3.169 \(\int \frac{1}{(a+b \text{sech}^2(c+d x))^4} \, dx\)

Optimal. Leaf size=207 \[ -\frac{\sqrt{b} \left (70 a^2 b+35 a^3+56 a b^2+16 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{16 a^4 d (a+b)^{7/2}}-\frac{b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{16 a^3 d (a+b)^3 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac{b (11 a+6 b) \tanh (c+d x)}{24 a^2 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )^2}+\frac{x}{a^4}-\frac{b \tanh (c+d x)}{6 a d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3} \]

[Out]

x/a^4 - (Sqrt[b]*(35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(16*a^4
*(a + b)^(7/2)*d) - (b*Tanh[c + d*x])/(6*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^3) - (b*(11*a + 6*b)*Tanh[c +
d*x])/(24*a^2*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2)^2) - (b*(19*a^2 + 22*a*b + 8*b^2)*Tanh[c + d*x])/(16*a^
3*(a + b)^3*d*(a + b - b*Tanh[c + d*x]^2))

________________________________________________________________________________________

Rubi [A]  time = 0.345628, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4128, 414, 527, 522, 206, 208} \[ -\frac{\sqrt{b} \left (70 a^2 b+35 a^3+56 a b^2+16 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{16 a^4 d (a+b)^{7/2}}-\frac{b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{16 a^3 d (a+b)^3 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac{b (11 a+6 b) \tanh (c+d x)}{24 a^2 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )^2}+\frac{x}{a^4}-\frac{b \tanh (c+d x)}{6 a d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sech[c + d*x]^2)^(-4),x]

[Out]

x/a^4 - (Sqrt[b]*(35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(16*a^4
*(a + b)^(7/2)*d) - (b*Tanh[c + d*x])/(6*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^3) - (b*(11*a + 6*b)*Tanh[c +
d*x])/(24*a^2*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2)^2) - (b*(19*a^2 + 22*a*b + 8*b^2)*Tanh[c + d*x])/(16*a^
3*(a + b)^3*d*(a + b - b*Tanh[c + d*x]^2))

