3.158 \(\int \frac{\tanh ^6(c+d x)}{(a+b \text{sech}^2(c+d x))^3} \, dx\)

Optimal. Leaf size=148 \[ \frac{(3 a-4 b) (a+b) \tanh (c+d x)}{8 a^2 b^2 d \left (a-b \tanh ^2(c+d x)+b\right )}-\frac{\sqrt{a+b} \left (3 a^2-4 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^3 b^{5/2} d}+\frac{x}{a^3}-\frac{(a+b) \tanh ^3(c+d x)}{4 a b d \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.319562, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4141, 1975, 470, 578, 522, 206, 208} \[ \frac{(3 a-4 b) (a+b) \tanh (c+d x)}{8 a^2 b^2 d \left (a-b \tanh ^2(c+d x)+b\right )}-\frac{\sqrt{a+b} \left (3 a^2-4 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^3 b^{5/2} d}+\frac{x}{a^3}-\frac{(a+b) \tanh ^3(c+d x)}{4 a b d \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[c + d*x]^6/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

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

Rule 4141

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[((d*ff*x)^m*(a + b*(1 + ff^2*x^2)^(n/2))^p)/(1 + ff^
2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] ||
EqQ[n, 2])

Rule 1975

Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q
, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0]
&&  !BinomialMatchQ[{u, v}, x]

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2
*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2
*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +
(a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 578

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(g^(n - 1)*(b*e - a*f)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c -
a*d)*(p + 1)), x] - Dist[g^n/(b*n*(b*c - a*d)*(p + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*S
imp[c*(b*e - a*f)*(m - n + 1) + (d*(b*e - a*f)*(m + n*q + 1) - b*n*(c*f - d*e)*(p + 1))*x^n, x], x], x] /; Fre
eQ[{a, b, c, d, e, f, g, q}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, 0]

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

Mathematica [B]  time = 6.0547, size = 515, normalized size = 3.48 \[ \frac{\text{sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\text{sech}(2 c) \left (-6 a^2 b^2 \sinh (2 (c+2 d x))+20 a^2 b^2 \sinh (4 c+2 d x)+4 a^2 b^2 d x \cosh (2 (c+2 d x))+16 a^2 b^2 d x \cosh (4 c+2 d x)+4 a^2 b^2 d x \cosh (6 c+4 d x)+8 b^2 d x \left (3 a^2+8 a b+8 b^2\right ) \cosh (2 c)+18 a^2 b^2 \sinh (2 c)-28 a^2 b^2 \sinh (2 d x)-3 a^3 b \sinh (2 (c+2 d x))+a^3 b \sinh (4 c+2 d x)-15 a^3 b \sinh (2 c)+13 a^3 b \sinh (2 d x)+3 a^4 \sinh (2 (c+2 d x))-3 a^4 \sinh (4 c+2 d x)-9 a^4 \sinh (2 c)+9 a^4 \sinh (2 d x)+16 a b^3 \sinh (4 c+2 d x)+32 a b^3 d x \cosh (4 c+2 d x)+72 a b^3 \sinh (2 c)-32 a b^3 \sinh (2 d x)+16 a b^2 d x (a+2 b) \cosh (2 d x)+48 b^4 \sinh (2 c)\right )-\frac{2 \left (-a^2 b+3 a^3+4 a b^2+8 b^3\right ) (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b)^2 \tanh ^{-1}\left (\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right )}{\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right )}{128 a^3 b^2 d \left (a+b \text{sech}^2(c+d x)\right )^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Tanh[c + d*x]^6/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*((-2*(3*a^3 - a^2*b + 4*a*b^2 + 8*b^3)*ArcTanh[(Sech[d*x]*(Co
sh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]
*(a + 2*b + a*Cosh[2*(c + d*x)])^2*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + Sech
[2*c]*(8*b^2*(3*a^2 + 8*a*b + 8*b^2)*d*x*Cosh[2*c] + 16*a*b^2*(a + 2*b)*d*x*Cosh[2*d*x] + 4*a^2*b^2*d*x*Cosh[2
*(c + 2*d*x)] + 16*a^2*b^2*d*x*Cosh[4*c + 2*d*x] + 32*a*b^3*d*x*Cosh[4*c + 2*d*x] + 4*a^2*b^2*d*x*Cosh[6*c + 4
*d*x] - 9*a^4*Sinh[2*c] - 15*a^3*b*Sinh[2*c] + 18*a^2*b^2*Sinh[2*c] + 72*a*b^3*Sinh[2*c] + 48*b^4*Sinh[2*c] +
9*a^4*Sinh[2*d*x] + 13*a^3*b*Sinh[2*d*x] - 28*a^2*b^2*Sinh[2*d*x] - 32*a*b^3*Sinh[2*d*x] + 3*a^4*Sinh[2*(c + 2
*d*x)] - 3*a^3*b*Sinh[2*(c + 2*d*x)] - 6*a^2*b^2*Sinh[2*(c + 2*d*x)] - 3*a^4*Sinh[4*c + 2*d*x] + a^3*b*Sinh[4*
c + 2*d*x] + 20*a^2*b^2*Sinh[4*c + 2*d*x] + 16*a*b^3*Sinh[4*c + 2*d*x])))/(128*a^3*b^2*d*(a + b*Sech[c + d*x]^
2)^3)

