3.162 \(\int \frac{\tanh ^2(c+d x)}{(a+b \text{sech}^2(c+d x))^3} \, dx\)
Optimal. Leaf size=139 \[ -\frac{\left (3 a^2+12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^3 \sqrt{b} d (a+b)^{3/2}}-\frac{(3 a+4 b) \tanh (c+d x)}{8 a^2 d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}+\frac{x}{a^3}-\frac{\tanh (c+d x)}{4 a d \left (a-b \tanh ^2(c+d x)+b\right )^2} \]
[Out]
x/a^3 - ((3*a^2 + 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^3*Sqrt[b]*(a + b)^(3/2)*d
) - Tanh[c + d*x]/(4*a*d*(a + b - b*Tanh[c + d*x]^2)^2) - ((3*a + 4*b)*Tanh[c + d*x])/(8*a^2*(a + b)*d*(a + b
- b*Tanh[c + d*x]^2))
________________________________________________________________________________________
Rubi [A] time = 0.246268, antiderivative size = 139, 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, 471, 527, 522, 206, 208} \[ -\frac{\left (3 a^2+12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^3 \sqrt{b} d (a+b)^{3/2}}-\frac{(3 a+4 b) \tanh (c+d x)}{8 a^2 d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}+\frac{x}{a^3}-\frac{\tanh (c+d x)}{4 a d \left (a-b \tanh ^2(c+d x)+b\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[Tanh[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]
[Out]
x/a^3 - ((3*a^2 + 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^3*Sqrt[b]*(a + b)^(3/2)*d
) - Tanh[c + d*x]/(4*a*d*(a + b - b*Tanh[c + d*x]^2)^2) - ((3*a + 4*b)*Tanh[c + d*x])/(8*a^2*(a + b)*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 471
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 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] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 527
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
Rule 522
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{\tanh ^2(c+d x)}{\left (a+b \text{sech}^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{\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^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{\tanh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{1+3 x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a d}\\ &=-\frac{\tanh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{(3 a+4 b) \tanh (c+d x)}{8 a^2 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-5 a-4 b+(-3 a-4 b) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a+b) d}\\ &=-\frac{\tanh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{(3 a+4 b) \tanh (c+d x)}{8 a^2 (a+b) 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 (3 a^2+12 a b+8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^3 (a+b) d}\\ &=\frac{x}{a^3}-\frac{\left (3 a^2+12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{8 a^3 \sqrt{b} (a+b)^{3/2} d}-\frac{\tanh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac{(3 a+4 b) \tanh (c+d x)}{8 a^2 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 14.5523, size = 1317, normalized size = 9.47 \[ \frac{(\cosh (2 (c+d x)) a+a+2 b)^3 \text{sech}^6(c+d x) \left (\frac{6 a (a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{(a+b)^{5/2}}-\frac{4 \left (3 a^2+8 b a+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{(a+b)^{5/2}}+\frac{4 a \sqrt{b} \left (3 a^2+16 b a+3 (a+2 b) \cosh (2 (c+d x)) a+16 b^2\right ) \sinh (2 (c+d x))}{(a+b)^2 (\cosh (2 (c+d x)) a+a+2 b)^2}-\frac{2 \sqrt{b} \left (3 a^3+14 b a^2+24 b^2 a+\left (3 a^2+4 b a+4 b^2\right ) \cosh (2 (c+d x)) a+16 b^3\right ) \sinh (2 (c+d x))}{(a+b)^2 (\cosh (2 (c+d x)) a+a+2 b)^2}+\frac{\sqrt{b} \left (\frac{\text{sech}(2 