Rule 4128

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, 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{1}{\left (a+b \text{sech}^2(c+d x)\right )^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \left (a+b-b x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{b \tanh (c+d x)}{6 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^3}-\frac{\operatorname{Subst}\left (\int \frac{-6 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 )}{6 a (a+b) d}\\ &=-\frac{b \tanh (c+d x)}{6 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^3}-\frac{b (11 a+6 b) \tanh (c+d x)}{24 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{3 \left (8 a^2+5 a b+2 b^2\right )+3 b (11 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 )}{24 a^2 (a+b)^2 d}\\ &=-\frac{b \tanh (c+d x)}{6 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^3}-\frac{b (11 a+6 b) \tanh (c+d x)}{24 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{16 a^3 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-3 \left (16 a^3+29 a^2 b+26 a b^2+8 b^3\right )-3 b \left (19 a^2+22 a b+8 b^2\right ) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{48 a^3 (a+b)^3 d}\\ &=-\frac{b \tanh (c+d x)}{6 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^3}-\frac{b (11 a+6 b) \tanh (c+d x)}{24 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{16 a^3 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{a^4 d}-\frac{\left (b \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{16 a^4 (a+b)^3 d}\\ &=\frac{x}{a^4}-\frac{\sqrt{b} \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{16 a^4 (a+b)^{7/2} d}-\frac{b \tanh (c+d x)}{6 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^3}-\frac{b (11 a+6 b) \tanh (c+d x)}{24 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{16 a^3 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 6.85889, size = 1405, normalized size = 6.79 \[ \frac{\left (35 a^3+70 b a^2+56 b^2 a+16 b^3\right ) (\cosh (2 c+2 d x) a+a+2 b)^4 \left (\frac{i b \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)}{256 a^4 \sqrt{a+b} d \sqrt{b \cosh (4 c)-b \sinh (4 c)}}-\frac{i b \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)}{256 a^4 \sqrt{a+b} d \sqrt{b \cosh (4 c)-b \sinh (4 c)}}\right ) \text{sech}^8(c+d x)}{(a+b)^3 \left (b \text{sech}^2(c+d x)+a\right )^4}+\frac{(\cosh (2 c+2 d x) a+a+2 b) \text{sech}(2 c) \left (480 d x \cosh (2 c) a^6+360 d x \cosh (2 d x) a^6+360 d x \cosh (4 c+2 d x) a^6+144 d x \cosh (2 c+4 d x) a^6+144 d x \cosh (6 c+4 d x) a^6+24 d x \cosh (4 c+6 d x) a^6+24 d x \cosh (8 c+6 d x) a^6+3168 b d x \cosh (2 c) a^5+2232 b d x \cosh (2 d x) a^5+2232 b d x \cosh (4 c+2 d x) a^5+720 b d x \cosh (2 c+4 d x) a^5+720 b d x \cosh (6 c+4 d x) a^5+72 b d x \cosh (4 c+6 d x) a^5+72 b d x \cosh (8 c+6 d x) a^5+870 b \sinh (2 c) a^5-870 b \sinh (2 d x) a^5+435 b \sinh (4 c+2 d x) a^5-435 b \sinh (2 c+4 d x) a^5+87 b \sinh (6 c+4 d x) a^5-87 b \sinh (4 c+6 d x) a^5+8928 b^2 d x \cosh (2 c) a^4+5688 b^2 d x \cosh (2 d x) a^4+5688 b^2 d x \cosh (4 c+2 d x) a^4+1296 b^2 d x \cosh (2 c+4 d x) a^4+1296 b^2 d x \cosh (6 c+4 d x) a^4+72 b^2 d x \cosh (4 c+6 d x) a^4+72 b^2 d x \cosh (8 c+6 d x) a^4+4292 b^2 \sinh (2 c) a^4-3792 b^2 \sinh (2 d x) a^4+2124 b^2 \sinh (4 c+2 d x) a^4-1374 b^2 \sinh (2 c+4 d x) a^4+366 b^2 \sinh (6 c+4 d x) a^4-116 b^2 \sinh (4 c+6 d x) a^4+14112 b^3 d x \cosh (2 c) a^3+7272 b^3 d x \cosh (2 d x) a^3+7272 b^3 d x \cosh (4 c+2 d x) a^3+1008 b^3 d x \cosh (2 c+4 