________________________________________________________________________________________

Maple [B]  time = 0.106, size = 1713, normalized size = 11.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*x+c)^6/(a+b*sech(d*x+c)^2)^3,x)

[Out]

3/16/d/b^(5/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))-5/4
/d/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)
^2*tanh(1/2*d*x+1/2*c)^7-19/4/d/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)^2*tanh(1/2*d*x+1/2*c)^5-19/4/d/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)^2*tanh(1/2*d*x+1/2*c)^3-5/4/d/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)^2*tanh(1/2*d*x+1/2*c)+1
/2/d/(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)
^2/b*tanh(1/2*d*x+1/2*c)^7-7/2/d/(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)^2/b*tanh(1/2*d*x+1/2*c)^5-7/2/d/(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)^2/b*tanh(1/2*d*x+1/2*c)^3+1/2/d/(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)^2/b*tanh(1/2*d*x+1/2*c)-3
/16/d/b^(5/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))+1/d*
b/a^2/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b
)^2*tanh(1/2*d*x+1/2*c)^3-1/d*b/a^2/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a
-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)-1/2/d*b^(1/2)/a^3/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*
x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))+3/4/d*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)^2/b^2*tanh(1/2*d*x+1/2*c)^7+9/4/d*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)^2/b^2*tanh(1/2*d*x+1
/2*c)^5+1/16/d/a/b^(3/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^
(1/2))-1/4/d/a^2/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^
(1/2))-1/16/d/a/b^(3/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(
1/2))+1/4/d/a^2/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(
1/2))+9/4/d*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)^2/b^2*tanh(1/2*d*x+1/2*c)^3+3/4/d*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)^2/b^2*tanh(1/2*d*x+1/2*c)-1/d/a^3*ln(tanh(1/2*d*x+1/2*c)-1)+1/d*b/a^
2/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*
tanh(1/2*d*x+1/2*c)^5+1/d/a^3*ln(tanh(1/2*d*x+1/2*c)+1)+1/2/d/a^3*b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*
d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))-1/d*b/a^2/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*
c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^7

