\(\int \frac {\tanh ^4(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx\) [192]

Optimal result

Integrand size = 23, antiderivative size = 137 \[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {x}{(a+b)^3}-\frac {\left (a^2+6 a b-3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 \sqrt {a} b^{3/2} (a+b)^3 d}+\frac {a \tanh (c+d x)}{4 b (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(a+5 b) \tanh (c+d x)}{8 b (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )} \]

[Out]

x/(a+b)^3-1/8*(a^2+6*a*b-3*b^2)*arctan(b^(1/2)*tanh(d*x+c)/a^(1/2))/b^(3/2)/(a+b)^3/d/a^(1/2)+1/4*a*tanh(d*x+c
)/b/(a+b)/d/(a+b*tanh(d*x+c)^2)^2-1/8*(a+5*b)*tanh(d*x+c)/b/(a+b)^2/d/(a+b*tanh(d*x+c)^2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3751, 481, 541, 536, 212, 211} \[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {\left (a^2+6 a b-3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 \sqrt {a} b^{3/2} d (a+b)^3}-\frac {(a+5 b) \tanh (c+d x)}{8 b d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}+\frac {a \tanh (c+d x)}{4 b d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {x}{(a+b)^3} \]

[In]

Int[Tanh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

x/(a + b)^3 - ((a^2 + 6*a*b - 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*Sqrt[a]*b^(3/2)*(a + b)^3*d)
+ (a*Tanh[c + d*x])/(4*b*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) - ((a + 5*b)*Tanh[c + d*x])/(8*b*(a + b)^2*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 481

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 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 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((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[(d*ff*(x/c))^m*((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, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

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

Mathematica [A] (verified)

Time = 1.27 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.99 \[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {8 (c+d x)-\frac {\left (a^2+6 a b-3 b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {4 a (a+b) \sinh (2 (c+d x))}{(a-b+(a+b) \cosh (2 (c+d x)))^2}+\frac {(a-5 b) (a+b) \sinh (2 (c+d x))}{b (a-b+(a+b) \cosh (2 (c+d x)))}}{8 (a+b)^3 d} \]

[In]

Integrate[Tanh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(\frac {-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{3}}-\frac {\frac {\left (\frac {1}{8} a^{2}+\frac {3}{4} a b +\frac {5}{8} b^{2}\right ) \tanh \left (d x +c \right )^{3}-\frac {a \left (a^{2}-2 a b -3 b^{2}\right ) \tanh \left (d x +c \right )}{8 b}}{\left (a +b \tanh \left (d x +c \right )^{2}\right )^{2}}+\frac {\left (a^{2}+6 a b -3 b^{2}\right ) \arctan \left (\frac {b \tanh \left (d x +c \right )}{\sqrt {a b}}\right )}{8 b \sqrt {a b}}}{\left (a +b \right )^{3}}}{d}\)	\(148\)
default	\(\frac {-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{3}}-\frac {\frac {\left (\frac {1}{8} a^{2}+\frac {3}{4} a b +\frac {5}{8} b^{2}\right ) \tanh \left (d x +c \right )^{3}-\frac {a \left (a^{2}-2 a b -3 b^{2}\right ) \tanh \left (d x +c \right )}{8 b}}{\left (a +b \tanh \left (d x +c \right )^{2}\right )^{2}}+\frac {\left (a^{2}+6 a b -3 b^{2}\right ) \arctan \left (\frac {b \tanh \left (d x +c \right )}{\sqrt {a b}}\right )}{8 b \sqrt {a b}}}{\left (a +b \right )^{3}}}{d}\)	\(148\)
risch	\(\frac {x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {a^{3} {\mathrm e}^{6 d x +6 c}-9 a^{2} b \,{\mathrm e}^{6 d x +6 c}-5 a \,b^{2} {\mathrm e}^{6 d x +6 c}+5 \,{\mathrm e}^{6 d x +6 c} b^{3}+3 a^{3} {\mathrm e}^{4 d x +4 c}-17 a^{2} b \,{\mathrm e}^{4 d x +4 c}+13 a \,b^{2} {\mathrm e}^{4 d x +4 c}-15 \,{\mathrm e}^{4 d x +4 c} b^{3}+3 a^{3} {\mathrm e}^{2 d x +2 c}-11 a^{2} b \,{\mathrm e}^{2 d x +2 c}+{\mathrm e}^{2 d x +2 c} a \,b^{2}+15 \,{\mathrm e}^{2 d x +2 c} b^{3}+a^{3}-3 a^{2} b -9 a \,b^{2}-5 b^{3}}{4 b \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} d \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a^{2}}{16 \sqrt {-a b}\, \left (a +b \right )^{3} d b}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a}{8 \sqrt {-a b}\, \left (a +b \right )^{3} d}+\frac {3 b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{16 \sqrt {-a b}\, \left (a +b \right )^{3} d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a^{2}}{16 \sqrt {-a b}\, \left (a +b \right )^{3} d b}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a}{8 \sqrt {-a b}\, \left (a +b \right )^{3} d}-\frac {3 b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{16 \sqrt {-a b}\, \left (a +b \right )^{3} d}\)	\(677\)

