\(\int \frac {1}{(a+b \tanh ^2(c+d x))^4} \, dx\) [201]

Optimal result

Integrand size = 14, antiderivative size = 201 \[ \int \frac {1}{\left (a+b \tanh ^2(c+d x)\right )^4} \, dx=\frac {x}{(a+b)^4}+\frac {\sqrt {b} \left (35 a^3+35 a^2 b+21 a b^2+5 b^3\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{16 a^{7/2} (a+b)^4 d}+\frac {b \tanh (c+d x)}{6 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^3}+\frac {b (11 a+5 b) \tanh (c+d x)}{24 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b \left (19 a^2+16 a b+5 b^2\right ) \tanh (c+d x)}{16 a^3 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )} \]

[Out]

x/(a+b)^4+1/16*(35*a^3+35*a^2*b+21*a*b^2+5*b^3)*arctan(b^(1/2)*tanh(d*x+c)/a^(1/2))*b^(1/2)/a^(7/2)/(a+b)^4/d+
1/6*b*tanh(d*x+c)/a/(a+b)/d/(a+b*tanh(d*x+c)^2)^3+1/24*b*(11*a+5*b)*tanh(d*x+c)/a^2/(a+b)^2/d/(a+b*tanh(d*x+c)
^2)^2+1/16*b*(19*a^2+16*a*b+5*b^2)*tanh(d*x+c)/a^3/(a+b)^3/d/(a+b*tanh(d*x+c)^2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3742, 425, 541, 536, 212, 211} \[ \int \frac {1}{\left (a+b \tanh ^2(c+d x)\right )^4} \, dx=\frac {b (11 a+5 b) \tanh (c+d x)}{24 a^2 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b \left (19 a^2+16 a b+5 b^2\right ) \tanh (c+d x)}{16 a^3 d (a+b)^3 \left (a+b \tanh ^2(c+d x)\right )}+\frac {\sqrt {b} \left (35 a^3+35 a^2 b+21 a b^2+5 b^3\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{16 a^{7/2} d (a+b)^4}+\frac {b \tanh (c+d x)}{6 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )^3}+\frac {x}{(a+b)^4} \]

[In]

Int[(a + b*Tanh[c + d*x]^2)^(-4),x]

[Out]

x/(a + b)^4 + (Sqrt[b]*(35*a^3 + 35*a^2*b + 21*a*b^2 + 5*b^3)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(16*a^(
7/2)*(a + b)^4*d) + (b*Tanh[c + d*x])/(6*a*(a + b)*d*(a + b*Tanh[c + d*x]^2)^3) + (b*(11*a + 5*b)*Tanh[c + d*x
])/(24*a^2*(a + b)^2*d*(a + b*Tanh[c + d*x]^2)^2) + (b*(19*a^2 + 16*a*b + 5*b^2)*Tanh[c + d*x])/(16*a^3*(a + b
)^3*d*(a + b*Tanh[c + d*x]^2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 212

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

Rule 425

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]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 3742