________________________________________________________________________________________

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(tanh(d*x+c)^6/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.89494, size = 12740, normalized size = 86.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)^6/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(16*a^2*b^2*d*x*cosh(d*x + c)^8 + 128*a^2*b^2*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + 16*a^2*b^2*d*x*sinh(d*
x + c)^8 - 4*(3*a^4 - a^3*b - 20*a^2*b^2 - 16*a*b^3 - 16*(a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c)^6 + 4*(112*a^2
*b^2*d*x*cosh(d*x + c)^2 - 3*a^4 + a^3*b + 20*a^2*b^2 + 16*a*b^3 + 16*(a^2*b^2 + 2*a*b^3)*d*x)*sinh(d*x + c)^6
+ 16*a^2*b^2*d*x + 8*(112*a^2*b^2*d*x*cosh(d*x + c)^3 - 3*(3*a^4 - a^3*b - 20*a^2*b^2 - 16*a*b^3 - 16*(a^2*b^
2 + 2*a*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 - 4*(9*a^4 + 15*a^3*b - 18*a^2*b^2 - 72*a*b^3 - 48*b^4 - 8*(3
*a^2*b^2 + 8*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c)^4 + 4*(280*a^2*b^2*d*x*cosh(d*x + c)^4 - 9*a^4 - 15*a^3*b + 18*
a^2*b^2 + 72*a*b^3 + 48*b^4 + 8*(3*a^2*b^2 + 8*a*b^3 + 8*b^4)*d*x - 15*(3*a^4 - a^3*b - 20*a^2*b^2 - 16*a*b^3
- 16*(a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 12*a^4 + 12*a^3*b + 24*a^2*b^2 + 16*(56*a^2*b
^2*d*x*cosh(d*x + c)^5 - 5*(3*a^4 - a^3*b - 20*a^2*b^2 - 16*a*b^3 - 16*(a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c)^
3 - (9*a^4 + 15*a^3*b - 18*a^2*b^2 - 72*a*b^3 - 48*b^4 - 8*(3*a^2*b^2 + 8*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c))*s
inh(d*x + c)^3 - 4*(9*a^4 + 13*a^3*b - 28*a^2*b^2 - 32*a*b^3 - 16*(a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c)^2 + 4
*(112*a^2*b^2*d*x*cosh(d*x + c)^6 - 15*(3*a^4 - a^3*b - 20*a^2*b^2 - 16*a*b^3 - 16*(a^2*b^2 + 2*a*b^3)*d*x)*co
sh(d*x + c)^4 - 9*a^4 - 13*a^3*b + 28*a^2*b^2 + 32*a*b^3 + 16*(a^2*b^2 + 2*a*b^3)*d*x - 6*(9*a^4 + 15*a^3*b -
18*a^2*b^2 - 72*a*b^3 - 48*b^4 - 8*(3*a^2*b^2 + 8*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((3*a
^4 - 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^8 + 8*(3*a^4 - 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (3
*a^4 - 4*a^3*b + 8*a^2*b^2)*sinh(d*x + c)^8 + 4*(3*a^4 + 2*a^3*b + 16*a*b^3)*cosh(d*x + c)^6 + 4*(3*a^4 + 2*a^
3*b + 16*a*b^3 + 7*(3*a^4 - 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3*a^4 - 4*a^3*b + 8*
a^2*b^2)*cosh(d*x + c)^3 + 3*(3*a^4 + 2*a^3*b + 16*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(9*a^4 + 12*a^3*b
+ 16*a^2*b^2 + 32*a*b^3 + 64*b^4)*cosh(d*x + c)^4 + 2*(35*(3*a^4 - 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^4 + 9*a
^4 + 12*a^3*b + 16*a^2*b^2 + 32*a*b^3 + 64*b^4 + 30*(3*a^4 + 2*a^3*b + 16*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c
)^4 + 3*a^4 - 4*a^3*b + 8*a^2*b^2 + 8*(7*(3*a^4 - 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^5 + 10*(3*a^4 + 2*a^3*b +
16*a*b^3)*cosh(d*x + c)^3 + (9*a^4 + 12*a^3*b + 16*a^2*b^2 + 32*a*b^3 + 64*b^4)*cosh(d*x + c))*sinh(d*x + c)^
3 + 4*(3*a^4 + 2*a^3*b + 16*a*b^3)*cosh(d*x + c)^2 + 4*(7*(3*a^4 - 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^6 + 15*(
3*a^4 + 2*a^3*b + 16*a*b^3)*cosh(d*x + c)^4 + 3*a^4 + 2*a^3*b + 16*a*b^3 + 3*(9*a^4 + 12*a^3*b + 16*a^2*b^2 +
32*a*b^3 + 64*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a^4 - 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^7 + 3*(3*
a^4 + 2*a^3*b + 16*a*b^3)*cosh(d*x + c)^5 + (9*a^4 + 12*a^3*b + 16*a^2*b^2 + 32*a*b^3 + 64*b^4)*cosh(d*x + c)^
