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

Optimal. Leaf size=154 \[ -\frac {b (7 a+3 b) \cosh (c+d x)}{8 a^2 d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )}+\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{5/2} d (a+b)^3}-\frac {b \cosh ^3(c+d x)}{4 a d (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {\tanh ^{-1}(\cosh (c+d x))}{d (a+b)^3} \]

[Out]

-arctanh(cosh(d*x+c))/d/(a+b)^3-1/4*b*cosh(d*x+c)^3/a/(a+b)/d/(b+a*cosh(d*x+c)^2)^2-1/8*b*(7*a+3*b)*cosh(d*x+c
)/a^2/(a+b)^2/d/(b+a*cosh(d*x+c)^2)+1/8*(15*a^2+10*a*b+3*b^2)*arctan(cosh(d*x+c)*a^(1/2)/b^(1/2))*b^(1/2)/a^(5
/2)/(a+b)^3/d

________________________________________________________________________________________

Rubi [A]  time = 0.22, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4133, 470, 578, 522, 206, 205} \[ \frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{5/2} d (a+b)^3}-\frac {b (7 a+3 b) \cosh (c+d x)}{8 a^2 d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )}-\frac {b \cosh ^3(c+d x)}{4 a d (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {\tanh ^{-1}(\cosh (c+d x))}{d (a+b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(8*a^(5/2)*(a + b)^3*d) - ArcTanh[
Cosh[c + d*x]]/((a + b)^3*d) - (b*Cosh[c + d*x]^3)/(4*a*(a + b)*d*(b + a*Cosh[c + d*x]^2)^2) - (b*(7*a + 3*b)*
Cosh[c + d*x])/(8*a^2*(a + b)^2*d*(b + a*Cosh[c + d*x]^2))

Rule 205

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

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 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 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 4133

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = F
reeFactors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[((1 - ff^2*x^2)^((m - 1)/2)*(b + a*(ff*x)^n)^p)/(ff*x)^(n*
p), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p
]

Rubi steps

\begin {align*} \int \frac {\text {csch}(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 (b+a x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {b \cosh ^3(c+d x)}{4 a (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 b+(-4 a-3 b) x^2\right )}{\left (1-x^2\right ) \left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{4 a (a+b) d}\\ &=-\frac {b \cosh ^3(c+d x)}{4 a (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {b (7 a+3 b) \cosh (c+d x)}{8 a^2 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {b (7 a+3 b)+\left (-8 a^2-7 a b-3 b^2\right ) x^2}{\left (1-x^2\right ) \left (b+a x^2\right )} \, dx,x,\cosh (c+d x)\right )}{8 a^2 (a+b)^2 d}\\ &=-\frac {b \cosh ^3(c+d x)}{4 a (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {b (7 a+3 b) \cosh (c+d x)}{8 a^2 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{(a+b)^3 d}+\frac {\left (b \left (15 a^2+10 a b+3 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^2 (a+b)^3 d}\\ &=\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{5/2} (a+b)^3 d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{(a+b)^3 d}-\frac {b \cosh ^3(c+d x)}{4 a (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {b (7 a+3 b) \cosh (c+d x)}{8 a^2 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 2.50, size = 440, normalized size = 2.86 \[ \frac {\text {sech}^5(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\frac {8 b^2 (a+b)^2}{a^2}-\frac {2 b (9 a+5 b) (a+b) (a \cosh (2 (c+d x))+a+2 b)}{a^2}+\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \text {sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left (\frac {\sinh (c) \tanh \left (\frac {d x}{2}\right ) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )}{a^{5/2}}+\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \text {sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left (\frac {\sinh (c) \tanh \left (\frac {d x}{2}\right ) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )}{a^{5/2}}-8 \text {sech}(c+d x) \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right ) (a \cosh (2 (c+d x))+a+2 b)^2+8 \text {sech}(c+d x) \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right ) (a \cosh (2 (c+d x))+a+2 b)^2\right )}{64 d (a+b)^3 \left (a+b \text {sech}^2(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^5*((8*b^2*(a + b)^2)/a^2 - (2*b*(a + b)*(9*a + 5*b)*(a + 2*b +
a*Cosh[2*(c + d*x)]))/a^2 + (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c]
- Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)
/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x])/a^(5/2) + (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*A
rcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt
[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x])/
a^(5/2) - 8*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Log[Cosh[(c + d*x)/2]]*Sech[c + d*x] + 8*(a + 2*b + a*Cosh[2*(c
+ d*x)])^2*Log[Sinh[(c + d*x)/2]]*Sech[c + d*x]))/(64*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^3)