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[c*(ff/f), Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a+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 \tanh ^2(c+d x)\right )^3}-\frac {\text {Subst}\left (\int \frac {b-6 (a+b)+5 b x^2}{\left (1-x^2\right ) \left (a+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 \tanh ^2(c+d x)\right )^3}+\frac {b (11 a+5 b) \tanh (c+d x)}{24 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {3 \left (8 a^2+11 a b+5 b^2\right )-3 b (11 a+5 b) x^2}{\left (1-x^2\right ) \left (a+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 \tanh ^2(c+d x)\right )^3}+\frac {b (11 a+5 b) \tanh (c+d x)}{24 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b \left (19 a^2+16 a b+5 b^2\right ) \tanh (c+d x)}{16 a^3 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-3 \left (16 a^3+19 a^2 b+16 a b^2+5 b^3\right )+3 b \left (19 a^2+16 a b+5 b^2\right ) x^2}{\left (1-x^2\right ) \left (a+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 \tanh ^2(c+d x)\right )^3}+\frac {b (11 a+5 b) \tanh (c+d x)}{24 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b \left (19 a^2+16 a b+5 b^2\right ) \tanh (c+d x)}{16 a^3 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{(a+b)^4 d}+\frac {\left (b \left (35 a^3+35 a^2 b+21 a b^2+5 b^3\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{16 a^3 (a+b)^4 d} \\ & = \frac {x}{(a+b)^4}+\frac {\sqrt {b} \left (35 a^3+35 a^2 b+21 a b^2+5 b^3\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{16 a^{7/2} (a+b)^4 d}+\frac {b \tanh (c+d x)}{6 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^3}+\frac {b (11 a+5 b) \tanh (c+d x)}{24 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b \left (19 a^2+16 a b+5 b^2\right ) \tanh (c+d x)}{16 a^3 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\left (a+b \tanh ^2(c+d x)\right )^4} \, dx=\frac {\frac {3 \sqrt {b} \left (35 a^3+35 a^2 b+21 a b^2+5 b^3\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{7/2}}-24 \log (1-\tanh (c+d x))+24 \log (1+\tanh (c+d x))+\frac {8 b (a+b)^3 \tanh (c+d x)}{a \left (a+b \tanh ^2(c+d x)\right )^3}+\frac {2 b (a+b)^2 (11 a+5 b) \tanh (c+d x)}{a^2 \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {3 b (a+b) \left (19 a^2+16 a b+5 b^2\right ) \tanh (c+d x)}{a^3 \left (a+b \tanh ^2(c+d x)\right )}}{48 (a+b)^4 d} \]

[In]

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

[Out]

((3*Sqrt[b]*(35*a^3 + 35*a^2*b + 21*a*b^2 + 5*b^3)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/a^(7/2) - 24*Log[1
- Tanh[c + d*x]] + 24*Log[1 + Tanh[c + d*x]] + (8*b*(a + b)^3*Tanh[c + d*x])/(a*(a + b*Tanh[c + d*x]^2)^3) +
(2*b*(a + b)^2*(11*a + 5*b)*Tanh[c + d*x])/(a^2*(a + b*Tanh[c + d*x]^2)^2) + (3*b*(a + b)*(19*a^2 + 16*a*b + 5
*b^2)*Tanh[c + d*x])/(a^3*(a + b*Tanh[c + d*x]^2)))/(48*(a + b)^4*d)

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{4}}-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{4}}+\frac {b \left (\frac {\frac {b^{2} \left (19 a^{3}+35 a^{2} b +21 a \,b^{2}+5 b^{3}\right ) \tanh \left (d x +c \right )^{5}}{16 a^{3}}+\frac {b \left (17 a^{3}+33 a^{2} b +21 a \,b^{2}+5 b^{3}\right ) \tanh \left (d x +c \right )^{3}}{6 a^{2}}+\frac {\left (29 a^{3}+61 a^{2} b +43 a \,b^{2}+11 b^{3}\right ) \tanh \left (d x +c \right )}{16 a}}{\left (a +b \tanh \left (d x +c \right )^{2}\right )^{3}}+\frac {\left (35 a^{3}+35 a^{2} b +21 a \,b^{2}+5 b^{3}\right ) \arctan \left (\frac {b \tanh \left (d x +c \right )}{\sqrt {a b}}\right )}{16 a^{3} \sqrt {a b}}\right )}{\left (a +b \right )^{4}}}{d}\)	\(219\)
default	\(\frac {\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{4}}-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{4}}+\frac {b \left (\frac {\frac {b^{2} \left (19 a^{3}+35 a^{2} b +21 a \,b^{2}+5 b^{3}\right ) \tanh \left (d x +c \right )^{5}}{16 a^{3}}+\frac {b \left (17 a^{3}+33 a^{2} b +21 a \,b^{2}+5 b^{3}\right ) \tanh \left (d x +c \right )^{3}}{6 a^{2}}+\frac {\left (29 a^{3}+61 a^{2} b +43 a \,b^{2}+11 b^{3}\right ) \tanh \left (d x +c \right )}{16 a}}{\left (a +b \tanh \left (d x +c \right )^{2}\right )^{3}}+\frac {\left (35 a^{3}+35 a^{2} b +21 a \,b^{2}+5 b^{3}\right ) \arctan \left (\frac {b \tanh \left (d x +c \right )}{\sqrt {a b}}\right )}{16 a^{3} \sqrt {a b}}\right )}{\left (a +b \right )^{4}}}{d}\)	\(219\)
risch	\(\text {Expression too large to display}\)	\(1031\)