3 + (3*a^4 + 2*a^3*b + 16*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a + b)/b)*log((a^2*cosh(d*x + c)^4 + 4*a^
2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x +
c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b + 8*b^2 + 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x +
c))*sinh(d*x + c) + 4*(a*b*cosh(d*x + c)^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) + a*b*sinh(d*x + c)^2 + a*b + 2
*b^2)*sqrt((a + b)/b))/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b
)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(
d*x + c))*sinh(d*x + c) + a)) + 8*(16*a^2*b^2*d*x*cosh(d*x + c)^7 - 3*(3*a^4 - a^3*b - 20*a^2*b^2 - 16*a*b^3 -
16*(a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c)^5 - 2*(9*a^4 + 15*a^3*b - 18*a^2*b^2 - 72*a*b^3 - 48*b^4 - 8*(3*a^2
*b^2 + 8*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c)^3 - (9*a^4 + 13*a^3*b - 28*a^2*b^2 - 32*a*b^3 - 16*(a^2*b^2 + 2*a*b
^3)*d*x)*cosh(d*x + c))*sinh(d*x + c))/(a^5*b^2*d*cosh(d*x + c)^8 + 8*a^5*b^2*d*cosh(d*x + c)*sinh(d*x + c)^7
+ a^5*b^2*d*sinh(d*x + c)^8 + a^5*b^2*d + 4*(a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^6 + 4*(7*a^5*b^2*d*cosh(d*x
+ c)^2 + (a^5*b^2 + 2*a^4*b^3)*d)*sinh(d*x + c)^6 + 2*(3*a^5*b^2 + 8*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^4 +
8*(7*a^5*b^2*d*cosh(d*x + c)^3 + 3*(a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*a^5*b^2*d*co
sh(d*x + c)^4 + 30*(a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^2 + (3*a^5*b^2 + 8*a^4*b^3 + 8*a^3*b^4)*d)*sinh(d*x +
c)^4 + 4*(a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^2 + 8*(7*a^5*b^2*d*cosh(d*x + c)^5 + 10*(a^5*b^2 + 2*a^4*b^3)*
d*cosh(d*x + c)^3 + (3*a^5*b^2 + 8*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*a^5*b^2*d*cosh
(d*x + c)^6 + 15*(a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^4 + 3*(3*a^5*b^2 + 8*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x +
c)^2 + (a^5*b^2 + 2*a^4*b^3)*d)*sinh(d*x + c)^2 + 8*(a^5*b^2*d*cosh(d*x + c)^7 + 3*(a^5*b^2 + 2*a^4*b^3)*d*cos
h(d*x + c)^5 + (3*a^5*b^2 + 8*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^3 + (a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c))*
sinh(d*x + c)), 1/8*(8*a^2*b^2*d*x*cosh(d*x + c)^8 + 64*a^2*b^2*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + 8*a^2*b^2*
d*x*sinh(d*x + c)^8 - 2*(3*a^4 - a^3*b - 20*a^2*b^2 - 16*a*b^3 - 16*(a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c)^6 +
2*(112*a^2*b^2*d*x*cosh(d*x + c)^2 - 3*a^4 + a^3*b + 20*a^2*b^2 + 16*a*b^3 + 16*(a^2*b^2 + 2*a*b^3)*d*x)*sinh
(d*x + c)^6 + 8*a^2*b^2*d*x + 4*(112*a^2*b^2*d*x*cosh(d*x + c)^3 - 3*(3*a^4 - a^3*b - 20*a^2*b^2 - 16*a*b^3 -
16*(a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(9*a^4 + 15*a^3*b - 18*a^2*b^2 - 72*a*b^3 - 48*
b^4 - 8*(3*a^2*b^2 + 8*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c)^4 + 2*(280*a^2*b^2*d*x*cosh(d*x + c)^4 - 9*a^4 - 15*a
^3*b + 18*a^2*b^2 + 72*a*b^3 + 48*b^4 + 8*(3*a^2*b^2 + 8*a*b^3 + 8*b^4)*d*x - 15*(3*a^4 - a^3*b - 20*a^2*b^2 -
16*a*b^3 - 16*(a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 6*a^4 + 6*a^3*b + 12*a^2*b^2 + 8*(5
6*a^2*b^2*d*x*cosh(d*x + c)^5 - 5*(3*a^4 - a^3*b - 20*a^2*b^2 - 16*a*b^3 - 16*(a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*
x + c)^3 - (9*a^4 + 15*a^3*b - 18*a^2*b^2 - 72*a*b^3 - 48*b^4 - 8*(3*a^2*b^2 + 8*a*b^3 + 8*b^4)*d*x)*cosh(d*x
+ c))*sinh(d*x + c)^3 - 2*(9*a^4 + 13*a^3*b - 28*a^2*b^2 - 32*a*b^3 - 16*(a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c
)^2 + 2*(112*a^2*b^2*d*x*cosh(d*x + c)^6 - 15*(3*a^4 - a^3*b - 20*a^2*b^2 - 16*a*b^3 - 16*(a^2*b^2 + 2*a*b^3)*
