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

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

[Out]

x/a^4-1/16*(35*a^3+70*a^2*b+56*a*b^2+16*b^3)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))*b^(1/2)/a^4/(a+b)^(7/2)/
d-1/6*b*tanh(d*x+c)/a/(a+b)/d/(a+b-b*tanh(d*x+c)^2)^3-1/24*b*(11*a+6*b)*tanh(d*x+c)/a^2/(a+b)^2/d/(a+b-b*tanh(
d*x+c)^2)^2-1/16*b*(19*a^2+22*a*b+8*b^2)*tanh(d*x+c)/a^3/(a+b)^3/d/(a+b-b*tanh(d*x+c)^2)

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Rubi [A]  time = 0.35, antiderivative size = 207, normalized size of antiderivative = 1.00, 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 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]

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 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 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 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]

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*}

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Mathematica [C]  time = 6.93, 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]*(((-1/2*I)*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*Sqrt[
a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) - ((I/256)*b*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqr
t[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*Co
sh[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*C
osh[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*C
osh[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*Co
sh[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*Co
sh[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*Sin
h[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] - 13
74*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)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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giac [B]  time = 0.85, size = 594, normalized size = 2.87 \[ -\frac {\frac {3 \, {\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 )}{{\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} \sqrt {-a b - b^{2}}} - \frac {2 \, {\left (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}\right )}}{{\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\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 {48 \, {\left (d x + c\right )}}{a^{4}}}{48 \, d} \]

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]

-1/48*(3*(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 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*sqrt(-a*b - b^2)) - 2*(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) + 212
4*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 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*(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) - 48*(d*x + c)/a^4)/d

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maple [B]  time = 0.45, size = 2880, normalized size = 13.91 \[ \text {output too large to display} \]

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

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maxima [B]  time = 0.84, size = 718, normalized size = 3.47 \[ \frac {{\left (35 \, a^{3} b + 70 \, a^{2} b^{2} + 56 \, 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 )}{32 \, {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {87 \, a^{5} b + 116 \, a^{4} b^{2} + 44 \, a^{3} b^{3} + 3 \, {\left (145 \, a^{5} b + 458 \, a^{4} b^{2} + 416 \, a^{3} b^{3} + 128 \, a^{2} b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, {\left (145 \, a^{5} b + 632 \, a^{4} b^{2} + 1072 \, a^{3} b^{3} + 768 \, a^{2} b^{4} + 208 \, a b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, {\left (435 \, a^{5} b + 2146 \, a^{4} b^{2} + 4396 \, a^{3} b^{3} + 4968 \, a^{2} b^{4} + 2912 \, a b^{5} + 704 \, b^{6}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + 3 \, {\left (145 \, a^{5} b + 708 \, a^{4} b^{2} + 1324 \, a^{3} b^{3} + 1024 \, a^{2} b^{4} + 288 \, a b^{5}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + 3 \, {\left (29 \, a^{5} b + 122 \, a^{4} b^{2} + 136 \, a^{3} b^{3} + 48 \, a^{2} b^{4}\right )} e^{\left (-10 \, d x - 10 \, c\right )}}{24 \, {\left (a^{10} + 3 \, a^{9} b + 3 \, a^{8} b^{2} + a^{7} b^{3} + 6 \, {\left (a^{10} + 5 \, a^{9} b + 9 \, a^{8} b^{2} + 7 \, a^{7} b^{3} + 2 \, a^{6} b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (5 \, a^{10} + 31 \, a^{9} b + 79 \, a^{8} b^{2} + 101 \, a^{7} b^{3} + 64 \, a^{6} b^{4} + 16 \, a^{5} b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (5 \, a^{10} + 33 \, a^{9} b + 93 \, a^{8} b^{2} + 147 \, a^{7} b^{3} + 138 \, a^{6} b^{4} + 72 \, a^{5} b^{5} + 16 \, a^{4} b^{6}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + 3 \, {\left (5 \, a^{10} + 31 \, a^{9} b + 79 \, a^{8} b^{2} + 101 \, a^{7} b^{3} + 64 \, a^{6} b^{4} + 16 \, a^{5} b^{5}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, {\left (a^{10} + 5 \, a^{9} b + 9 \, a^{8} b^{2} + 7 \, a^{7} b^{3} + 2 \, a^{6} b^{4}\right )} e^{\left (-10 \, d x - 10 \, c\right )} + {\left (a^{10} + 3 \, a^{9} b + 3 \, a^{8} b^{2} + a^{7} b^{3}\right )} e^{\left (-12 \, d x - 12 \, c\right )}\right )} d} + \frac {d x + c}{a^{4} d} \]

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]

1/32*(35*a^3*b + 70*a^2*b^2 + 56*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^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*sqrt((a + b)*b)*d) - 1/24
*(87*a^5*b + 116*a^4*b^2 + 44*a^3*b^3 + 3*(145*a^5*b + 458*a^4*b^2 + 416*a^3*b^3 + 128*a^2*b^4)*e^(-2*d*x - 2*
c) + 6*(145*a^5*b + 632*a^4*b^2 + 1072*a^3*b^3 + 768*a^2*b^4 + 208*a*b^5)*e^(-4*d*x - 4*c) + 2*(435*a^5*b + 21
46*a^4*b^2 + 4396*a^3*b^3 + 4968*a^2*b^4 + 2912*a*b^5 + 704*b^6)*e^(-6*d*x - 6*c) + 3*(145*a^5*b + 708*a^4*b^2
+ 1324*a^3*b^3 + 1024*a^2*b^4 + 288*a*b^5)*e^(-8*d*x - 8*c) + 3*(29*a^5*b + 122*a^4*b^2 + 136*a^3*b^3 + 48*a^
2*b^4)*e^(-10*d*x - 10*c))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3 + 6*(a^10 + 5*a^9*b + 9*a^8*b^2 + 7*a^7*b^3
+ 2*a^6*b^4)*e^(-2*d*x - 2*c) + 3*(5*a^10 + 31*a^9*b + 79*a^8*b^2 + 101*a^7*b^3 + 64*a^6*b^4 + 16*a^5*b^5)*e^(
-4*d*x - 4*c) + 4*(5*a^10 + 33*a^9*b + 93*a^8*b^2 + 147*a^7*b^3 + 138*a^6*b^4 + 72*a^5*b^5 + 16*a^4*b^6)*e^(-6
*d*x - 6*c) + 3*(5*a^10 + 31*a^9*b + 79*a^8*b^2 + 101*a^7*b^3 + 64*a^6*b^4 + 16*a^5*b^5)*e^(-8*d*x - 8*c) + 6*
(a^10 + 5*a^9*b + 9*a^8*b^2 + 7*a^7*b^3 + 2*a^6*b^4)*e^(-10*d*x - 10*c) + (a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^
3)*e^(-12*d*x - 12*c))*d) + (d*x + c)/(a^4*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/cosh(c + d*x)^2)^4,x)

[Out]

int(1/(a + b/cosh(c + d*x)^2)^4, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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