cosh(d*x + c)^6 + 3*a^5*d*cosh(d*x + c)^2 + 15*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^4
)*sinh(d*x + c)^2 + 4*(2*a^5*d*cosh(d*x + c)^7 + a^5*d*cosh(d*x + c)^3 + 3*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^5)*
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)**4/(a+b*sech(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.18934, size = 454, normalized size = 2.34 \begin{align*} \frac{3 \,{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )}{\left (d x + c\right )}}{8 \, a^{4} d} - \frac{{\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 144 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 144 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, a b e^{\left (2 \, d x + 2 \, c\right )} + a^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, a^{4} d} - \frac{3 \,{\left (a^{2} b + 3 \, a b^{2} + 2 \, 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 )}{2 \, \sqrt{-a b - b^{2}} a^{4} d} + \frac{a^{2} d e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} d e^{\left (2 \, d x + 2 \, c\right )} - 16 \, a b d e^{\left (2 \, d x + 2 \, c\right )}}{64 \, a^{4} d^{2}} + \frac{a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} b + a b^{2}}{{\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 )} a^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")

[Out]

3/8*(a^2 + 8*a*b + 8*b^2)*(d*x + c)/(a^4*d) - 1/64*(18*a^2*e^(4*d*x + 4*c) + 144*a*b*e^(4*d*x + 4*c) + 144*b^2
*e^(4*d*x + 4*c) - 8*a^2*e^(2*d*x + 2*c) - 16*a*b*e^(2*d*x + 2*c) + a^2)*e^(-4*d*x - 4*c)/(a^4*d) - 3/2*(a^2*b
+ 3*a*b^2 + 2*b^3)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*a^4*d) + 1/64
*(a^2*d*e^(4*d*x + 4*c) - 8*a^2*d*e^(2*d*x + 2*c) - 16*a*b*d*e^(2*d*x + 2*c))/(a^4*d^2) + (a^2*b*e^(2*d*x + 2*
c) + 3*a*b^2*e^(2*d*x + 2*c) + 2*b^3*e^(2*d*x + 2*c) + a^2*b + a*b^2)/((a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c
) + 4*b*e^(2*d*x + 2*c) + a)*a^4*d)