d*x)*cosh(d*x + c)^4 - 9*a^4 - 13*a^3*b + 28*a^2*b^2 + 32*a*b^3 + 16*(a^2*b^2 + 2*a*b^3)*d*x - 6*(9*a^4 + 15*a
^3*b - 18*a^2*b^2 - 72*a*b^3 - 48*b^4 - 8*(3*a^2*b^2 + 8*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2
- ((3*a^4 - 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^8 + 8*(3*a^4 - 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)
^7 + (3*a^4 - 4*a^3*b + 8*a^2*b^2)*sinh(d*x + c)^8 + 4*(3*a^4 + 2*a^3*b + 16*a*b^3)*cosh(d*x + c)^6 + 4*(3*a^4
+ 2*a^3*b + 16*a*b^3 + 7*(3*a^4 - 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3*a^4 - 4*a^3
*b + 8*a^2*b^2)*cosh(d*x + c)^3 + 3*(3*a^4 + 2*a^3*b + 16*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(9*a^4 + 1
2*a^3*b + 16*a^2*b^2 + 32*a*b^3 + 64*b^4)*cosh(d*x + c)^4 + 2*(35*(3*a^4 - 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^
4 + 9*a^4 + 12*a^3*b + 16*a^2*b^2 + 32*a*b^3 + 64*b^4 + 30*(3*a^4 + 2*a^3*b + 16*a*b^3)*cosh(d*x + c)^2)*sinh(
d*x + c)^4 + 3*a^4 - 4*a^3*b + 8*a^2*b^2 + 8*(7*(3*a^4 - 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^5 + 10*(3*a^4 + 2*
a^3*b + 16*a*b^3)*cosh(d*x + c)^3 + (9*a^4 + 12*a^3*b + 16*a^2*b^2 + 32*a*b^3 + 64*b^4)*cosh(d*x + c))*sinh(d*
x + c)^3 + 4*(3*a^4 + 2*a^3*b + 16*a*b^3)*cosh(d*x + c)^2 + 4*(7*(3*a^4 - 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^6
+ 15*(3*a^4 + 2*a^3*b + 16*a*b^3)*cosh(d*x + c)^4 + 3*a^4 + 2*a^3*b + 16*a*b^3 + 3*(9*a^4 + 12*a^3*b + 16*a^2
*b^2 + 32*a*b^3 + 64*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a^4 - 4*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^7
+ 3*(3*a^4 + 2*a^3*b + 16*a*b^3)*cosh(d*x + c)^5 + (9*a^4 + 12*a^3*b + 16*a^2*b^2 + 32*a*b^3 + 64*b^4)*cosh(d*
x + c)^3 + (3*a^4 + 2*a^3*b + 16*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-(a + b)/b)*arctan(1/2*(a*cosh(d*x
+ c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a + 2*b)*sqrt(-(a + b)/b)/(a + b)) + 4*(16*a^2*
b^2*d*x*cosh(d*x + c)^7 - 3*(3*a^4 - a^3*b - 20*a^2*b^2 - 16*a*b^3 - 16*(a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c)
^5 - 2*(9*a^4 + 15*a^3*b - 18*a^2*b^2 - 72*a*b^3 - 48*b^4 - 8*(3*a^2*b^2 + 8*a*b^3 + 8*b^4)*d*x)*cosh(d*x + c)
^3 - (9*a^4 + 13*a^3*b - 28*a^2*b^2 - 32*a*b^3 - 16*(a^2*b^2 + 2*a*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c))/(a^
5*b^2*d*cosh(d*x + c)^8 + 8*a^5*b^2*d*cosh(d*x + c)*sinh(d*x + c)^7 + a^5*b^2*d*sinh(d*x + c)^8 + a^5*b^2*d +
4*(a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^6 + 4*(7*a^5*b^2*d*cosh(d*x + c)^2 + (a^5*b^2 + 2*a^4*b^3)*d)*sinh(d*x
+ c)^6 + 2*(3*a^5*b^2 + 8*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^4 + 8*(7*a^5*b^2*d*cosh(d*x + c)^3 + 3*(a^5*b^
2 + 2*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*a^5*b^2*d*cosh(d*x + c)^4 + 30*(a^5*b^2 + 2*a^4*b^3)*d
*cosh(d*x + c)^2 + (3*a^5*b^2 + 8*a^4*b^3 + 8*a^3*b^4)*d)*sinh(d*x + c)^4 + 4*(a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x
+ c)^2 + 8*(7*a^5*b^2*d*cosh(d*x + c)^5 + 10*(a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^3 + (3*a^5*b^2 + 8*a^4*b^3
+ 8*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*a^5*b^2*d*cosh(d*x + c)^6 + 15*(a^5*b^2 + 2*a^4*b^3)*d*c
osh(d*x + c)^4 + 3*(3*a^5*b^2 + 8*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^2 + (a^5*b^2 + 2*a^4*b^3)*d)*sinh(d*x +
c)^2 + 8*(a^5*b^2*d*cosh(d*x + c)^7 + 3*(a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c)^5 + (3*a^5*b^2 + 8*a^4*b^3 + 8*
a^3*b^4)*d*cosh(d*x + c)^3 + (a^5*b^2 + 2*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c))]