c) \left (-9 \sinh (2 c) a^6+9 \sinh (2 d x) a^6+3 \sinh (2 (c+2 d x)) a^6-3 \sinh (4 c+2 d x) a^6+12 b \sinh (2 c) a^5-14 b \sinh (2 d x) a^5-12 b \sinh (2 (c+2 d x)) a^5+10 b \sinh (4 c+2 d x) a^5+128 b^2 d x \cosh (2 (c+2 d x)) a^4+512 b^2 d x \cosh (4 c+2 d x) a^4+128 b^2 d x \cosh (6 c+4 d x) a^4+684 b^2 \sinh (2 c) a^4-608 b^2 \sinh (2 d x) a^4-204 b^2 \sinh (2 (c+2 d x)) a^4+304 b^2 \sinh (4 c+2 d x) a^4+256 b^3 d x \cosh (2 (c+2 d x)) a^3+2048 b^3 d x \cosh (4 c+2 d x) a^3+256 b^3 d x \cosh (6 c+4 d x) a^3+2880 b^3 \sinh (2 c) a^3-2112 b^3 \sinh (2 d x) a^3-384 b^3 \sinh (2 (c+2 d x)) a^3+1056 b^3 \sinh (4 c+2 d x) a^3+128 b^4 d x \cosh (2 (c+2 d x)) a^2+2560 b^4 d x \cosh (4 c+2 d x) a^2+128 b^4 d x \cosh (6 c+4 d x) a^2+5280 b^4 \sinh (2 c) a^2-2560 b^4 \sinh (2 d x) a^2-192 b^4 \sinh (2 (c+2 d x)) a^2+1280 b^4 \sinh (4 c+2 d x) a^2+512 b^2 (a+b)^2 (a+2 b) d x \cosh (2 d x) a+1024 b^5 d x \cosh (4 c+2 d x) a+4608 b^5 \sinh (2 c) a-1024 b^5 \sinh (2 d x) a+512 b^5 \sinh (4 c+2 d x) a+256 b^2 (a+b)^2 \left (3 a^2+8 b a+8 b^2\right ) d x \cosh (2 c)+1536 b^6 \sinh (2 c)\right )}{(\cosh (2 (c+d x)) a+a+2 b)^2}-\frac{2 \left (3 a^5-10 b a^4+80 b^2 a^3+480 b^3 a^2+640 b^4 a+256 b^5\right ) \tanh ^{-1}\left (\frac{\text{sech}(d x) (\cosh (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}}\right ) (\cosh (2 c)-\sinh (2 c))}{\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right )}{a^3 (a+b)^2}+\frac{2 \sqrt{b} \left (\frac{6 \tanh ^{-1}\left (\frac{\text{sech}(d x) (\cosh (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}}\right ) (\cosh (2 c)-\sinh (2 c)) a^2}{\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}+\frac{a \text{sech}(2 c) \left (\left (-9 a^4-16 b a^3+48 b^2 a^2+128 b^3 a+64 b^4\right ) \sinh (2 d x)+a \left (-3 a^3+2 b a^2+24 b^2 a+16 b^3\right ) \sinh (2 (c+2 d x))+\left (3 a^4-64 b^2 a^2-128 b^3 a-64 b^4\right ) \sinh (4 c+2 d x)\right )+\left (9 a^5+18 b a^4-64 b^2 a^3-256 b^3 a^2-320 b^4 a-128 b^5\right ) \tanh (2 c)}{(\cosh (2 (c+d x)) a+a+2 b)^2 a^2}\right )}{(a+b)^2}\right )}{4096 b^{5/2} d \left (b \text{sech}^2(c+d x)+a\right )^3} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Tanh[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]
[Out]
((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*((6*a*(a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]
])/(a + b)^(5/2) - (4*(3*a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) + (4
*a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b
+ a*Cosh[2*(c + d*x)])^2) - (2*Sqrt[b]*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a*b + 4*b^2)*Cosh[
2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2) + (Sqrt[b]*((-2*(3*a^5 - 10*a^4
*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 + 256*b^5)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sin
h[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a +
b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (Sech[2*c]*(256*b^2*(a + b)^2*(3*a^2 + 8*a*b + 8*b^2)*d*x*Cosh[2*c] + 512
*a*b^2*(a + b)^2*(a + 2*b)*d*x*Cosh[2*d*x] + 128*a^4*b^2*d*x*Cosh[2*(c + 2*d*x)] + 256*a^3*b^3*d*x*Cosh[2*(c +
2*d*x)] + 128*a^2*b^4*d*x*Cosh[2*(c + 2*d*x)] + 512*a^4*b^2*d*x*Cosh[4*c + 2*d*x] + 2048*a^3*b^3*d*x*Cosh[4*c
+ 2*d*x] + 2560*a^2*b^4*d*x*Cosh[4*c + 2*d*x] + 1024*a*b^5*d*x*Cosh[4*c + 2*d*x] + 128*a^4*b^2*d*x*Cosh[6*c +
4*d*x] + 256*a^3*b^3*d*x*Cosh[6*c + 4*d*x] + 128*a^2*b^4*d*x*Cosh[6*c + 4*d*x] - 9*a^6*Sinh[2*c] + 12*a^5*b*S