[In]

int(1/(a+b*tanh(d*x+c)^2)^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/2/(a+b)^4*ln(tanh(d*x+c)+1)-1/2/(a+b)^4*ln(tanh(d*x+c)-1)+1/(a+b)^4*b*((1/16*b^2*(19*a^3+35*a^2*b+21*a*
b^2+5*b^3)/a^3*tanh(d*x+c)^5+1/6*b*(17*a^3+33*a^2*b+21*a*b^2+5*b^3)/a^2*tanh(d*x+c)^3+1/16*(29*a^3+61*a^2*b+43
*a*b^2+11*b^3)/a*tanh(d*x+c))/(a+b*tanh(d*x+c)^2)^3+1/16*(35*a^3+35*a^2*b+21*a*b^2+5*b^3)/a^3/(a*b)^(1/2)*arct
an(b*tanh(d*x+c)/(a*b)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 9849 vs. \(2 (185) = 370\).

Time = 0.51 (sec) , antiderivative size = 20020, normalized size of antiderivative = 99.60 \[ \int \frac {1}{\left (a+b \tanh ^2(c+d x)\right )^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \tanh ^2(c+d x)\right )^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 925 vs. \(2 (185) = 370\).

Time = 0.50 (sec) , antiderivative size = 925, normalized size of antiderivative = 4.60 \[ \int \frac {1}{\left (a+b \tanh ^2(c+d x)\right )^4} \, dx=-\frac {{\left (35 \, a^{3} b + 35 \, a^{2} b^{2} + 21 \, a b^{3} + 5 \, b^{4}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{16 \, {\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} \sqrt {a b} d} + \frac {87 \, a^{5} b + 319 \, a^{4} b^{2} + 450 \, a^{3} b^{3} + 306 \, a^{2} b^{4} + 103 \, a b^{5} + 15 \, b^{6} + 3 \, {\left (145 \, a^{5} b + 267 \, a^{4} b^{2} + 34 \, a^{3} b^{3} - 178 \, a^{2} b^{4} - 115 \, a b^{5} - 25 \, b^{6}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, {\left (145 \, a^{5} b + 93 \, a^{4} b^{2} - 6 \, a^{3} b^{3} + 106 \, a^{2} b^{4} + 85 \, a b^{5} + 25 \, b^{6}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, {\left (435 \, a^{5} b + 29 \, a^{4} b^{2} + 162 \, a^{3} b^{3} - 306 \, a^{2} b^{4} - 245 \, a b^{5} - 75 \, b^{6}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + 3 \, {\left (145 \, a^{5} b + 17 \, a^{4} b^{2} - 58 \, a^{3} b^{3} + 150 \, a^{2} b^{4} + 105 \, a b^{5} + 25 \, b^{6}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + 3 \, {\left (29 \, a^{5} b + 23 \, a^{4} b^{2} - 62 \, a^{3} b^{3} - 82 \, a^{2} b^{4} - 31 \, a b^{5} - 5 \, b^{6}\right )} e^{\left (-10 \, d x - 10 \, c\right )}}{24 \, {\left (a^{10} + 7 \, a^{9} b + 21 \, a^{8} b^{2} + 35 \, a^{7} b^{3} + 35 \, a^{6} b^{4} + 21 \, a^{5} b^{5} + 7 \, a^{4} b^{6} + a^{3} b^{7} + 6 \, {\left (a^{10} + 5 \, a^{9} b + 9 \, a^{8} b^{2} + 5 \, a^{7} b^{3} - 5 \, a^{6} b^{4} - 9 \, a^{5} b^{5} - 5 \, a^{4} b^{6} - a^{3} b^{7}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (5 \, a^{10} + 19 \, a^{9} b + 25 \, a^{8} b^{2} + 15 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 25 \, a^{5} b^{5} + 19 \, a^{4} b^{6} + 5 \, a^{3} b^{7}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (5 \, a^{10} + 17 \, a^{9} b + 21 \, a^{8} b^{2} + 9 \, a^{7} b^{3} - 9 \, a^{6} b^{4} - 21 \, a^{5} b^{5} - 17 \, a^{4} b^{6} - 5 \, a^{3} b^{7}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + 3 \, {\left (5 \, a^{10} + 19 \, a^{9} b + 25 \, a^{8} b^{2} + 15 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 25 \, a^{5} b^{5} + 19 \, a^{4} b^{6} + 5 \, a^{3} b^{7}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, {\left (a^{10} + 5 \, a^{9} b + 9 \, a^{8} b^{2} + 5 \, a^{7} b^{3} - 5 \, a^{6} b^{4} - 9 \, a^{5} b^{5} - 5 \, a^{4} b^{6} - a^{3} b^{7}\right )} e^{\left (-10 \, d x - 10 \, c\right )} + {\left (a^{10} + 7 \, a^{9} b + 21 \, a^{8} b^{2} + 35 \, a^{7} b^{3} + 35 \, a^{6} b^{4} + 21 \, a^{5} b^{5} + 7 \, a^{4} b^{6} + a^{3} b^{7}\right )} e^{\left (-12 \, d x - 12 \, c\right )}\right )} d} + \frac {d x + c}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d} \]