________________________________________________________________________________________

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(tanh(d*x+c)**6/(a+b*sech(d*x+c)**2)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 4.31904, size = 501, normalized size = 3.39 \begin{align*} \frac{\frac{8 \, d x}{a^{3}} - \frac{{\left (3 \, a^{3} e^{\left (2 \, c\right )} - a^{2} b e^{\left (2 \, c\right )} + 4 \, a b^{2} e^{\left (2 \, c\right )} + 8 \, b^{3} e^{\left (2 \, c\right )}\right )} \arctan \left (\frac{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt{-a b - b^{2}}}\right ) e^{\left (-2 \, c\right )}}{\sqrt{-a b - b^{2}} a^{3} b^{2}} - \frac{2 \,{\left (3 \, a^{4} e^{\left (6 \, d x + 6 \, c\right )} - a^{3} b e^{\left (6 \, d x + 6 \, c\right )} - 20 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 16 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 9 \, a^{4} e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} - 18 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 72 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 48 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, a^{4} e^{\left (2 \, d x + 2 \, c\right )} + 13 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} - 28 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 32 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2}\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 )}^{2} a^{3} b^{2}}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)^6/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/8*(8*d*x/a^3 - (3*a^3*e^(2*c) - a^2*b*e^(2*c) + 4*a*b^2*e^(2*c) + 8*b^3*e^(2*c))*arctan(1/2*(a*e^(2*d*x + 2*
c) + a + 2*b)/sqrt(-a*b - b^2))*e^(-2*c)/(sqrt(-a*b - b^2)*a^3*b^2) - 2*(3*a^4*e^(6*d*x + 6*c) - a^3*b*e^(6*d*
x + 6*c) - 20*a^2*b^2*e^(6*d*x + 6*c) - 16*a*b^3*e^(6*d*x + 6*c) + 9*a^4*e^(4*d*x + 4*c) + 15*a^3*b*e^(4*d*x +
4*c) - 18*a^2*b^2*e^(4*d*x + 4*c) - 72*a*b^3*e^(4*d*x + 4*c) - 48*b^4*e^(4*d*x + 4*c) + 9*a^4*e^(2*d*x + 2*c)
+ 13*a^3*b*e^(2*d*x + 2*c) - 28*a^2*b^2*e^(2*d*x + 2*c) - 32*a*b^3*e^(2*d*x + 2*c) + 3*a^4 - 3*a^3*b - 6*a^2*
b^2)/((a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)^2*a^3*b^2))/d