d x) a^3+1008 b^3 d x \cosh (6 c+4 d x) a^3+24 b^3 d x \cosh (4 c+6 d x) a^3+24 b^3 d x \cosh (8 c+6 d x) a^3+8792 b^3 \sinh (2 c) a^3-6432 b^3 \sinh (2 d x) a^3+3972 b^3 \sinh (4 c+2 d x) a^3-1248 b^3 \sinh (2 c+4 d x) a^3+408 b^3 \sinh (6 c+4 d x) a^3-44 b^3 \sinh (4 c+6 d x) a^3+13248 b^4 d x \cosh (2 c) a^2+4608 b^4 d x \cosh (2 d x) a^2+4608 b^4 d x \cosh (4 c+2 d x) a^2+288 b^4 d x \cosh (2 c+4 d x) a^2+288 b^4 d x \cosh (6 c+4 d x) a^2+9936 b^4 \sinh (2 c) a^2-4608 b^4 \sinh (2 d x) a^2+3072 b^4 \sinh (4 c+2 d x) a^2-384 b^4 \sinh (2 c+4 d x) a^2+144 b^4 \sinh (6 c+4 d x) a^2+6912 b^5 d x \cosh (2 c) a+1152 b^5 d x \cosh (2 d x) a+1152 b^5 d x \cosh (4 c+2 d x) a+5824 b^5 \sinh (2 c) a-1248 b^5 \sinh (2 d x) a+864 b^5 \sinh (4 c+2 d x) a+1536 b^6 d x \cosh (2 c)+1408 b^6 \sinh (2 c)\right ) \text{sech}^8(c+d x)}{3072 a^4 (a+b)^3 d \left (b \text{sech}^2(c+d x)+a\right )^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*(a + 2*b + a*Cosh[2*c + 2*d*x])^4*Sech[c + d*x]^8*(((I/256)*b*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]*S
qrt[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*Sqrt[a
+ b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) - ((I/256)*b*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*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/((a + b)
^3*(a + b*Sech[c + d*x]^2)^4) + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Sech[2*c]*Sech[c + d*x]^8*(480*a^6*d*x*Cosh[2
*c] + 3168*a^5*b*d*x*Cosh[2*c] + 8928*a^4*b^2*d*x*Cosh[2*c] + 14112*a^3*b^3*d*x*Cosh[2*c] + 13248*a^2*b^4*d*x*
Cosh[2*c] + 6912*a*b^5*d*x*Cosh[2*c] + 1536*b^6*d*x*Cosh[2*c] + 360*a^6*d*x*Cosh[2*d*x] + 2232*a^5*b*d*x*Cosh[
2*d*x] + 5688*a^4*b^2*d*x*Cosh[2*d*x] + 7272*a^3*b^3*d*x*Cosh[2*d*x] + 4608*a^2*b^4*d*x*Cosh[2*d*x] + 1152*a*b
^5*d*x*Cosh[2*d*x] + 360*a^6*d*x*Cosh[4*c + 2*d*x] + 2232*a^5*b*d*x*Cosh[4*c + 2*d*x] + 5688*a^4*b^2*d*x*Cosh[
4*c + 2*d*x] + 7272*a^3*b^3*d*x*Cosh[4*c + 2*d*x] + 4608*a^2*b^4*d*x*Cosh[4*c + 2*d*x] + 1152*a*b^5*d*x*Cosh[4
*c + 2*d*x] + 144*a^6*d*x*Cosh[2*c + 4*d*x] + 720*a^5*b*d*x*Cosh[2*c + 4*d*x] + 1296*a^4*b^2*d*x*Cosh[2*c + 4*
d*x] + 1008*a^3*b^3*d*x*Cosh[2*c + 4*d*x] + 288*a^2*b^4*d*x*Cosh[2*c + 4*d*x] + 144*a^6*d*x*Cosh[6*c + 4*d*x]
+ 720*a^5*b*d*x*Cosh[6*c + 4*d*x] + 1296*a^4*b^2*d*x*Cosh[6*c + 4*d*x] + 1008*a^3*b^3*d*x*Cosh[6*c + 4*d*x] +
288*a^2*b^4*d*x*Cosh[6*c + 4*d*x] + 24*a^6*d*x*Cosh[4*c + 6*d*x] + 72*a^5*b*d*x*Cosh[4*c + 6*d*x] + 72*a^4*b^2
*d*x*Cosh[4*c + 6*d*x] + 24*a^3*b^3*d*x*Cosh[4*c + 6*d*x] + 24*a^6*d*x*Cosh[8*c + 6*d*x] + 72*a^5*b*d*x*Cosh[8
*c + 6*d*x] + 72*a^4*b^2*d*x*Cosh[8*c + 6*d*x] + 24*a^3*b^3*d*x*Cosh[8*c + 6*d*x] + 870*a^5*b*Sinh[2*c] + 4292
*a^4*b^2*Sinh[2*c] + 8792*a^3*b^3*Sinh[2*c] + 9936*a^2*b^4*Sinh[2*c] + 5824*a*b^5*Sinh[2*c] + 1408*b^6*Sinh[2*
c] - 870*a^5*b*Sinh[2*d*x] - 3792*a^4*b^2*Sinh[2*d*x] - 6432*a^3*b^3*Sinh[2*d*x] - 4608*a^2*b^4*Sinh[2*d*x] -
1248*a*b^5*Sinh[2*d*x] + 435*a^5*b*Sinh[4*c + 2*d*x] + 2124*a^4*b^2*Sinh[4*c + 2*d*x] + 3972*a^3*b^3*Sinh[4*c
+ 2*d*x] + 3072*a^2*b^4*Sinh[4*c + 2*d*x] + 864*a*b^5*Sinh[4*c + 2*d*x] - 435*a^5*b*Sinh[2*c + 4*d*x] - 1374*a
^4*b^2*Sinh[2*c + 4*d*x] - 1248*a^3*b^3*Sinh[2*c + 4*d*x] - 384*a^2*b^4*Sinh[2*c + 4*d*x] + 87*a^5*b*Sinh[6*c
+ 4*d*x] + 366*a^4*b^2*Sinh[6*c + 4*d*x] + 408*a^3*b^3*Sinh[6*c + 4*d*x] + 144*a^2*b^4*Sinh[6*c + 4*d*x] - 87*
a^5*b*Sinh[4*c + 6*d*x] - 116*a^4*b^2*Sinh[4*c + 6*d*x] - 44*a^3*b^3*Sinh[4*c + 6*d*x]))/(3072*a^4*(a + b)^3*d
*(a + b*Sech[c + d*x]^2)^4)

