3.160 \(\int \frac {\tanh ^4(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\)

Optimal. Leaf size=139 \[ \frac {x}{a^3}+\frac {(a-4 b) \tanh (c+d x)}{8 a^2 b d \left (a-b \tanh ^2(c+d x)+b\right )}+\frac {\left (a^2-4 a b-8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^3 b^{3/2} d \sqrt {a+b}}-\frac {(a+b) \tanh (c+d x)}{4 a b d \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.29, antiderivative size = 139, normalized size of antiderivative = 1.00, 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, 527, 522, 206, 208} \[ \frac {\left (a^2-4 a b-8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^3 b^{3/2} d \sqrt {a+b}}+\frac {(a-4 b) \tanh (c+d x)}{8 a^2 b d \left (a-b \tanh ^2(c+d x)+b\right )}+\frac {x}{a^3}-\frac {(a+b) \tanh (c+d x)}{4 a b d \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 206

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

Rule 208

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

Rule 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 522

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

Rule 527

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

Rule 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 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])

Rubi steps

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

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Mathematica [B]  time = 13.09, size = 1457, normalized size = 10.48 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*(((3*a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/S
qrt[a + b]])/(a + b)^(5/2) - (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)))/(1024*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3) - ((a + 2*
b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-3*a*(a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a +
b)^(5/2) + (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)))/(2048*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3) + ((
a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-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)*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]) + (Sec
h[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*Sinh[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] + 1536*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)] - 3
84*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))/(4096*a^3*b^2*(a + b)^2*d*(a + b*Sech[c + d*
x]^2)^3) - ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((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)))/(2048*b^2*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3)