inh[2*c] + 684*a^4*b^2*Sinh[2*c] + 2880*a^3*b^3*Sinh[2*c] + 5280*a^2*b^4*Sinh[2*c] + 4608*a*b^5*Sinh[2*c] + 15
36*b^6*Sinh[2*c] + 9*a^6*Sinh[2*d*x] - 14*a^5*b*Sinh[2*d*x] - 608*a^4*b^2*Sinh[2*d*x] - 2112*a^3*b^3*Sinh[2*d*
x] - 2560*a^2*b^4*Sinh[2*d*x] - 1024*a*b^5*Sinh[2*d*x] + 3*a^6*Sinh[2*(c + 2*d*x)] - 12*a^5*b*Sinh[2*(c + 2*d*
x)] - 204*a^4*b^2*Sinh[2*(c + 2*d*x)] - 384*a^3*b^3*Sinh[2*(c + 2*d*x)] - 192*a^2*b^4*Sinh[2*(c + 2*d*x)] - 3*
a^6*Sinh[4*c + 2*d*x] + 10*a^5*b*Sinh[4*c + 2*d*x] + 304*a^4*b^2*Sinh[4*c + 2*d*x] + 1056*a^3*b^3*Sinh[4*c + 2
*d*x] + 1280*a^2*b^4*Sinh[4*c + 2*d*x] + 512*a*b^5*Sinh[4*c + 2*d*x]))/(a + 2*b + a*Cosh[2*(c + d*x)])^2))/(a^
3*(a + b)^2) + (2*Sqrt[b]*((6*a^2*ArcTanh[(Sech[d*x]*(Cosh[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])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c]
- Sinh[c])^4]) + (a*Sech[2*c]*((-9*a^4 - 16*a^3*b + 48*a^2*b^2 + 128*a*b^3 + 64*b^4)*Sinh[2*d*x] + a*(-3*a^3
+ 2*a^2*b + 24*a*b^2 + 16*b^3)*Sinh[2*(c + 2*d*x)] + (3*a^4 - 64*a^2*b^2 - 128*a*b^3 - 64*b^4)*Sinh[4*c + 2*d*
x]) + (9*a^5 + 18*a^4*b - 64*a^3*b^2 - 256*a^2*b^3 - 320*a*b^4 - 128*b^5)*Tanh[2*c])/(a^2*(a + 2*b + a*Cosh[2*
(c + d*x)])^2)))/(a + b)^2))/(4096*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3)
________________________________________________________________________________________
Maple [B] time = 0.104, size = 1173, normalized size = 8.4 \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)^2/(a+b*sech(d*x+c)^2)^3,x)
[Out]
1/d/a^3*ln(tanh(1/2*d*x+1/2*c)+1)-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)^7-1/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)^7-15/4/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/(a+b)*tanh(1/
2*d*x+1/2*c)^5-15/4/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)^2/(a+b)*tanh(1/2*d*x+1/2*c)^5+1/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)^2/(a+b)*tanh(1/2*d*x+1/2*c)^5-15/4/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/(a+b)*tanh(1/2*
d*x+1/2*c)^3-15/4/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)^2/(a+b)*tanh(1/2*d*x+1/2*c)^3+1/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)^2/(a+b)*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/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)-3/16/d/a/(a+b)^(3/2)/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*b^(1/2)/a^2/(a+b)^(3/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d
*x+1/2*c)*b^(1/2)+(a+b)^(1/2))-1/2/d*b^(3/2)/a^3/(a+b)^(3/2)*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))+3/16/d/a/(a+b)^(3/2)/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*b^(1/2)/a^2/(a+b)^(3/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*
x+1/2*c)*b^(1/2)+(a+b)^(1/2))+1/2/d*b^(3/2)/a^3/(a+b)^(3/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*
x+1/2*c)*b^(1/2)+(a+b)^(1/2))-1/d/a^3*ln(tanh(1/2*d*x+1/2*c)-1)
________________________________________________________________________________________
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)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 3.31096, size = 16496, normalized size = 118.68 \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)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
[Out]
[1/16*(16*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^8 + 128*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x +
c)*sinh(d*x + c)^7 + 16*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*sinh(d*x + c)^8 + 4*(5*a^4*b + 25*a^3*b^2 + 36*a^2*