________________________________________________________________________________________

Maple [B]  time = 0.119, size = 2892, normalized size = 14. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sech(d*x+c)^2)^4,x)

[Out]

-2/d*b^5/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)^3/(a+b)/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7-145/4/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)^3/(a+b)/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7-1
/2/d*b^4/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)^3/(a+b)/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7-37/4/d*b^3/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)^3/(a+b)/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7-
145/8/d*b/(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)^3/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^9-145/8/d*b/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tan
h(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3-35/16/d*b^(3/2)/a^
2/(a^3+3*a^2*b+3*a*b^2+b^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))-37/4/d*b^3/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)^3/(a^3+3*a^2*b+3*a*b^2+b^3)*tanh(1/2*d*x+1/2*c)^5+11/4/d*b^3/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)^3/(a^2+2*a*b+b^2)*tanh(1/2*d*x
+1/2*c)^9-1/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)^3/(a+b)*tanh(1/2*d*x+1/2*c)^11-35/32/d*b^(1/2)/a/(a^3+3*a^2*b+3*a*b^2+b^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))+1/d/a^4*ln(tanh(1/2*d*x+1/2*c)+1)-1
/d/a^4*ln(tanh(1/2*d*x+1/2*c)-1)-37/d*b^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)^3/(a^3+3*a^2*b+3*a*b^2+b^3)*tanh(1/2*d*x+1/2*c)^5-1/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)^3/(a+b)*tanh(1
/2*d*x+1/2*c)+35/16/d*b^(3/2)/a^2/(a^3+3*a^2*b+3*a*b^2+b^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))+7/4/d*b^(5/2)/a^3/(a^3+3*a^2*b+3*a*b^2+b^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))+1/2/d*b^(7/2)/a^4/(a^3+3*a^2*b+3*a*b^2+b
^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))-145/4/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)^3/(a^
3+3*a^2*b+3*a*b^2+b^3)*tanh(1/2*d*x+1/2*c)^5-29/8/d*b/a/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tan
h(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)*tanh(1/2*d*x+1/2*c)^11-2/d*b^5/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)^3/(a^3+3*a^2*b+3*a*
b^2+b^3)*tanh(1/2*d*x+1/2*c)^5+3/d*b^4/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)^3/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3-1/2/d*b^(7/2)/a^4/(a^3+3*a^2*b+
3*a*b^2+b^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))-29/8/
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
)^3/(a+b)*tanh(1/2*d*x+1/2*c)-13/4/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)^3/(a+b)*tanh(1/2*d*x+1/2*c)^11+35/32/d*b^(1/2)/a/(a^3+3*a^2*b+3*a*b^2
+b^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))-37/d*b^2/(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)^3/(a+b
)/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7-281/24/d*b^2/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)^3/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3+11/4/d*b^3/a^2/(tan
h(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)^3/(a^2+2
*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3-1/2/d*b^4/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)^3/(a^3+3*a^2*b+3*a*b^2+b^3)*tanh(1/2*d*x+1/2*c)^5+3/d*b^4/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)^3/(a^2+2*a
*b+b^2)*tanh(1/2*d*x+1/2*c)^9-281/24/d*b^2/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)^3/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^9-13/4/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)^3/(a+b)*tanh(1/2*d
*x+1/2*c)-7/4/d*b^(5/2)/a^3/(a^3+3*a^2*b+3*a*b^2+b^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))

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [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(1/(a+b*sech(d*x+c)^2)^4,x, algorithm="fricas")

[Out]

Timed out

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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(1/(a+b*sech(d*x+c)**2)**4,x)

[Out]