[In]

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

[Out]

-1/16*(35*a^3*b + 35*a^2*b^2 + 21*a*b^3 + 5*b^4)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/((a^
7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*sqrt(a*b)*d) + 1/24*(87*a^5*b + 319*a^4*b^2 + 450*a^3*b^3 + 306
*a^2*b^4 + 103*a*b^5 + 15*b^6 + 3*(145*a^5*b + 267*a^4*b^2 + 34*a^3*b^3 - 178*a^2*b^4 - 115*a*b^5 - 25*b^6)*e^
(-2*d*x - 2*c) + 6*(145*a^5*b + 93*a^4*b^2 - 6*a^3*b^3 + 106*a^2*b^4 + 85*a*b^5 + 25*b^6)*e^(-4*d*x - 4*c) + 2
*(435*a^5*b + 29*a^4*b^2 + 162*a^3*b^3 - 306*a^2*b^4 - 245*a*b^5 - 75*b^6)*e^(-6*d*x - 6*c) + 3*(145*a^5*b + 1
7*a^4*b^2 - 58*a^3*b^3 + 150*a^2*b^4 + 105*a*b^5 + 25*b^6)*e^(-8*d*x - 8*c) + 3*(29*a^5*b + 23*a^4*b^2 - 62*a^
3*b^3 - 82*a^2*b^4 - 31*a*b^5 - 5*b^6)*e^(-10*d*x - 10*c))/((a^10 + 7*a^9*b + 21*a^8*b^2 + 35*a^7*b^3 + 35*a^6
*b^4 + 21*a^5*b^5 + 7*a^4*b^6 + a^3*b^7 + 6*(a^10 + 5*a^9*b + 9*a^8*b^2 + 5*a^7*b^3 - 5*a^6*b^4 - 9*a^5*b^5 -
5*a^4*b^6 - a^3*b^7)*e^(-2*d*x - 2*c) + 3*(5*a^10 + 19*a^9*b + 25*a^8*b^2 + 15*a^7*b^3 + 15*a^6*b^4 + 25*a^5*b
^5 + 19*a^4*b^6 + 5*a^3*b^7)*e^(-4*d*x - 4*c) + 4*(5*a^10 + 17*a^9*b + 21*a^8*b^2 + 9*a^7*b^3 - 9*a^6*b^4 - 21
*a^5*b^5 - 17*a^4*b^6 - 5*a^3*b^7)*e^(-6*d*x - 6*c) + 3*(5*a^10 + 19*a^9*b + 25*a^8*b^2 + 15*a^7*b^3 + 15*a^6*
b^4 + 25*a^5*b^5 + 19*a^4*b^6 + 5*a^3*b^7)*e^(-8*d*x - 8*c) + 6*(a^10 + 5*a^9*b + 9*a^8*b^2 + 5*a^7*b^3 - 5*a^
6*b^4 - 9*a^5*b^5 - 5*a^4*b^6 - a^3*b^7)*e^(-10*d*x - 10*c) + (a^10 + 7*a^9*b + 21*a^8*b^2 + 35*a^7*b^3 + 35*a
^6*b^4 + 21*a^5*b^5 + 7*a^4*b^6 + a^3*b^7)*e^(-12*d*x - 12*c))*d) + (d*x + c)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*
a*b^3 + b^4)*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 750 vs. \(2 (185) = 370\).