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fricas [B]  time = 0.57, size = 6464, normalized size = 46.50 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(16*(a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^8 + 128*(a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)*sinh(d*x + c)^7
+ 16*(a^3*b^2 + a^2*b^3)*d*x*sinh(d*x + c)^8 + 4*(a^4*b + 13*a^3*b^2 + 28*a^2*b^3 + 16*a*b^4 + 16*(a^3*b^2 + 3
*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^6 + 4*(a^4*b + 13*a^3*b^2 + 28*a^2*b^3 + 16*a*b^4 + 112*(a^3*b^2 + a^2*
b^3)*d*x*cosh(d*x + c)^2 + 16*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*d*x)*sinh(d*x + c)^6 + 8*(112*(a^3*b^2 + a^2*b^3
)*d*x*cosh(d*x + c)^3 + 3*(a^4*b + 13*a^3*b^2 + 28*a^2*b^3 + 16*a*b^4 + 16*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*d*x
)*cosh(d*x + c))*sinh(d*x + c)^5 + 4*a^4*b + 28*a^3*b^2 + 24*a^2*b^3 + 4*(3*a^4*b + 29*a^3*b^2 + 82*a^2*b^3 +
104*a*b^4 + 48*b^5 + 8*(3*a^3*b^2 + 11*a^2*b^3 + 16*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^4 + 4*(280*(a^3*b^2 + a^
2*b^3)*d*x*cosh(d*x + c)^4 + 3*a^4*b + 29*a^3*b^2 + 82*a^2*b^3 + 104*a*b^4 + 48*b^5 + 8*(3*a^3*b^2 + 11*a^2*b^
3 + 16*a*b^4 + 8*b^5)*d*x + 15*(a^4*b + 13*a^3*b^2 + 28*a^2*b^3 + 16*a*b^4 + 16*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4
)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(56*(a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^5 + 5*(a^4*b + 13*a^3*b
^2 + 28*a^2*b^3 + 16*a*b^4 + 16*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^3 + (3*a^4*b + 29*a^3*b^2 +
82*a^2*b^3 + 104*a*b^4 + 48*b^5 + 8*(3*a^3*b^2 + 11*a^2*b^3 + 16*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c))*sinh(d*x
+ c)^3 + 16*(a^3*b^2 + a^2*b^3)*d*x + 4*(3*a^4*b + 23*a^3*b^2 + 52*a^2*b^3 + 32*a*b^4 + 16*(a^3*b^2 + 3*a^2*b^
3 + 2*a*b^4)*d*x)*cosh(d*x + c)^2 + 4*(112*(a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^6 + 3*a^4*b + 23*a^3*b^2 + 52
*a^2*b^3 + 32*a*b^4 + 15*(a^4*b + 13*a^3*b^2 + 28*a^2*b^3 + 16*a*b^4 + 16*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*d*x)
*cosh(d*x + c)^4 + 16*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*d*x + 6*(3*a^4*b + 29*a^3*b^2 + 82*a^2*b^3 + 104*a*b^4 +
48*b^5 + 8*(3*a^3*b^2 + 11*a^2*b^3 + 16*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((a^4 - 4*a^3*
b - 8*a^2*b^2)*cosh(d*x + c)^8 + 8*(a^4 - 4*a^3*b - 8*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 - 4*a^3*b
- 8*a^2*b^2)*sinh(d*x + c)^8 + 4*(a^4 - 2*a^3*b - 16*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^6 + 4*(a^4 - 2*a^3*b -
16*a^2*b^2 - 16*a*b^3 + 7*(a^4 - 4*a^3*b - 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^4 - 4*a^3*b -
8*a^2*b^2)*cosh(d*x + c)^3 + 3*(a^4 - 2*a^3*b - 16*a^2*b^2 - 16*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*
a^4 - 4*a^3*b - 48*a^2*b^2 - 96*a*b^3 - 64*b^4)*cosh(d*x + c)^4 + 2*(35*(a^4 - 4*a^3*b - 8*a^2*b^2)*cosh(d*x +
c)^4 + 3*a^4 - 4*a^3*b - 48*a^2*b^2 - 96*a*b^3 - 64*b^4 + 30*(a^4 - 2*a^3*b - 16*a^2*b^2 - 16*a*b^3)*cosh(d*x
+ c)^2)*sinh(d*x + c)^4 + a^4 - 4*a^3*b - 8*a^2*b^2 + 8*(7*(a^4 - 4*a^3*b - 8*a^2*b^2)*cosh(d*x + c)^5 + 10*(
a^4 - 2*a^3*b - 16*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^3 + (3*a^4 - 4*a^3*b - 48*a^2*b^2 - 96*a*b^3 - 64*b^4)*co
sh(d*x + c))*sinh(d*x + c)^3 + 4*(a^4 - 2*a^3*b - 16*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^2 + 4*(7*(a^4 - 4*a^3*b
- 8*a^2*b^2)*cosh(d*x + c)^6 + 15*(a^4 - 2*a^3*b - 16*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^4 + a^4 - 2*a^3*b - 1
6*a^2*b^2 - 16*a*b^3 + 3*(3*a^4 - 4*a^3*b - 48*a^2*b^2 - 96*a*b^3 - 64*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 +
8*((a^4 - 4*a^3*b - 8*a^2*b^2)*cosh(d*x + c)^7 + 3*(a^4 - 2*a^3*b - 16*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^5 +
(3*a^4 - 4*a^3*b - 48*a^2*b^2 - 96*a*b^3 - 64*b^4)*cosh(d*x + c)^3 + (a^4 - 2*a^3*b - 16*a^2*b^2 - 16*a*b^3)*c