________________________________________________________________________________________

fricas [B]  time = 0.64, size = 8742, normalized size = 56.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(4*(9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^7 + 28*(9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)
*sinh(d*x + c)^6 + 4*(9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*sinh(d*x + c)^7 + 4*(27*a^3*b + 70*a^2*b^2 + 55*a*b^3 +
12*b^4)*cosh(d*x + c)^5 + 4*(27*a^3*b + 70*a^2*b^2 + 55*a*b^3 + 12*b^4 + 21*(9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*c
osh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(7*(9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^3 + (27*a^3*b + 70*a^2*
b^2 + 55*a*b^3 + 12*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(27*a^3*b + 70*a^2*b^2 + 55*a*b^3 + 12*b^4)*cosh(d
*x + c)^3 + 4*(35*(9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^4 + 27*a^3*b + 70*a^2*b^2 + 55*a*b^3 + 12*b^4
+ 10*(27*a^3*b + 70*a^2*b^2 + 55*a*b^3 + 12*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(21*(9*a^3*b + 14*a^2*b
^2 + 5*a*b^3)*cosh(d*x + c)^5 + 10*(27*a^3*b + 70*a^2*b^2 + 55*a*b^3 + 12*b^4)*cosh(d*x + c)^3 + 3*(27*a^3*b +
70*a^2*b^2 + 55*a*b^3 + 12*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 - ((15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x +
c)^8 + 8*(15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (15*a^4 + 10*a^3*b + 3*a^2*b^2)*sinh(
d*x + c)^8 + 4*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^6 + 4*(15*a^4 + 40*a^3*b + 23*a^2*b^2
+ 6*a*b^3 + 7*(15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(15*a^4 + 10*a^3*b + 3*a
^2*b^2)*cosh(d*x + c)^3 + 3*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(45*
a^4 + 150*a^3*b + 209*a^2*b^2 + 104*a*b^3 + 24*b^4)*cosh(d*x + c)^4 + 2*(35*(15*a^4 + 10*a^3*b + 3*a^2*b^2)*co
sh(d*x + c)^4 + 45*a^4 + 150*a^3*b + 209*a^2*b^2 + 104*a*b^3 + 24*b^4 + 30*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6
*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 15*a^4 + 10*a^3*b + 3*a^2*b^2 + 8*(7*(15*a^4 + 10*a^3*b + 3*a^2*b^2
)*cosh(d*x + c)^5 + 10*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^3 + (45*a^4 + 150*a^3*b + 209*
a^2*b^2 + 104*a*b^3 + 24*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*co
sh(d*x + c)^2 + 4*(7*(15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^6 + 15*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*
a*b^3)*cosh(d*x + c)^4 + 15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3 + 3*(45*a^4 + 150*a^3*b + 209*a^2*b^2 + 104*
a*b^3 + 24*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^7 + 3*(15*
a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^5 + (45*a^4 + 150*a^3*b + 209*a^2*b^2 + 104*a*b^3 + 24*b^
4)*cosh(d*x + c)^3 + (15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/a)*log((
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)
+ 4*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 + a*cosh(d*x + c) + (3*a*cosh(d
*x + c)^2 + a)*sinh(d*x + c))*sqrt(-b/a) + a)/(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)) + 4*(9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cosh(d*x + c) + 16*(
a^4*cosh(d*x + c)^8 + 8*a^4*cosh(d*x + c)*sinh(d*x + c)^7 + a^4*sinh(d*x + c)^8 + 4*(a^4 + 2*a^3*b)*cosh(d*x +
c)^6 + 4*(7*a^4*cosh(d*x + c)^2 + a^4 + 2*a^3*b)*sinh(d*x + c)^6 + 8*(7*a^4*cosh(d*x + c)^3 + 3*(a^4 + 2*a^3*
b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^4 + 2*(35*a^4*cosh(d*x + c)^
4 + 3*a^4 + 8*a^3*b + 8*a^2*b^2 + 30*(a^4 + 2*a^3*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^4 + 8*(7*a^4*cosh(d*