Time = 0.36 (sec) , antiderivative size = 750, normalized size of antiderivative = 3.73 \[ \int \frac {1}{\left (a+b \tanh ^2(c+d x)\right )^4} \, dx=\frac {\frac {3 \, {\left (35 \, a^{3} b + 35 \, a^{2} b^{2} + 21 \, a b^{3} + 5 \, b^{4}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{{\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} \sqrt {a b}} + \frac {48 \, {\left (d x + c\right )}}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {2 \, {\left (87 \, a^{5} b e^{\left (10 \, d x + 10 \, c\right )} + 69 \, a^{4} b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 186 \, a^{3} b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 246 \, a^{2} b^{4} e^{\left (10 \, d x + 10 \, c\right )} - 93 \, a b^{5} e^{\left (10 \, d x + 10 \, c\right )} - 15 \, b^{6} e^{\left (10 \, d x + 10 \, c\right )} + 435 \, a^{5} b e^{\left (8 \, d x + 8 \, c\right )} + 51 \, a^{4} b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 174 \, a^{3} b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 450 \, a^{2} b^{4} e^{\left (8 \, d x + 8 \, c\right )} + 315 \, a b^{5} e^{\left (8 \, d x + 8 \, c\right )} + 75 \, b^{6} e^{\left (8 \, d x + 8 \, c\right )} + 870 \, a^{5} b e^{\left (6 \, d x + 6 \, c\right )} + 58 \, a^{4} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 324 \, a^{3} b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 612 \, a^{2} b^{4} e^{\left (6 \, d x + 6 \, c\right )} - 490 \, a b^{5} e^{\left (6 \, d x + 6 \, c\right )} - 150 \, b^{6} e^{\left (6 \, d x + 6 \, c\right )} + 870 \, a^{5} b e^{\left (4 \, d x + 4 \, c\right )} + 558 \, a^{4} b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 36 \, a^{3} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 636 \, a^{2} b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 510 \, a b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 150 \, b^{6} e^{\left (4 \, d x + 4 \, c\right )} + 435 \, a^{5} b e^{\left (2 \, d x + 2 \, c\right )} + 801 \, a^{4} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 102 \, a^{3} b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 534 \, a^{2} b^{4} e^{\left (2 \, d x + 2 \, c\right )} - 345 \, a b^{5} e^{\left (2 \, d x + 2 \, c\right )} - 75 \, b^{6} e^{\left (2 \, d x + 2 \, c\right )} + 87 \, a^{5} b + 319 \, a^{4} b^{2} + 450 \, a^{3} b^{3} + 306 \, a^{2} b^{4} + 103 \, a b^{5} + 15 \, b^{6}\right )}}{{\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}^{3}}}{48 \, d} \]

[In]

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

[Out]