osh(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^3*b
^2 + a^2*b^3)*d*x*cosh(d*x + c)^7 + 3*(a^4*b + 13*a^3*b^2 + 28*a^2*b^3 + 16*a*b^4 + 16*(a^3*b^2 + 3*a^2*b^3 +
2*a*b^4)*d*x)*cosh(d*x + c)^5 + 2*(3*a^4*b + 29*a^3*b^2 + 82*a^2*b^3 + 104*a*b^4 + 48*b^5 + 8*(3*a^3*b^2 + 11*
a^2*b^3 + 16*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^3 + (3*a^4*b + 23*a^3*b^2 + 52*a^2*b^3 + 32*a*b^4 + 16*(a^3*b^2
+ 3*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^8 + 8*(a^6*b^2
+ a^5*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^6*b^2 + a^5*b^3)*d*sinh(d*x + c)^8 + 4*(a^6*b^2 + 3*a^5*b^3 +
2*a^4*b^4)*d*cosh(d*x + c)^6 + 4*(7*(a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^2 + (a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)
*d)*sinh(d*x + c)^6 + 2*(3*a^6*b^2 + 11*a^5*b^3 + 16*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^6*b^2 +
a^5*b^3)*d*cosh(d*x + c)^3 + 3*(a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^6
*b^2 + a^5*b^3)*d*cosh(d*x + c)^4 + 30*(a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^2 + (3*a^6*b^2 + 11*a
^5*b^3 + 16*a^4*b^4 + 8*a^3*b^5)*d)*sinh(d*x + c)^4 + 4*(a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^2 +
8*(7*(a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^5 + 10*(a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^3 + (3*a^6*b
^2 + 11*a^5*b^3 + 16*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^6*b^2 + a^5*b^3)*d*cosh(d
*x + c)^6 + 15*(a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^4 + 3*(3*a^6*b^2 + 11*a^5*b^3 + 16*a^4*b^4 +
8*a^3*b^5)*d*cosh(d*x + c)^2 + (a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d)*sinh(d*x + c)^2 + (a^6*b^2 + a^5*b^3)*d +
8*((a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^7 + 3*(a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^5 + (3*a^6*b^2
+ 11*a^5*b^3 + 16*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^3 + (a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c))*
sinh(d*x + c)), 1/8*(8*(a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^8 + 64*(a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)*sinh
(d*x + c)^7 + 8*(a^3*b^2 + a^2*b^3)*d*x*sinh(d*x + c)^8 + 2*(a^4*b + 13*a^3*b^2 + 28*a^2*b^3 + 16*a*b^4 + 16*(
a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^6 + 2*(a^4*b + 13*a^3*b^2 + 28*a^2*b^3 + 16*a*b^4 + 112*(a^3
*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^2 + 16*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*d*x)*sinh(d*x + c)^6 + 4*(112*(a^3*b^
2 + a^2*b^3)*d*x*cosh(d*x + c)^3 + 3*(a^4*b + 13*a^3*b^2 + 28*a^2*b^3 + 16*a*b^4 + 16*(a^3*b^2 + 3*a^2*b^3 + 2
*a*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*a^4*b + 14*a^3*b^2 + 12*a^2*b^3 + 2*(3*a^4*b + 29*a^3*b^2 + 82
*a^2*b^3 + 104*a*b^4 + 48*b^5 + 8*(3*a^3*b^2 + 11*a^2*b^3 + 16*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^4 + 2*(280*(a
^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^4 + 3*a^4*b + 29*a^3*b^2 + 82*a^2*b^3 + 104*a*b^4 + 48*b^5 + 8*(3*a^3*b^2
+ 11*a^2*b^3 + 16*a*b^4 + 8*b^5)*d*x + 15*(a^4*b + 13*a^3*b^2 + 28*a^2*b^3 + 16*a*b^4 + 16*(a^3*b^2 + 3*a^2*b^
3 + 2*a*b^4)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(56*(a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^5 + 5*(a^4*b
+ 13*a^3*b^2 + 28*a^2*b^3 + 16*a*b^4 + 16*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^3 + (3*a^4*b + 29
*a^3*b^2 + 82*a^2*b^3 + 104*a*b^4 + 48*b^5 + 8*(3*a^3*b^2 + 11*a^2*b^3 + 16*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c))
*sinh(d*x + c)^3 + 8*(a^3*b^2 + a^2*b^3)*d*x + 2*(3*a^4*b + 23*a^3*b^2 + 52*a^2*b^3 + 32*a*b^4 + 16*(a^3*b^2 +
3*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^2 + 2*(112*(a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^6 + 3*a^4*b + 23*a^3