b^3 + 16*a*b^4 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^6 + 4*(5*a^4*b + 25*a^3*b^2 +
36*a^2*b^3 + 16*a*b^4 + 112*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^2 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2
*b^3 + 2*a*b^4)*d*x)*sinh(d*x + c)^6 + 8*(112*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^3 + 3*(5*a^4*b +
25*a^3*b^2 + 36*a^2*b^3 + 16*a*b^4 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c))*sinh(d*
x + c)^5 + 20*a^4*b + 44*a^3*b^2 + 24*a^2*b^3 + 4*(15*a^4*b + 73*a^3*b^2 + 146*a^2*b^3 + 136*a*b^4 + 48*b^5 +
8*(3*a^4*b + 14*a^3*b^2 + 27*a^2*b^3 + 24*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^4 + 4*(280*(a^4*b + 2*a^3*b^2 + a^
2*b^3)*d*x*cosh(d*x + c)^4 + 15*a^4*b + 73*a^3*b^2 + 146*a^2*b^3 + 136*a*b^4 + 48*b^5 + 8*(3*a^4*b + 14*a^3*b^
2 + 27*a^2*b^3 + 24*a*b^4 + 8*b^5)*d*x + 15*(5*a^4*b + 25*a^3*b^2 + 36*a^2*b^3 + 16*a*b^4 + 16*(a^4*b + 4*a^3*
b^2 + 5*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(56*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*co
sh(d*x + c)^5 + 5*(5*a^4*b + 25*a^3*b^2 + 36*a^2*b^3 + 16*a*b^4 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)
*d*x)*cosh(d*x + c)^3 + (15*a^4*b + 73*a^3*b^2 + 146*a^2*b^3 + 136*a*b^4 + 48*b^5 + 8*(3*a^4*b + 14*a^3*b^2 +
27*a^2*b^3 + 24*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 16*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x + 4*
(15*a^4*b + 59*a^3*b^2 + 76*a^2*b^3 + 32*a*b^4 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x +
c)^2 + 4*(112*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^6 + 15*a^4*b + 59*a^3*b^2 + 76*a^2*b^3 + 32*a*b^
4 + 15*(5*a^4*b + 25*a^3*b^2 + 36*a^2*b^3 + 16*a*b^4 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)*d*x)*cosh(
d*x + c)^4 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)*d*x + 6*(15*a^4*b + 73*a^3*b^2 + 146*a^2*b^3 + 136*a
*b^4 + 48*b^5 + 8*(3*a^4*b + 14*a^3*b^2 + 27*a^2*b^3 + 24*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2
+ ((3*a^4 + 12*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^8 + 8*(3*a^4 + 12*a^3*b + 8*a^2*b^2)*cosh(d*x + c)*sinh(d*x +
c)^7 + (3*a^4 + 12*a^3*b + 8*a^2*b^2)*sinh(d*x + c)^8 + 4*(3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^3)*cosh(d*x
+ c)^6 + 4*(3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^3 + 7*(3*a^4 + 12*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2)*sinh
(d*x + c)^6 + 8*(7*(3*a^4 + 12*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^3 + 3*(3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^
3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(9*a^4 + 60*a^3*b + 144*a^2*b^2 + 160*a*b^3 + 64*b^4)*cosh(d*x + c)^4 +
2*(35*(3*a^4 + 12*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^4 + 9*a^4 + 60*a^3*b + 144*a^2*b^2 + 160*a*b^3 + 64*b^4 + 3
0*(3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 3*a^4 + 12*a^3*b + 8*a^2*b^2 +
8*(7*(3*a^4 + 12*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^5 + 10*(3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^3)*cosh(d*x
+ c)^3 + (9*a^4 + 60*a^3*b + 144*a^2*b^2 + 160*a*b^3 + 64*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(3*a^4 + 18*
a^3*b + 32*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^2 + 4*(7*(3*a^4 + 12*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^6 + 15*(3*a
^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^4 + 3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^3 + 3*(9*a^4 +
60*a^3*b + 144*a^2*b^2 + 160*a*b^3 + 64*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a^4 + 12*a^3*b + 8*a^2*