[In]

int(tanh(d*x+c)^4/(a+b*tanh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2/(a+b)^3*ln(tanh(d*x+c)-1)+1/2/(a+b)^3*ln(tanh(d*x+c)+1)-1/(a+b)^3*(((1/8*a^2+3/4*a*b+5/8*b^2)*tanh(d
*x+c)^3-1/8*a*(a^2-2*a*b-3*b^2)/b*tanh(d*x+c))/(a+b*tanh(d*x+c)^2)^2+1/8*(a^2+6*a*b-3*b^2)/b/(a*b)^(1/2)*arcta
n(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. 3727 vs. \(2 (123) = 246\).

Time = 0.40 (sec) , antiderivative size = 7757, normalized size of antiderivative = 56.62 \[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(tanh(d*x+c)^4/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(tanh(d*x+c)**4/(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2432 vs. \(2 (123) = 246\).

Time = 0.95 (sec) , antiderivative size = 2432, normalized size of antiderivative = 17.75 \[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(tanh(d*x+c)^4/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

-1/128*(a^4 + 24*a^3*b - 54*a^2*b^2 - 16*a*b^3 - 3*b^4)*arctan(1/2*((a + b)*e^(2*d*x + 2*c) + a - b)/sqrt(a*b)
)/((a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*sqrt(a*b)*d) + 1/128*(a^4 + 24*a^3*b - 54*a^2*b^2 - 16*a*b^3 - 3*
b^4)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/((a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*sqrt(
a*b)*d) - 1/64*(a^5 - 33*a^4*b - 54*a^3*b^2 - 2*a^2*b^3 + 21*a*b^4 + 3*b^5 + (a^5 - 71*a^4*b + 98*a^3*b^2 + 15
4*a^2*b^3 - 19*a*b^4 - 3*b^5)*e^(6*d*x + 6*c) + (3*a^5 - 171*a^4*b + 310*a^3*b^2 - 254*a^2*b^3 + 39*a*b^4 + 9*
b^5)*e^(4*d*x + 4*c) + (3*a^5 - 133*a^4*b + 86*a^3*b^2 + 190*a^2*b^3 - 41*a*b^4 - 9*b^5)*e^(2*d*x + 2*c))/((a^
7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6 + (a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4
+ 5*a^3*b^5 + a^2*b^6)*e^(8*d*x + 8*c) + 4*(a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*
e^(6*d*x + 6*c) + 2*(3*a^7*b + 7*a^6*b^2 + 6*a^5*b^3 + 6*a^4*b^4 + 7*a^3*b^5 + 3*a^2*b^6)*e^(4*d*x + 4*c) + 4*
(a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*e^(2*d*x + 2*c))*d) + 1/64*(a^5 - 33*a^4*b -
54*a^3*b^2 - 2*a^2*b^3 + 21*a*b^4 + 3*b^5 + (3*a^5 - 133*a^4*b + 86*a^3*b^2 + 190*a^2*b^3 - 41*a*b^4 - 9*b^5)
*e^(-2*d*x - 2*c) + (3*a^5 - 171*a^4*b + 310*a^3*b^2 - 254*a^2*b^3 + 39*a*b^4 + 9*b^5)*e^(-4*d*x - 4*c) + (a^5
- 71*a^4*b + 98*a^3*b^2 + 154*a^2*b^3 - 19*a*b^4 - 3*b^5)*e^(-6*d*x - 6*c))/((a^7*b + 5*a^6*b^2 + 10*a^5*b^3
+ 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6 + 4*(a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*e^(-2
*d*x - 2*c) + 2*(3*a^7*b + 7*a^6*b^2 + 6*a^5*b^3 + 6*a^4*b^4 + 7*a^3*b^5 + 3*a^2*b^6)*e^(-4*d*x - 4*c) + 4*(a^