*b^2 + 52*a^2*b^3 + 32*a*b^4 + 15*(a^4*b + 13*a^3*b^2 + 28*a^2*b^3 + 16*a*b^4 + 16*(a^3*b^2 + 3*a^2*b^3 + 2*a*
b^4)*d*x)*cosh(d*x + c)^4 + 16*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*d*x + 6*(3*a^4*b + 29*a^3*b^2 + 82*a^2*b^3 + 10
4*a*b^4 + 48*b^5 + 8*(3*a^3*b^2 + 11*a^2*b^3 + 16*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((a^4
- 4*a^3*b - 8*a^2*b^2)*cosh(d*x + c)^8 + 8*(a^4 - 4*a^3*b - 8*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 -
4*a^3*b - 8*a^2*b^2)*sinh(d*x + c)^8 + 4*(a^4 - 2*a^3*b - 16*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^6 + 4*(a^4 - 2
*a^3*b - 16*a^2*b^2 - 16*a*b^3 + 7*(a^4 - 4*a^3*b - 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^4 -
4*a^3*b - 8*a^2*b^2)*cosh(d*x + c)^3 + 3*(a^4 - 2*a^3*b - 16*a^2*b^2 - 16*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^
5 + 2*(3*a^4 - 4*a^3*b - 48*a^2*b^2 - 96*a*b^3 - 64*b^4)*cosh(d*x + c)^4 + 2*(35*(a^4 - 4*a^3*b - 8*a^2*b^2)*c
osh(d*x + c)^4 + 3*a^4 - 4*a^3*b - 48*a^2*b^2 - 96*a*b^3 - 64*b^4 + 30*(a^4 - 2*a^3*b - 16*a^2*b^2 - 16*a*b^3)
*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^4 - 4*a^3*b - 8*a^2*b^2 + 8*(7*(a^4 - 4*a^3*b - 8*a^2*b^2)*cosh(d*x + c)
^5 + 10*(a^4 - 2*a^3*b - 16*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^3 + (3*a^4 - 4*a^3*b - 48*a^2*b^2 - 96*a*b^3 - 6
4*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^4 - 2*a^3*b - 16*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^2 + 4*(7*(a^4
- 4*a^3*b - 8*a^2*b^2)*cosh(d*x + c)^6 + 15*(a^4 - 2*a^3*b - 16*a^2*b^2 - 16*a*b^3)*cosh(d*x + c)^4 + a^4 - 2*
a^3*b - 16*a^2*b^2 - 16*a*b^3 + 3*(3*a^4 - 4*a^3*b - 48*a^2*b^2 - 96*a*b^3 - 64*b^4)*cosh(d*x + c)^2)*sinh(d*x
+ c)^2 + 8*((a^4 - 4*a^3*b - 8*a^2*b^2)*cosh(d*x + c)^7 + 3*(a^4 - 2*a^3*b - 16*a^2*b^2 - 16*a*b^3)*cosh(d*x
+ c)^5 + (3*a^4 - 4*a^3*b - 48*a^2*b^2 - 96*a*b^3 - 64*b^4)*cosh(d*x + c)^3 + (a^4 - 2*a^3*b - 16*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^3*b^2 + a^2*b^3)*d*x*cosh(d*x
+ c)^7 + 3*(a^4*b + 13*a^3*b^2 + 28*a^2*b^3 + 16*a*b^4 + 16*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c
)^5 + 2*(3*a^4*b + 29*a^3*b^2 + 82*a^2*b^3 + 104*a*b^4 + 48*b^5 + 8*(3*a^3*b^2 + 11*a^2*b^3 + 16*a*b^4 + 8*b^5
)*d*x)*cosh(d*x + c)^3 + (3*a^4*b + 23*a^3*b^2 + 52*a^2*b^3 + 32*a*b^4 + 16*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*d*
x)*cosh(d*x + c))*sinh(d*x + c))/((a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^8 + 8*(a^6*b^2 + a^5*b^3)*d*cosh(d*x + c
)*sinh(d*x + c)^7 + (a^6*b^2 + a^5*b^3)*d*sinh(d*x + c)^8 + 4*(a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c
)^6 + 4*(7*(a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^2 + (a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d)*sinh(d*x + c)^6 + 2*(3
*a^6*b^2 + 11*a^5*b^3 + 16*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^3
+ 3*(a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^6*b^2 + a^5*b^3)*d*cosh(d*x
+ c)^4 + 30*(a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^2 + (3*a^6*b^2 + 11*a^5*b^3 + 16*a^4*b^4 + 8*a^
3*b^5)*d)*sinh(d*x + c)^4 + 4*(a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^2 + 8*(7*(a^6*b^2 + a^5*b^3)*d
*cosh(d*x + c)^5 + 10*(a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^3 + (3*a^6*b^2 + 11*a^5*b^3 + 16*a^4*b
^4 + 8*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^6 + 15*(a^6*b^2 +
3*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^4 + 3*(3*a^6*b^2 + 11*a^5*b^3 + 16*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)
^2 + (a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d)*sinh(d*x + c)^2 + (a^6*b^2 + a^5*b^3)*d + 8*((a^6*b^2 + a^5*b^3)*d*c
osh(d*x + c)^7 + 3*(a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^5 + (3*a^6*b^2 + 11*a^5*b^3 + 16*a^4*b^4
+ 8*a^3*b^5)*d*cosh(d*x + c)^3 + (a^6*b^2 + 3*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c))*sinh(d*x + c))]