1/48*(3*(35*a^3*b + 35*a^2*b^2 + 21*a*b^3 + 5*b^4)*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/
sqrt(a*b))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*sqrt(a*b)) + 48*(d*x + c)/(a^4 + 4*a^3*b + 6*a^2
*b^2 + 4*a*b^3 + b^4) - 2*(87*a^5*b*e^(10*d*x + 10*c) + 69*a^4*b^2*e^(10*d*x + 10*c) - 186*a^3*b^3*e^(10*d*x +
10*c) - 246*a^2*b^4*e^(10*d*x + 10*c) - 93*a*b^5*e^(10*d*x + 10*c) - 15*b^6*e^(10*d*x + 10*c) + 435*a^5*b*e^(
8*d*x + 8*c) + 51*a^4*b^2*e^(8*d*x + 8*c) - 174*a^3*b^3*e^(8*d*x + 8*c) + 450*a^2*b^4*e^(8*d*x + 8*c) + 315*a*
b^5*e^(8*d*x + 8*c) + 75*b^6*e^(8*d*x + 8*c) + 870*a^5*b*e^(6*d*x + 6*c) + 58*a^4*b^2*e^(6*d*x + 6*c) + 324*a^
3*b^3*e^(6*d*x + 6*c) - 612*a^2*b^4*e^(6*d*x + 6*c) - 490*a*b^5*e^(6*d*x + 6*c) - 150*b^6*e^(6*d*x + 6*c) + 87
0*a^5*b*e^(4*d*x + 4*c) + 558*a^4*b^2*e^(4*d*x + 4*c) - 36*a^3*b^3*e^(4*d*x + 4*c) + 636*a^2*b^4*e^(4*d*x + 4*
c) + 510*a*b^5*e^(4*d*x + 4*c) + 150*b^6*e^(4*d*x + 4*c) + 435*a^5*b*e^(2*d*x + 2*c) + 801*a^4*b^2*e^(2*d*x +
2*c) + 102*a^3*b^3*e^(2*d*x + 2*c) - 534*a^2*b^4*e^(2*d*x + 2*c) - 345*a*b^5*e^(2*d*x + 2*c) - 75*b^6*e^(2*d*x
+ 2*c) + 87*a^5*b + 319*a^4*b^2 + 450*a^3*b^3 + 306*a^2*b^4 + 103*a*b^5 + 15*b^6)/((a^7 + 4*a^6*b + 6*a^5*b^2
+ 4*a^4*b^3 + a^3*b^4)*(a*e^(4*d*x + 4*c) + b*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + a
+ b)^3))/d

Mupad [B] (verification not implemented)

Time = 1.26 (sec) , antiderivative size = 3685, normalized size of antiderivative = 18.33 \[ \int \frac {1}{\left (a+b \tanh ^2(c+d x)\right )^4} \, dx=\text {Too large to display} \]

[In]

int(1/(a + b*tanh(c + d*x)^2)^4,x)

[Out]