b^2)*cosh(d*x + c)^7 + 3*(3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^5 + (9*a^4 + 60*a^3*b + 144*
a^2*b^2 + 160*a*b^3 + 64*b^4)*cosh(d*x + c)^3 + (3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^3)*cosh(d*x + c))*sinh
(d*x + c))*sqrt(a*b + b^2)*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*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^2))/(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^4*b + 2*a^3*b^2 + a^2
*b^3)*d*x*cosh(d*x + c)^7 + 3*(5*a^4*b + 25*a^3*b^2 + 36*a^2*b^3 + 16*a*b^4 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^
3 + 2*a*b^4)*d*x)*cosh(d*x + c)^5 + 2*(15*a^4*b + 73*a^3*b^2 + 146*a^2*b^3 + 136*a*b^4 + 48*b^5 + 8*(3*a^4*b +
14*a^3*b^2 + 27*a^2*b^3 + 24*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^3 + (15*a^4*b + 59*a^3*b^2 + 76*a^2*b^3 + 32*a
*b^4 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^7*b + 2*a^6*b^2 + a
^5*b^3)*d*cosh(d*x + c)^8 + 8*(a^7*b + 2*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^7*b + 2*a^6*b
^2 + a^5*b^3)*d*sinh(d*x + c)^8 + 4*(a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^6 + 4*(7*(a^7*
b + 2*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^2 + (a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*d)*sinh(d*x + c)^6 +
2*(3*a^7*b + 14*a^6*b^2 + 27*a^5*b^3 + 24*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^7*b + 2*a^6*b^2 + a
^5*b^3)*d*cosh(d*x + c)^3 + 3*(a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2
*(35*(a^7*b + 2*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^4 + 30*(a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*d*cosh(d
*x + c)^2 + (3*a^7*b + 14*a^6*b^2 + 27*a^5*b^3 + 24*a^4*b^4 + 8*a^3*b^5)*d)*sinh(d*x + c)^4 + 4*(a^7*b + 4*a^6
*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^2 + 8*(7*(a^7*b + 2*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^5 + 10*(a
^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^3 + (3*a^7*b + 14*a^6*b^2 + 27*a^5*b^3 + 24*a^4*b^4
+ 8*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7*b + 2*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^6 + 15*(a^7
*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^4 + 3*(3*a^7*b + 14*a^6*b^2 + 27*a^5*b^3 + 24*a^4*b^4
+ 8*a^3*b^5)*d*cosh(d*x + c)^2 + (a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*d)*sinh(d*x + c)^2 + (a^7*b + 2*a
^6*b^2 + a^5*b^3)*d + 8*((a^7*b + 2*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^7 + 3*(a^7*b + 4*a^6*b^2 + 5*a^5*b^3 +
2*a^4*b^4)*d*cosh(d*x + c)^5 + (3*a^7*b + 14*a^6*b^2 + 27*a^5*b^3 + 24*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^3
+ (a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(8*(a^4*b + 2*a^3*b^2 + a^2
*b^3)*d*x*cosh(d*x + c)^8 + 64*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + 8*(a^4*b + 2*
a^3*b^2 + a^2*b^3)*d*x*sinh(d*x + c)^8 + 2*(5*a^4*b + 25*a^3*b^2 + 36*a^2*b^3 + 16*a*b^4 + 16*(a^4*b + 4*a^3*b
^2 + 5*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^6 + 2*(5*a^4*b + 25*a^3*b^2 + 36*a^2*b^3 + 16*a*b^4 + 112*(a^4*b
+ 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^2 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)*d*x)*sinh(d*x + c)^6
+ 4*(112*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^3 + 3*(5*a^4*b + 25*a^3*b^2 + 36*a^2*b^3 + 16*a*b^4
+ 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + 10*a^4*b + 22*a^3*b^2 + 1
2*a^2*b^3 + 2*(15*a^4*b + 73*a^3*b^2 + 146*a^2*b^3 + 136*a*b^4 + 48*b^5 + 8*(3*a^4*b + 14*a^3*b^2 + 27*a^2*b^3