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giac [B]  time = 1.61, size = 295, normalized size = 2.12 \[ \frac {\frac {8 \, d x}{a^{3}} + \frac {{\left (a^{2} e^{\left (2 \, c\right )} - 4 \, a b e^{\left (2 \, c\right )} - 8 \, b^{2} e^{\left (2 \, c\right )}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right ) e^{\left (-2 \, c\right )}}{\sqrt {-a b - b^{2}} a^{3} b} + \frac {2 \, {\left (a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 12 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 26 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 56 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 20 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 32 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{3} + 6 \, a^{2} b\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2} a^{3} b}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(8*d*x/a^3 + (a^2*e^(2*c) - 4*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)/(sqrt(-a*b - b^2)*a^3*b) + 2*(a^3*e^(6*d*x + 6*c) + 12*a^2*b*e^(6*d*x + 6*c) + 16*a*b^2*e
^(6*d*x + 6*c) + 3*a^3*e^(4*d*x + 4*c) + 26*a^2*b*e^(4*d*x + 4*c) + 56*a*b^2*e^(4*d*x + 4*c) + 48*b^3*e^(4*d*x
+ 4*c) + 3*a^3*e^(2*d*x + 2*c) + 20*a^2*b*e^(2*d*x + 2*c) + 32*a*b^2*e^(2*d*x + 2*c) + a^3 + 6*a^2*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*a^3*b))/d