log(tanh(c + d*x) + 1)/(2*a^4*d + 2*b^4*d + 12*a^2*b^2*d + 8*a*b^3*d + 8*a^3*b*d) + ((tanh(c + d*x)^3*(16*a*b^
3 + 5*b^4 + 17*a^2*b^2))/(6*a^2*(3*a*b^2 + 3*a^2*b + a^3 + b^3)) + (tanh(c + d*x)*(32*a*b^2 + 29*a^2*b + 11*b^
3))/(16*a*(3*a*b^2 + 3*a^2*b + a^3 + b^3)) + (b^2*tanh(c + d*x)^5*(16*a*b^2 + 19*a^2*b + 5*b^3))/(16*a^2*(a*b^
3 + 3*a^3*b + a^4 + 3*a^2*b^2)))/(a^3*d + b^3*d*tanh(c + d*x)^6 + 3*a^2*b*d*tanh(c + d*x)^2 + 3*a*b^2*d*tanh(c
+ d*x)^4) - log(tanh(c + d*x) - 1)/(2*d*(a + b)^4) - (atan((((-a^7*b)^(1/2)*((tanh(c + d*x)*(210*a*b^8 + 25*b
^9 + 791*a^2*b^7 + 1820*a^3*b^6 + 2695*a^4*b^5 + 2450*a^5*b^4 + 1481*a^6*b^3))/(128*(a^12*d^2 + 6*a^11*b*d^2 +
a^6*b^6*d^2 + 6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)) + ((((5*a^3*b^13*d^2)/4 + 1
4*a^4*b^12*d^2 + (287*a^5*b^11*d^2)/4 + 224*a^6*b^10*d^2 + (953*a^7*b^9*d^2)/2 + 728*a^8*b^8*d^2 + (1631*a^9*b
^7*d^2)/2 + 668*a^10*b^6*d^2 + (1561*a^11*b^5*d^2)/4 + 154*a^12*b^4*d^2 + (147*a^13*b^3*d^2)/4 + 4*a^14*b^2*d^
2)/(a^15*d^3 + 9*a^14*b*d^3 + a^6*b^9*d^3 + 9*a^7*b^8*d^3 + 36*a^8*b^7*d^3 + 84*a^9*b^6*d^3 + 126*a^10*b^5*d^3
+ 126*a^11*b^4*d^3 + 84*a^12*b^3*d^3 + 36*a^13*b^2*d^3) - (tanh(c + d*x)*(-a^7*b)^(1/2)*(21*a*b^2 + 35*a^2*b
+ 35*a^3 + 5*b^3)*(1024*a^6*b^11*d^2 + 7168*a^7*b^10*d^2 + 20480*a^8*b^9*d^2 + 28672*a^9*b^8*d^2 + 14336*a^10*
b^7*d^2 - 14336*a^11*b^6*d^2 - 28672*a^12*b^5*d^2 - 20480*a^13*b^4*d^2 - 7168*a^14*b^3*d^2 - 1024*a^15*b^2*d^2
))/(4096*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)*(a^12*d^2 + 6*a^11*b*d^2 + a^6*b^6*d^2
+ 6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)))*(-a^7*b)^(1/2)*(21*a*b^2 + 35*a^2*b + 3
5*a^3 + 5*b^3))/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)))*(21*a*b^2 + 35*a^2*b + 35*
a^3 + 5*b^3)*1i)/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)) + ((-a^7*b)^(1/2)*((tanh(c
+ d*x)*(210*a*b^8 + 25*b^9 + 791*a^2*b^7 + 1820*a^3*b^6 + 2695*a^4*b^5 + 2450*a^5*b^4 + 1481*a^6*b^3))/(128*(
a^12*d^2 + 6*a^11*b*d^2 + a^6*b^6*d^2 + 6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)) -
((((5*a^3*b^13*d^2)/4 + 14*a^4*b^12*d^2 + (287*a^5*b^11*d^2)/4 + 224*a^6*b^10*d^2 + (953*a^7*b^9*d^2)/2 + 728*
a^8*b^8*d^2 + (1631*a^9*b^7*d^2)/2 + 668*a^10*b^6*d^2 + (1561*a^11*b^5*d^2)/4 + 154*a^12*b^4*d^2 + (147*a^13*b
^3*d^2)/4 + 4*a^14*b^2*d^2)/(a^15*d^3 + 9*a^14*b*d^3 + a^6*b^9*d^3 + 9*a^7*b^8*d^3 + 36*a^8*b^7*d^3 + 84*a^9*b
^6*d^3 + 126*a^10*b^5*d^3 + 126*a^11*b^4*d^3 + 84*a^12*b^3*d^3 + 36*a^13*b^2*d^3) + (tanh(c + d*x)*(-a^7*b)^(1
/2)*(21*a*b^2 + 35*a^2*b + 35*a^3 + 5*b^3)*(1024*a^6*b^11*d^2 + 7168*a^7*b^10*d^2 + 20480*a^8*b^9*d^2 + 28672*
a^9*b^8*d^2 + 14336*a^10*b^7*d^2 - 14336*a^11*b^6*d^2 - 28672*a^12*b^5*d^2 - 20480*a^13*b^4*d^2 - 7168*a^14*b^
3*d^2 - 1024*a^15*b^2*d^2))/(4096*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)*(a^12*d^2 + 6*
a^11*b*d^2 + a^6*b^6*d^2 + 6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)))*(-a^7*b)^(1/2)
*(21*a*b^2 + 35*a^2*b + 35*a^3 + 5*b^3))/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)))*(