7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*e^(-6*d*x - 6*c) + (a^7*b + 5*a^6*b^2 + 10*a^5*
b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*e^(-8*d*x - 8*c))*d) - 1/16*(a^4 - 4*a^3*b - 14*a^2*b^2 - 12*a*b^3 - 3
*b^4 + (a^4 - 26*a^3*b - 20*a^2*b^2 + 10*a*b^3 + 3*b^4)*e^(6*d*x + 6*c) + (3*a^4 - 52*a^3*b + 6*a^2*b^2 - 12*a
*b^3 - 9*b^4)*e^(4*d*x + 4*c) + (3*a^4 - 30*a^3*b - 28*a^2*b^2 + 14*a*b^3 + 9*b^4)*e^(2*d*x + 2*c))/((a^6*b +
4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5 + (a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*e^(8*d*x +
8*c) + 4*(a^6*b + 2*a^5*b^2 - 2*a^3*b^4 - a^2*b^5)*e^(6*d*x + 6*c) + 2*(3*a^6*b + 4*a^5*b^2 + 2*a^4*b^3 + 4*a
^3*b^4 + 3*a^2*b^5)*e^(4*d*x + 4*c) + 4*(a^6*b + 2*a^5*b^2 - 2*a^3*b^4 - a^2*b^5)*e^(2*d*x + 2*c))*d) + 1/16*(
a^4 - 4*a^3*b - 14*a^2*b^2 - 12*a*b^3 - 3*b^4 + (3*a^4 - 30*a^3*b - 28*a^2*b^2 + 14*a*b^3 + 9*b^4)*e^(-2*d*x -
2*c) + (3*a^4 - 52*a^3*b + 6*a^2*b^2 - 12*a*b^3 - 9*b^4)*e^(-4*d*x - 4*c) + (a^4 - 26*a^3*b - 20*a^2*b^2 + 10
*a*b^3 + 3*b^4)*e^(-6*d*x - 6*c))/((a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5 + 4*(a^6*b + 2*a^5*b^2
- 2*a^3*b^4 - a^2*b^5)*e^(-2*d*x - 2*c) + 2*(3*a^6*b + 4*a^5*b^2 + 2*a^4*b^3 + 4*a^3*b^4 + 3*a^2*b^5)*e^(-4*d
*x - 4*c) + 4*(a^6*b + 2*a^5*b^2 - 2*a^3*b^4 - a^2*b^5)*e^(-6*d*x - 6*c) + (a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*
a^3*b^4 + a^2*b^5)*e^(-8*d*x - 8*c))*d) + 3/32*(a^3 + 5*a^2*b + 7*a*b^2 + 3*b^3 + (3*a^3 + 13*a^2*b + a*b^2 -
9*b^3)*e^(-2*d*x - 2*c) + (3*a^3 + 7*a^2*b - 3*a*b^2 + 9*b^3)*e^(-4*d*x - 4*c) + (a^3 - a^2*b - 5*a*b^2 - 3*b^
3)*e^(-6*d*x - 6*c))/((a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4 + 4*(a^5*b + a^4*b^2 - a^3*b^3 - a^2*b^4)*e^(-2
*d*x - 2*c) + 2*(3*a^5*b + a^4*b^2 + a^3*b^3 + 3*a^2*b^4)*e^(-4*d*x - 4*c) + 4*(a^5*b + a^4*b^2 - a^3*b^3 - a^
2*b^4)*e^(-6*d*x - 6*c) + (a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*e^(-8*d*x - 8*c))*d) + 1/4*log((a + b)*e^(
4*d*x + 4*c) + 2*(a - b)*e^(2*d*x + 2*c) + a + b)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) - 1/4*log(2*(a - b)*e^(-
2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) - 1/32*(a + 3*b)*arctan(1
/2*((a + b)*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*a^2*b*d) + 1/32*(a + 3*b)*arctan(1/2*((a + b)*e^(-2
*d*x - 2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*a^2*b*d) + 3/64*(a - 3*b)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a
- b)/sqrt(a*b))/(sqrt(a*b)*a^2*b*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (123) = 246\).