+ 24*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^4 + 2*(280*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^4 + 15*a^4*
b + 73*a^3*b^2 + 146*a^2*b^3 + 136*a*b^4 + 48*b^5 + 8*(3*a^4*b + 14*a^3*b^2 + 27*a^2*b^3 + 24*a*b^4 + 8*b^5)*d
*x + 15*(5*a^4*b + 25*a^3*b^2 + 36*a^2*b^3 + 16*a*b^4 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)*d*x)*cosh
(d*x + c)^2)*sinh(d*x + c)^4 + 8*(56*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^5 + 5*(5*a^4*b + 25*a^3*b
^2 + 36*a^2*b^3 + 16*a*b^4 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^3 + (15*a^4*b + 7
3*a^3*b^2 + 146*a^2*b^3 + 136*a*b^4 + 48*b^5 + 8*(3*a^4*b + 14*a^3*b^2 + 27*a^2*b^3 + 24*a*b^4 + 8*b^5)*d*x)*c
osh(d*x + c))*sinh(d*x + c)^3 + 8*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x + 2*(15*a^4*b + 59*a^3*b^2 + 76*a^2*b^3 +
32*a*b^4 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^2 + 2*(112*(a^4*b + 2*a^3*b^2 + a^2
*b^3)*d*x*cosh(d*x + c)^6 + 15*a^4*b + 59*a^3*b^2 + 76*a^2*b^3 + 32*a*b^4 + 15*(5*a^4*b + 25*a^3*b^2 + 36*a^2*
b^3 + 16*a*b^4 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^4 + 16*(a^4*b + 4*a^3*b^2 + 5
*a^2*b^3 + 2*a*b^4)*d*x + 6*(15*a^4*b + 73*a^3*b^2 + 146*a^2*b^3 + 136*a*b^4 + 48*b^5 + 8*(3*a^4*b + 14*a^3*b^
2 + 27*a^2*b^3 + 24*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((3*a^4 + 12*a^3*b + 8*a^2*b^2)*cos
h(d*x + c)^8 + 8*(3*a^4 + 12*a^3*b + 8*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^4 + 12*a^3*b + 8*a^2*b^2)
*sinh(d*x + c)^8 + 4*(3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^6 + 4*(3*a^4 + 18*a^3*b + 32*a^2
*b^2 + 16*a*b^3 + 7*(3*a^4 + 12*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3*a^4 + 12*a^3*b +
8*a^2*b^2)*cosh(d*x + c)^3 + 3*(3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*
(9*a^4 + 60*a^3*b + 144*a^2*b^2 + 160*a*b^3 + 64*b^4)*cosh(d*x + c)^4 + 2*(35*(3*a^4 + 12*a^3*b + 8*a^2*b^2)*c
osh(d*x + c)^4 + 9*a^4 + 60*a^3*b + 144*a^2*b^2 + 160*a*b^3 + 64*b^4 + 30*(3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*
a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 3*a^4 + 12*a^3*b + 8*a^2*b^2 + 8*(7*(3*a^4 + 12*a^3*b + 8*a^2*b^2)*c
osh(d*x + c)^5 + 10*(3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^3 + (9*a^4 + 60*a^3*b + 144*a^2*b
^2 + 160*a*b^3 + 64*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^3)*cosh(d*
x + c)^2 + 4*(7*(3*a^4 + 12*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^6 + 15*(3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^3)
*cosh(d*x + c)^4 + 3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^3 + 3*(9*a^4 + 60*a^3*b + 144*a^2*b^2 + 160*a*b^3 +
64*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a^4 + 12*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^7 + 3*(3*a^4 + 18*a
^3*b + 32*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^5 + (9*a^4 + 60*a^3*b + 144*a^2*b^2 + 160*a*b^3 + 64*b^4)*cosh(d*x
+ c)^3 + (3*a^4 + 18*a^3*b + 32*a^2*b^2 + 16*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a*b - b^2)*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^2)/(a*b + b
^2)) + 4*(16*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^7 + 3*(5*a^4*b + 25*a^3*b^2 + 36*a^2*b^3 + 16*a*b
^4 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^5 + 2*(15*a^4*b + 73*a^3*b^2 + 146*a^2*b^
3 + 136*a*b^4 + 48*b^5 + 8*(3*a^4*b + 14*a^3*b^2 + 27*a^2*b^3 + 24*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^3 + (15*a
^4*b + 59*a^3*b^2 + 76*a^2*b^3 + 32*a*b^4 + 16*(a^4*b + 4*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c))*s