21*a*b^2 + 35*a^2*b + 35*a^3 + 5*b^3)*1i)/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)))/
(((185*a*b^7)/128 + (25*b^8)/128 + (303*a^2*b^6)/64 + (567*a^3*b^5)/64 + (1225*a^4*b^4)/128 + (665*a^5*b^3)/12
8)/(a^15*d^3 + 9*a^14*b*d^3 + a^6*b^9*d^3 + 9*a^7*b^8*d^3 + 36*a^8*b^7*d^3 + 84*a^9*b^6*d^3 + 126*a^10*b^5*d^3
+ 126*a^11*b^4*d^3 + 84*a^12*b^3*d^3 + 36*a^13*b^2*d^3) + ((-a^7*b)^(1/2)*((tanh(c + d*x)*(210*a*b^8 + 25*b^9
+ 791*a^2*b^7 + 1820*a^3*b^6 + 2695*a^4*b^5 + 2450*a^5*b^4 + 1481*a^6*b^3))/(128*(a^12*d^2 + 6*a^11*b*d^2 + a
^6*b^6*d^2 + 6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)) + ((((5*a^3*b^13*d^2)/4 + 14*
a^4*b^12*d^2 + (287*a^5*b^11*d^2)/4 + 224*a^6*b^10*d^2 + (953*a^7*b^9*d^2)/2 + 728*a^8*b^8*d^2 + (1631*a^9*b^7
*d^2)/2 + 668*a^10*b^6*d^2 + (1561*a^11*b^5*d^2)/4 + 154*a^12*b^4*d^2 + (147*a^13*b^3*d^2)/4 + 4*a^14*b^2*d^2)
/(a^15*d^3 + 9*a^14*b*d^3 + a^6*b^9*d^3 + 9*a^7*b^8*d^3 + 36*a^8*b^7*d^3 + 84*a^9*b^6*d^3 + 126*a^10*b^5*d^3 +
126*a^11*b^4*d^3 + 84*a^12*b^3*d^3 + 36*a^13*b^2*d^3) - (tanh(c + d*x)*(-a^7*b)^(1/2)*(21*a*b^2 + 35*a^2*b +
35*a^3 + 5*b^3)*(1024*a^6*b^11*d^2 + 7168*a^7*b^10*d^2 + 20480*a^8*b^9*d^2 + 28672*a^9*b^8*d^2 + 14336*a^10*b^
7*d^2 - 14336*a^11*b^6*d^2 - 28672*a^12*b^5*d^2 - 20480*a^13*b^4*d^2 - 7168*a^14*b^3*d^2 - 1024*a^15*b^2*d^2))
/(4096*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)*(a^12*d^2 + 6*a^11*b*d^2 + a^6*b^6*d^2 +
6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)))*(-a^7*b)^(1/2)*(21*a*b^2 + 35*a^2*b + 35*
a^3 + 5*b^3))/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)))*(21*a*b^2 + 35*a^2*b + 35*a^
3 + 5*b^3))/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)) - ((-a^7*b)^(1/2)*((tanh(c + d*
x)*(210*a*b^8 + 25*b^9 + 791*a^2*b^7 + 1820*a^3*b^6 + 2695*a^4*b^5 + 2450*a^5*b^4 + 1481*a^6*b^3))/(128*(a^12*
d^2 + 6*a^11*b*d^2 + a^6*b^6*d^2 + 6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)) - ((((5
*a^3*b^13*d^2)/4 + 14*a^4*b^12*d^2 + (287*a^5*b^11*d^2)/4 + 224*a^6*b^10*d^2 + (953*a^7*b^9*d^2)/2 + 728*a^8*b
^8*d^2 + (1631*a^9*b^7*d^2)/2 + 668*a^10*b^6*d^2 + (1561*a^11*b^5*d^2)/4 + 154*a^12*b^4*d^2 + (147*a^13*b^3*d^
2)/4 + 4*a^14*b^2*d^2)/(a^15*d^3 + 9*a^14*b*d^3 + a^6*b^9*d^3 + 9*a^7*b^8*d^3 + 36*a^8*b^7*d^3 + 84*a^9*b^6*d^
3 + 126*a^10*b^5*d^3 + 126*a^11*b^4*d^3 + 84*a^12*b^3*d^3 + 36*a^13*b^2*d^3) + (tanh(c + d*x)*(-a^7*b)^(1/2)*(
21*a*b^2 + 35*a^2*b + 35*a^3 + 5*b^3)*(1024*a^6*b^11*d^2 + 7168*a^7*b^10*d^2 + 20480*a^8*b^9*d^2 + 28672*a^9*b
^8*d^2 + 14336*a^10*b^7*d^2 - 14336*a^11*b^6*d^2 - 28672*a^12*b^5*d^2 - 20480*a^13*b^4*d^2 - 7168*a^14*b^3*d^2
- 1024*a^15*b^2*d^2))/(4096*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)*(a^12*d^2 + 6*a^11*
b*d^2 + a^6*b^6*d^2 + 6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)))*(-a^7*b)^(1/2)*(21*
a*b^2 + 35*a^2*b + 35*a^3 + 5*b^3))/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)))*(21*a*
b^2 + 35*a^2*b + 35*a^3 + 5*b^3))/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d))))*(-a^7*b
)^(1/2)*(21*a*b^2 + 35*a^2*b + 35*a^3 + 5*b^3)*1i)/(16*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^1
0*b*d))