x + c)^5 + 10*(a^4 + 2*a^3*b)*cosh(d*x + c)^3 + (3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 +
4*(a^4 + 2*a^3*b)*cosh(d*x + c)^2 + 4*(7*a^4*cosh(d*x + c)^6 + 15*(a^4 + 2*a^3*b)*cosh(d*x + c)^4 + a^4 + 2*a
^3*b + 3*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(a^4*cosh(d*x + c)^7 + 3*(a^4 + 2*
a^3*b)*cosh(d*x + c)^5 + (3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^3 + (a^4 + 2*a^3*b)*cosh(d*x + c))*sinh(d
*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 16*(a^4*cosh(d*x + c)^8 + 8*a^4*cosh(d*x + c)*sinh(d*x + c)^
7 + a^4*sinh(d*x + c)^8 + 4*(a^4 + 2*a^3*b)*cosh(d*x + c)^6 + 4*(7*a^4*cosh(d*x + c)^2 + a^4 + 2*a^3*b)*sinh(d
*x + c)^6 + 8*(7*a^4*cosh(d*x + c)^3 + 3*(a^4 + 2*a^3*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^4 + 8*a^3*b +
8*a^2*b^2)*cosh(d*x + c)^4 + 2*(35*a^4*cosh(d*x + c)^4 + 3*a^4 + 8*a^3*b + 8*a^2*b^2 + 30*(a^4 + 2*a^3*b)*cos
h(d*x + c)^2)*sinh(d*x + c)^4 + a^4 + 8*(7*a^4*cosh(d*x + c)^5 + 10*(a^4 + 2*a^3*b)*cosh(d*x + c)^3 + (3*a^4 +
8*a^3*b + 8*a^2*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^4 + 2*a^3*b)*cosh(d*x + c)^2 + 4*(7*a^4*cosh(d*x +
c)^6 + 15*(a^4 + 2*a^3*b)*cosh(d*x + c)^4 + a^4 + 2*a^3*b + 3*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2)*
sinh(d*x + c)^2 + 8*(a^4*cosh(d*x + c)^7 + 3*(a^4 + 2*a^3*b)*cosh(d*x + c)^5 + (3*a^4 + 8*a^3*b + 8*a^2*b^2)*c
osh(d*x + c)^3 + (a^4 + 2*a^3*b)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 4*(7*(
9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^6 + 5*(27*a^3*b + 70*a^2*b^2 + 55*a*b^3 + 12*b^4)*cosh(d*x + c)^
4 + 9*a^3*b + 14*a^2*b^2 + 5*a*b^3 + 3*(27*a^3*b + 70*a^2*b^2 + 55*a*b^3 + 12*b^4)*cosh(d*x + c)^2)*sinh(d*x +
c))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^8 + 8*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh
(d*x + c)*sinh(d*x + c)^7 + (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*sinh(d*x + c)^8 + 4*(a^7 + 5*a^6*b + 9*a^5
*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^6 + 4*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^2
+ (a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d)*sinh(d*x + c)^6 + 2*(3*a^7 + 17*a^6*b + 41*a^5*b^2 +
51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*
x + c)^3 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^7
+ 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^4 + 30*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d
*cosh(d*x + c)^2 + (3*a^7 + 17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d)*sinh(d*x + c)^4 +
4*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^2 + 8*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^
4*b^3)*d*cosh(d*x + c)^5 + 10*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^3 + (3*a^7 +
17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7 + 3
*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^6 + 15*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cos
h(d*x + c)^4 + 3*(3*a^7 + 17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c)^2 + (a^
7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d)*sinh(d*x + c)^2 + (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d
+ 8*((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^7 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^
3*b^4)*d*cosh(d*x + c)^5 + (3*a^7 + 17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x +
c)^3 + (a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)), -1/8*(2*(9*a^3*b +
14*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^7 + 14*(9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + 2
*(9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*sinh(d*x + c)^7 + 2*(27*a^3*b + 70*a^2*b^2 + 55*a*b^3 + 12*b^4)*cosh(d*x + c