Timed out

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Giac [B]  time = 1.18109, size = 811, normalized size = 3.92 \begin{align*} -\frac{{\left (35 \, a^{3} b + 70 \, a^{2} b^{2} + 56 \, a b^{3} + 16 \, b^{4}\right )} \arctan \left (\frac{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt{-a b - b^{2}}}\right )}{16 \,{\left (a^{7} d + 3 \, a^{6} b d + 3 \, a^{5} b^{2} d + a^{4} b^{3} d\right )} \sqrt{-a b - b^{2}}} + \frac{87 \, a^{5} b e^{\left (10 \, d x + 10 \, c\right )} + 366 \, a^{4} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 408 \, a^{3} b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 144 \, a^{2} b^{4} e^{\left (10 \, d x + 10 \, c\right )} + 435 \, a^{5} b e^{\left (8 \, d x + 8 \, c\right )} + 2124 \, a^{4} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 3972 \, a^{3} b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 3072 \, a^{2} b^{4} e^{\left (8 \, d x + 8 \, c\right )} + 864 \, a b^{5} e^{\left (8 \, d x + 8 \, c\right )} + 870 \, a^{5} b e^{\left (6 \, d x + 6 \, c\right )} + 4292 \, a^{4} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 8792 \, a^{3} b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 9936 \, a^{2} b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 5824 \, a b^{5} e^{\left (6 \, d x + 6 \, c\right )} + 1408 \, b^{6} e^{\left (6 \, d x + 6 \, c\right )} + 870 \, a^{5} b e^{\left (4 \, d x + 4 \, c\right )} + 3792 \, a^{4} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 6432 \, a^{3} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 4608 \, a^{2} b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 1248 \, a b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 435 \, a^{5} b e^{\left (2 \, d x + 2 \, c\right )} + 1374 \, a^{4} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 1248 \, a^{3} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 384 \, a^{2} b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 87 \, a^{5} b + 116 \, a^{4} b^{2} + 44 \, a^{3} b^{3}}{24 \,{\left (a^{7} d + 3 \, a^{6} b d + 3 \, a^{5} b^{2} d + a^{4} b^{3} 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 )}^{3}} + \frac{d x + c}{a^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sech(d*x+c)^2)^4,x, algorithm="giac")

[Out]

-1/16*(35*a^3*b + 70*a^2*b^2 + 56*a*b^3 + 16*b^4)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/(
(a^7*d + 3*a^6*b*d + 3*a^5*b^2*d + a^4*b^3*d)*sqrt(-a*b - b^2)) + 1/24*(87*a^5*b*e^(10*d*x + 10*c) + 366*a^4*b
^2*e^(10*d*x + 10*c) + 408*a^3*b^3*e^(10*d*x + 10*c) + 144*a^2*b^4*e^(10*d*x + 10*c) + 435*a^5*b*e^(8*d*x + 8*
c) + 2124*a^4*b^2*e^(8*d*x + 8*c) + 3972*a^3*b^3*e^(8*d*x + 8*c) + 3072*a^2*b^4*e^(8*d*x + 8*c) + 864*a*b^5*e^
(8*d*x + 8*c) + 870*a^5*b*e^(6*d*x + 6*c) + 4292*a^4*b^2*e^(6*d*x + 6*c) + 8792*a^3*b^3*e^(6*d*x + 6*c) + 9936
*a^2*b^4*e^(6*d*x + 6*c) + 5824*a*b^5*e^(6*d*x + 6*c) + 1408*b^6*e^(6*d*x + 6*c) + 870*a^5*b*e^(4*d*x + 4*c) +
3792*a^4*b^2*e^(4*d*x + 4*c) + 6432*a^3*b^3*e^(4*d*x + 4*c) + 4608*a^2*b^4*e^(4*d*x + 4*c) + 1248*a*b^5*e^(4*
d*x + 4*c) + 435*a^5*b*e^(2*d*x + 2*c) + 1374*a^4*b^2*e^(2*d*x + 2*c) + 1248*a^3*b^3*e^(2*d*x + 2*c) + 384*a^2
*b^4*e^(2*d*x + 2*c) + 87*a^5*b + 116*a^4*b^2 + 44*a^3*b^3)/((a^7*d + 3*a^6*b*d + 3*a^5*b^2*d + a^4*b^3*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)^3) + (d*x + c)/(a^4*d)