inh(d*x + c))/((a^7*b + 2*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^8 + 8*(a^7*b + 2*a^6*b^2 + a^5*b^3)*d*cosh(d*x +
c)*sinh(d*x + c)^7 + (a^7*b + 2*a^6*b^2 + a^5*b^3)*d*sinh(d*x + c)^8 + 4*(a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^
4*b^4)*d*cosh(d*x + c)^6 + 4*(7*(a^7*b + 2*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^2 + (a^7*b + 4*a^6*b^2 + 5*a^5*b
^3 + 2*a^4*b^4)*d)*sinh(d*x + c)^6 + 2*(3*a^7*b + 14*a^6*b^2 + 27*a^5*b^3 + 24*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x
+ c)^4 + 8*(7*(a^7*b + 2*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^3 + 3*(a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)
*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^7*b + 2*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^4 + 30*(a^7*b + 4*a^6*
b^2 + 5*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^2 + (3*a^7*b + 14*a^6*b^2 + 27*a^5*b^3 + 24*a^4*b^4 + 8*a^3*b^5)*
d)*sinh(d*x + c)^4 + 4*(a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^2 + 8*(7*(a^7*b + 2*a^6*b^2
+ a^5*b^3)*d*cosh(d*x + c)^5 + 10*(a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^3 + (3*a^7*b +
14*a^6*b^2 + 27*a^5*b^3 + 24*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7*b + 2*a^6*b^2 +
a^5*b^3)*d*cosh(d*x + c)^6 + 15*(a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^4 + 3*(3*a^7*b +
14*a^6*b^2 + 27*a^5*b^3 + 24*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^2 + (a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b
^4)*d)*sinh(d*x + c)^2 + (a^7*b + 2*a^6*b^2 + a^5*b^3)*d + 8*((a^7*b + 2*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^7
+ 3*(a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^5 + (3*a^7*b + 14*a^6*b^2 + 27*a^5*b^3 + 24*a^
4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^3 + (a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c))*sinh(d*x +
c))]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(d*x+c)**2/(a+b*sech(d*x+c)**2)**3,x)
[Out]
Timed out
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Giac [B] time = 2.40493, size = 417, normalized size = 3. \begin{align*} -\frac{\frac{{\left (3 \, a^{2} e^{\left (2 \, c\right )} + 12 \, a b e^{\left (2 \, c\right )} + 8 \, b^{2} 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 )}}{{\left (a^{4} + a^{3} b\right )} \sqrt{-a b - b^{2}}} - \frac{8 \, d x}{a^{3}} - \frac{2 \,{\left (5 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 20 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 15 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 58 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 88 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 44 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 32 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, a^{3} + 6 \, a^{2} b\right )}}{{\left (a^{4} + a^{3} b\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}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
[Out]
-1/8*((3*a^2*e^(2*c) + 12*a*b*e^(2*c) + 8*b^2*e^(2*c))*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^
2))*e^(-2*c)/((a^4 + a^3*b)*sqrt(-a*b - b^2)) - 8*d*x/a^3 - 2*(5*a^3*e^(6*d*x + 6*c) + 20*a^2*b*e^(6*d*x + 6*c
) + 16*a*b^2*e^(6*d*x + 6*c) + 15*a^3*e^(4*d*x + 4*c) + 58*a^2*b*e^(4*d*x + 4*c) + 88*a*b^2*e^(4*d*x + 4*c) +
48*b^3*e^(4*d*x + 4*c) + 15*a^3*e^(2*d*x + 2*c) + 44*a^2*b*e^(2*d*x + 2*c) + 32*a*b^2*e^(2*d*x + 2*c) + 5*a^3
+ 6*a^2*b)/((a^4 + a^3*b)*(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))/d