)^5 + 2*(27*a^3*b + 70*a^2*b^2 + 55*a*b^3 + 12*b^4 + 21*(9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^2)*sinh
(d*x + c)^5 + 10*(7*(9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^3 + (27*a^3*b + 70*a^2*b^2 + 55*a*b^3 + 12*
b^4)*cosh(d*x + c))*sinh(d*x + c)^4 + 2*(27*a^3*b + 70*a^2*b^2 + 55*a*b^3 + 12*b^4)*cosh(d*x + c)^3 + 2*(35*(9
*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cosh(d*x + c)^4 + 27*a^3*b + 70*a^2*b^2 + 55*a*b^3 + 12*b^4 + 10*(27*a^3*b + 70
*a^2*b^2 + 55*a*b^3 + 12*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(21*(9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cosh(d
*x + c)^5 + 10*(27*a^3*b + 70*a^2*b^2 + 55*a*b^3 + 12*b^4)*cosh(d*x + c)^3 + 3*(27*a^3*b + 70*a^2*b^2 + 55*a*b
^3 + 12*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + ((15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^8 + 8*(15*a^4 + 1
0*a^3*b + 3*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (15*a^4 + 10*a^3*b + 3*a^2*b^2)*sinh(d*x + c)^8 + 4*(15*a
^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^6 + 4*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3 + 7*(15*a^
4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c
)^3 + 3*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(45*a^4 + 150*a^3*b + 20
9*a^2*b^2 + 104*a*b^3 + 24*b^4)*cosh(d*x + c)^4 + 2*(35*(15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^4 + 45*a
^4 + 150*a^3*b + 209*a^2*b^2 + 104*a*b^3 + 24*b^4 + 30*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c
)^2)*sinh(d*x + c)^4 + 15*a^4 + 10*a^3*b + 3*a^2*b^2 + 8*(7*(15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^5 +
10*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^3 + (45*a^4 + 150*a^3*b + 209*a^2*b^2 + 104*a*b^3
+ 24*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^2 + 4*(7
*(15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^6 + 15*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)
^4 + 15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3 + 3*(45*a^4 + 150*a^3*b + 209*a^2*b^2 + 104*a*b^3 + 24*b^4)*cosh
(d*x + c)^2)*sinh(d*x + c)^2 + 8*((15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^7 + 3*(15*a^4 + 40*a^3*b + 23*
a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^5 + (45*a^4 + 150*a^3*b + 209*a^2*b^2 + 104*a*b^3 + 24*b^4)*cosh(d*x + c)^3 +
(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/a)*arctan(1/2*(a*cosh(d*x + c
)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 + (a + 4*b)*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 +
a + 4*b)*sinh(d*x + c))*sqrt(b/a)/b) - ((15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^8 + 8*(15*a^4 + 10*a^3*
b + 3*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (15*a^4 + 10*a^3*b + 3*a^2*b^2)*sinh(d*x + c)^8 + 4*(15*a^4 + 4
0*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^6 + 4*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3 + 7*(15*a^4 + 10
*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^3 +
3*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(45*a^4 + 150*a^3*b + 209*a^2*
b^2 + 104*a*b^3 + 24*b^4)*cosh(d*x + c)^4 + 2*(35*(15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^4 + 45*a^4 + 1
50*a^3*b + 209*a^2*b^2 + 104*a*b^3 + 24*b^4 + 30*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^2)*s
inh(d*x + c)^4 + 15*a^4 + 10*a^3*b + 3*a^2*b^2 + 8*(7*(15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^5 + 10*(15
*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^3 + (45*a^4 + 150*a^3*b + 209*a^2*b^2 + 104*a*b^3 + 24*b
^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^2 + 4*(7*(15*a