Time = 0.46 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.80 \[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {\frac {{\left (a^{2} + 6 \, a b - 3 \, b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{{\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \sqrt {a b}} - \frac {8 \, {\left (d x + c\right )}}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {2 \, {\left (a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 9 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 5 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 5 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 17 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 13 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 15 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 11 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 15 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right )}}{{\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}^{2}}}{8 \, d} \]

[In]

integrate(tanh(d*x+c)^4/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-1/8*((a^2 + 6*a*b - 3*b^2)*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/((a^3*b + 3*
a^2*b^2 + 3*a*b^3 + b^4)*sqrt(a*b)) - 8*(d*x + c)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 2*(a^3*e^(6*d*x + 6*c) - 9
*a^2*b*e^(6*d*x + 6*c) - 5*a*b^2*e^(6*d*x + 6*c) + 5*b^3*e^(6*d*x + 6*c) + 3*a^3*e^(4*d*x + 4*c) - 17*a^2*b*e^
(4*d*x + 4*c) + 13*a*b^2*e^(4*d*x + 4*c) - 15*b^3*e^(4*d*x + 4*c) + 3*a^3*e^(2*d*x + 2*c) - 11*a^2*b*e^(2*d*x
+ 2*c) + a*b^2*e^(2*d*x + 2*c) + 15*b^3*e^(2*d*x + 2*c) + a^3 - 3*a^2*b - 9*a*b^2 - 5*b^3)/((a^3*b + 3*a^2*b^2
+ 3*a*b^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)^
2))/d

Mupad [B] (verification not implemented)

Time = 2.37 (sec) , antiderivative size = 2574, normalized size of antiderivative = 18.79 \[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

int(tanh(c + d*x)^4/(a + b*tanh(c + d*x)^2)^3,x)

[Out]