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maple [B]  time = 0.48, size = 1306, normalized size = 9.40 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.02, size = 2201, normalized size = 15.83 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/256*(a^4 - 20*a^3*b - 120*a^2*b^2 - 160*a*b^3 - 64*b^4)*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b)
)/(a*e^(2*d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^5*b + 2*a^4*b^2 + a^3*b^3)*sqrt((a + b)*b)*d) + 1/64*
(a - 2*b)*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2*d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*
b)))/((a^2*b + 2*a*b^2 + b^3)*sqrt((a + b)*b)*d) - 1/256*(a^4 - 20*a^3*b - 120*a^2*b^2 - 160*a*b^3 - 64*b^4)*l
og((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^
5*b + 2*a^4*b^2 + a^3*b^3)*sqrt((a + b)*b)*d) - 3/128*(a + 4*b)*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a
+ b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^2*b + 2*a*b^2 + b^3)*sqrt((a + b)*b)*d) - 1/6
4*(a - 2*b)*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a +
b)*b)))/((a^2*b + 2*a*b^2 + b^3)*sqrt((a + b)*b)*d) + 1/64*(a^5 + 38*a^4*b + 88*a^3*b^2 + 48*a^2*b^3 + (a^5 +
76*a^4*b + 392*a^3*b^2 + 576*a^2*b^3 + 256*a*b^4)*e^(6*d*x + 6*c) + (3*a^5 + 186*a^4*b + 1024*a^3*b^2 + 2240*
a^2*b^3 + 2176*a*b^4 + 768*b^5)*e^(4*d*x + 4*c) + (3*a^5 + 148*a^4*b + 648*a^3*b^2 + 896*a^2*b^3 + 384*a*b^4)*
e^(2*d*x + 2*c))/((a^7*b + 2*a^6*b^2 + a^5*b^3 + (a^7*b + 2*a^6*b^2 + a^5*b^3)*e^(8*d*x + 8*c) + 4*(a^7*b + 4*
a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*e^(6*d*x + 6*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)*e^(4*d*x + 4*c) + 4*(a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*e^(2*d*x + 2*c))*d) - 1/64*(a^5 + 38*a^4*b
+ 88*a^3*b^2 + 48*a^2*b^3 + (3*a^5 + 148*a^4*b + 648*a^3*b^2 + 896*a^2*b^3 + 384*a*b^4)*e^(-2*d*x - 2*c) + (3
*a^5 + 186*a^4*b + 1024*a^3*b^2 + 2240*a^2*b^3 + 2176*a*b^4 + 768*b^5)*e^(-4*d*x - 4*c) + (a^5 + 76*a^4*b + 39
2*a^3*b^2 + 576*a^2*b^3 + 256*a*b^4)*e^(-6*d*x - 6*c))/((a^7*b + 2*a^6*b^2 + a^5*b^3 + 4*(a^7*b + 4*a^6*b^2 +
5*a^5*b^3 + 2*a^4*b^4)*e^(-2*d*x - 2*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)*e^(-4
*d*x - 4*c) + 4*(a^7*b + 4*a^6*b^2 + 5*a^5*b^3 + 2*a^4*b^4)*e^(-6*d*x - 6*c) + (a^7*b + 2*a^6*b^2 + a^5*b^3)*e
^(-8*d*x - 8*c))*d) + 1/16*(a^4 + 8*a^3*b + 4*a^2*b^2 + (a^4 + 30*a^3*b + 64*a^2*b^2 + 32*a*b^3)*e^(6*d*x + 6*
c) + (3*a^4 + 64*a^3*b + 180*a^2*b^2 + 192*a*b^3 + 64*b^4)*e^(4*d*x + 4*c) + (3*a^4 + 42*a^3*b + 80*a^2*b^2 +
32*a*b^3)*e^(2*d*x + 2*c))/((a^6*b + 2*a^5*b^2 + a^4*b^3 + (a^6*b + 2*a^5*b^2 + a^4*b^3)*e^(8*d*x + 8*c) + 4*(
a^6*b + 4*a^5*b^2 + 5*a^4*b^3 + 2*a^3*b^4)*e^(6*d*x + 6*c) + 2*(3*a^6*b + 14*a^5*b^2 + 27*a^4*b^3 + 24*a^3*b^4
+ 8*a^2*b^5)*e^(4*d*x + 4*c) + 4*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 + 2*a^3*b^4)*e^(2*d*x + 2*c))*d) - 1/16*(a^4
+ 8*a^3*b + 4*a^2*b^2 + (3*a^4 + 42*a^3*b + 80*a^2*b^2 + 32*a*b^3)*e^(-2*d*x - 2*c) + (3*a^4 + 64*a^3*b + 180*
a^2*b^2 + 192*a*b^3 + 64*b^4)*e^(-4*d*x - 4*c) + (a^4 + 30*a^3*b + 64*a^2*b^2 + 32*a*b^3)*e^(-6*d*x - 6*c))/((
a^6*b + 2*a^5*b^2 + a^4*b^3 + 4*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 + 2*a^3*b^4)*e^(-2*d*x - 2*c) + 2*(3*a^6*b + 14
*a^5*b^2 + 27*a^4*b^3 + 24*a^3*b^4 + 8*a^2*b^5)*e^(-4*d*x - 4*c) + 4*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 + 2*a^3*b^
4)*e^(-6*d*x - 6*c) + (a^6*b + 2*a^5*b^2 + a^4*b^3)*e^(-8*d*x - 8*c))*d) - 3/32*(a^3 - 2*a^2*b + (3*a^3 - 4*a^
2*b - 16*a*b^2)*e^(-2*d*x - 2*c) + (3*a^3 + 2*a^2*b - 8*a*b^2 - 16*b^3)*e^(-4*d*x - 4*c) + (a^3 + 4*a^2*b)*e^(
-6*d*x - 6*c))/((a^5*b + 2*a^4*b^2 + a^3*b^3 + 4*(a^5*b + 4*a^4*b^2 + 5*a^3*b^3 + 2*a^2*b^4)*e^(-2*d*x - 2*c)
+ 2*(3*a^5*b + 14*a^4*b^2 + 27*a^3*b^3 + 24*a^2*b^4 + 8*a*b^5)*e^(-4*d*x - 4*c) + 4*(a^5*b + 4*a^4*b^2 + 5*a^3
*b^3 + 2*a^2*b^4)*e^(-6*d*x - 6*c) + (a^5*b + 2*a^4*b^2 + a^3*b^3)*e^(-8*d*x - 8*c))*d) + 1/4*log(a*e^(4*d*x +
4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/(a^3*d) - 1/4*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) +
a)/(a^3*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {tanh}\left (c+d\,x\right )}^4}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((cosh(c + d*x)^6*tanh(c + d*x)^4)/(b + a*cosh(c + d*x)^2)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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