log(tanh(c + d*x) + 1)/(2*a^3*d + 2*b^3*d + 6*a*b^2*d + 6*a^2*b*d) - ((tanh(c + d*x)^3*(a + 5*b))/(8*(2*a*b +
a^2 + b^2)) - (a*tanh(c + d*x)*(a - 3*b))/(8*b*(2*a*b + a^2 + b^2)))/(a^2*d + b^2*d*tanh(c + d*x)^4 + 2*a*b*d*
tanh(c + d*x)^2) - log(tanh(c + d*x) - 1)/(2*d*(a + b)^3) - (atan(((((tanh(c + d*x)*(12*a^3*b - 36*a*b^3 + a^4
+ 73*b^4 + 30*a^2*b^2))/(32*(b^5*d^2 + 4*a*b^4*d^2 + a^4*b*d^2 + 6*a^2*b^3*d^2 + 4*a^3*b^2*d^2)) + (((96*b^9*
d^2 + 544*a*b^8*d^2 + 1248*a^2*b^7*d^2 + 1440*a^3*b^6*d^2 + 800*a^4*b^5*d^2 + 96*a^5*b^4*d^2 - 96*a^6*b^3*d^2
- 32*a^7*b^2*d^2)/(64*(b^7*d^3 + 6*a*b^6*d^3 + a^6*b*d^3 + 15*a^2*b^5*d^3 + 20*a^3*b^4*d^3 + 15*a^4*b^3*d^3 +
6*a^5*b^2*d^3)) - (tanh(c + d*x)*(-a*b^3)^(1/2)*(6*a*b + a^2 - 3*b^2)*(256*b^10*d^2 + 1280*a*b^9*d^2 + 2304*a^
2*b^8*d^2 + 1280*a^3*b^7*d^2 - 1280*a^4*b^6*d^2 - 2304*a^5*b^5*d^2 - 1280*a^6*b^4*d^2 - 256*a^7*b^3*d^2))/(512
*(3*a^2*b^5*d + 3*a^3*b^4*d + a^4*b^3*d + a*b^6*d)*(b^5*d^2 + 4*a*b^4*d^2 + a^4*b*d^2 + 6*a^2*b^3*d^2 + 4*a^3*
b^2*d^2)))*(-a*b^3)^(1/2)*(6*a*b + a^2 - 3*b^2))/(16*(3*a^2*b^5*d + 3*a^3*b^4*d + a^4*b^3*d + a*b^6*d)))*(-a*b
^3)^(1/2)*(6*a*b + a^2 - 3*b^2)*1i)/(16*(3*a^2*b^5*d + 3*a^3*b^4*d + a^4*b^3*d + a*b^6*d)) + (((tanh(c + d*x)*
(12*a^3*b - 36*a*b^3 + a^4 + 73*b^4 + 30*a^2*b^2))/(32*(b^5*d^2 + 4*a*b^4*d^2 + a^4*b*d^2 + 6*a^2*b^3*d^2 + 4*
a^3*b^2*d^2)) - (((96*b^9*d^2 + 544*a*b^8*d^2 + 1248*a^2*b^7*d^2 + 1440*a^3*b^6*d^2 + 800*a^4*b^5*d^2 + 96*a^5
*b^4*d^2 - 96*a^6*b^3*d^2 - 32*a^7*b^2*d^2)/(64*(b^7*d^3 + 6*a*b^6*d^3 + a^6*b*d^3 + 15*a^2*b^5*d^3 + 20*a^3*b
^4*d^3 + 15*a^4*b^3*d^3 + 6*a^5*b^2*d^3)) + (tanh(c + d*x)*(-a*b^3)^(1/2)*(6*a*b + a^2 - 3*b^2)*(256*b^10*d^2
+ 1280*a*b^9*d^2 + 2304*a^2*b^8*d^2 + 1280*a^3*b^7*d^2 - 1280*a^4*b^6*d^2 - 2304*a^5*b^5*d^2 - 1280*a^6*b^4*d^
2 - 256*a^7*b^3*d^2))/(512*(3*a^2*b^5*d + 3*a^3*b^4*d + a^4*b^3*d + a*b^6*d)*(b^5*d^2 + 4*a*b^4*d^2 + a^4*b*d^
2 + 6*a^2*b^3*d^2 + 4*a^3*b^2*d^2)))*(-a*b^3)^(1/2)*(6*a*b + a^2 - 3*b^2))/(16*(3*a^2*b^5*d + 3*a^3*b^4*d + a^
4*b^3*d + a*b^6*d)))*(-a*b^3)^(1/2)*(6*a*b + a^2 - 3*b^2)*1i)/(16*(3*a^2*b^5*d + 3*a^3*b^4*d + a^4*b^3*d + a*b
^6*d)))/((27*a*b^2 + 11*a^2*b + a^3 - 15*b^3)/(32*(b^7*d^3 + 6*a*b^6*d^3 + a^6*b*d^3 + 15*a^2*b^5*d^3 + 20*a^3
*b^4*d^3 + 15*a^4*b^3*d^3 + 6*a^5*b^2*d^3)) + (((tanh(c + d*x)*(12*a^3*b - 36*a*b^3 + a^4 + 73*b^4 + 30*a^2*b^
2))/(32*(b^5*d^2 + 4*a*b^4*d^2 + a^4*b*d^2 + 6*a^2*b^3*d^2 + 4*a^3*b^2*d^2)) + (((96*b^9*d^2 + 544*a*b^8*d^2 +
1248*a^2*b^7*d^2 + 1440*a^3*b^6*d^2 + 800*a^4*b^5*d^2 + 96*a^5*b^4*d^2 - 96*a^6*b^3*d^2 - 32*a^7*b^2*d^2)/(64
*(b^7*d^3 + 6*a*b^6*d^3 + a^6*b*d^3 + 15*a^2*b^5*d^3 + 20*a^3*b^4*d^3 + 15*a^4*b^3*d^3 + 6*a^5*b^2*d^3)) - (ta
nh(c + d*x)*(-a*b^3)^(1/2)*(6*a*b + a^2 - 3*b^2)*(256*b^10*d^2 + 1280*a*b^9*d^2 + 2304*a^2*b^8*d^2 + 1280*a^3*
b^7*d^2 - 1280*a^4*b^6*d^2 - 2304*a^5*b^5*d^2 - 1280*a^6*b^4*d^2 - 256*a^7*b^3*d^2))/(512*(3*a^2*b^5*d + 3*a^3
*b^4*d + a^4*b^3*d + a*b^6*d)*(b^5*d^2 + 4*a*b^4*d^2 + a^4*b*d^2 + 6*a^2*b^3*d^2 + 4*a^3*b^2*d^2)))*(-a*b^3)^(
1/2)*(6*a*b + a^2 - 3*b^2))/(16*(3*a^2*b^5*d + 3*a^3*b^4*d + a^4*b^3*d + a*b^6*d)))*(-a*b^3)^(1/2)*(6*a*b + a^
2 - 3*b^2))/(16*(3*a^2*b^5*d + 3*a^3*b^4*d + a^4*b^3*d + a*b^6*d)) - (((tanh(c + d*x)*(12*a^3*b - 36*a*b^3 + a
^4 + 73*b^4 + 30*a^2*b^2))/(32*(b^5*d^2 + 4*a*b^4*d^2 + a^4*b*d^2 + 6*a^2*b^3*d^2 + 4*a^3*b^2*d^2)) - (((96*b^
9*d^2 + 544*a*b^8*d^2 + 1248*a^2*b^7*d^2 + 1440*a^3*b^6*d^2 + 800*a^4*b^5*d^2 + 96*a^5*b^4*d^2 - 96*a^6*b^3*d^
2 - 32*a^7*b^2*d^2)/(64*(b^7*d^3 + 6*a*b^6*d^3 + a^6*b*d^3 + 15*a^2*b^5*d^3 + 20*a^3*b^4*d^3 + 15*a^4*b^3*d^3
+ 6*a^5*b^2*d^3)) + (tanh(c + d*x)*(-a*b^3)^(1/2)*(6*a*b + a^2 - 3*b^2)*(256*b^10*d^2 + 1280*a*b^9*d^2 + 2304*
a^2*b^8*d^2 + 1280*a^3*b^7*d^2 - 1280*a^4*b^6*d^2 - 2304*a^5*b^5*d^2 - 1280*a^6*b^4*d^2 - 256*a^7*b^3*d^2))/(5
12*(3*a^2*b^5*d + 3*a^3*b^4*d + a^4*b^3*d + a*b^6*d)*(b^5*d^2 + 4*a*b^4*d^2 + a^4*b*d^2 + 6*a^2*b^3*d^2 + 4*a^
3*b^2*d^2)))*(-a*b^3)^(1/2)*(6*a*b + a^2 - 3*b^2))/(16*(3*a^2*b^5*d + 3*a^3*b^4*d + a^4*b^3*d + a*b^6*d)))*(-a
*b^3)^(1/2)*(6*a*b + a^2 - 3*b^2))/(16*(3*a^2*b^5*d + 3*a^3*b^4*d + a^4*b^3*d + a*b^6*d))))*(-a*b^3)^(1/2)*(6*
a*b + a^2 - 3*b^2)*1i)/(8*(3*a^2*b^5*d + 3*a^3*b^4*d + a^4*b^3*d + a*b^6*d))