^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^6 + 15*(15*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^4 + 1
5*a^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3 + 3*(45*a^4 + 150*a^3*b + 209*a^2*b^2 + 104*a*b^3 + 24*b^4)*cosh(d*x +
c)^2)*sinh(d*x + c)^2 + 8*((15*a^4 + 10*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^7 + 3*(15*a^4 + 40*a^3*b + 23*a^2*b^
2 + 6*a*b^3)*cosh(d*x + c)^5 + (45*a^4 + 150*a^3*b + 209*a^2*b^2 + 104*a*b^3 + 24*b^4)*cosh(d*x + c)^3 + (15*a
^4 + 40*a^3*b + 23*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/a)*arctan(1/2*(a*cosh(d*x + c) + a*
sinh(d*x + c))*sqrt(b/a)/b) + 2*(9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cosh(d*x + c) + 8*(a^4*cosh(d*x + c)^8 + 8*a^
4*cosh(d*x + c)*sinh(d*x + c)^7 + a^4*sinh(d*x + c)^8 + 4*(a^4 + 2*a^3*b)*cosh(d*x + c)^6 + 4*(7*a^4*cosh(d*x
+ c)^2 + a^4 + 2*a^3*b)*sinh(d*x + c)^6 + 8*(7*a^4*cosh(d*x + c)^3 + 3*(a^4 + 2*a^3*b)*cosh(d*x + c))*sinh(d*x
+ c)^5 + 2*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^4 + 2*(35*a^4*cosh(d*x + c)^4 + 3*a^4 + 8*a^3*b + 8*a^
2*b^2 + 30*(a^4 + 2*a^3*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^4 + 8*(7*a^4*cosh(d*x + c)^5 + 10*(a^4 + 2*a^3
*b)*cosh(d*x + c)^3 + (3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^4 + 2*a^3*b)*cosh(d*
x + c)^2 + 4*(7*a^4*cosh(d*x + c)^6 + 15*(a^4 + 2*a^3*b)*cosh(d*x + c)^4 + a^4 + 2*a^3*b + 3*(3*a^4 + 8*a^3*b
+ 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(a^4*cosh(d*x + c)^7 + 3*(a^4 + 2*a^3*b)*cosh(d*x + c)^5 + (
3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^3 + (a^4 + 2*a^3*b)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c)
+ sinh(d*x + c) + 1) - 8*(a^4*cosh(d*x + c)^8 + 8*a^4*cosh(d*x + c)*sinh(d*x + c)^7 + a^4*sinh(d*x + c)^8 + 4
*(a^4 + 2*a^3*b)*cosh(d*x + c)^6 + 4*(7*a^4*cosh(d*x + c)^2 + a^4 + 2*a^3*b)*sinh(d*x + c)^6 + 8*(7*a^4*cosh(d
*x + c)^3 + 3*(a^4 + 2*a^3*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^4
+ 2*(35*a^4*cosh(d*x + c)^4 + 3*a^4 + 8*a^3*b + 8*a^2*b^2 + 30*(a^4 + 2*a^3*b)*cosh(d*x + c)^2)*sinh(d*x + c)
^4 + a^4 + 8*(7*a^4*cosh(d*x + c)^5 + 10*(a^4 + 2*a^3*b)*cosh(d*x + c)^3 + (3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(
d*x + c))*sinh(d*x + c)^3 + 4*(a^4 + 2*a^3*b)*cosh(d*x + c)^2 + 4*(7*a^4*cosh(d*x + c)^6 + 15*(a^4 + 2*a^3*b)*
cosh(d*x + c)^4 + a^4 + 2*a^3*b + 3*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(a^4*co
sh(d*x + c)^7 + 3*(a^4 + 2*a^3*b)*cosh(d*x + c)^5 + (3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^3 + (a^4 + 2*a
^3*b)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(7*(9*a^3*b + 14*a^2*b^2 + 5*a*
b^3)*cosh(d*x + c)^6 + 5*(27*a^3*b + 70*a^2*b^2 + 55*a*b^3 + 12*b^4)*cosh(d*x + c)^4 + 9*a^3*b + 14*a^2*b^2 +
5*a*b^3 + 3*(27*a^3*b + 70*a^2*b^2 + 55*a*b^3 + 12*b^4)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^7 + 3*a^6*b + 3*a^
5*b^2 + a^4*b^3)*d*cosh(d*x + c)^8 + 8*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 +
(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*sinh(d*x + c)^8 + 4*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^
4)*d*cosh(d*x + c)^6 + 4*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^2 + (a^7 + 5*a^6*b + 9*a^5*b
^2 + 7*a^4*b^3 + 2*a^3*b^4)*d)*sinh(d*x + c)^6 + 2*(3*a^7 + 17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 +
8*a^2*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^3 + 3*(a^7 + 5*a^6*b
+ 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^
4*b^3)*d*cosh(d*x + c)^4 + 30*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^2 + (3*a^7 +
17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d)*sinh(d*x + c)^4 + 4*(a^7 + 5*a^6*b + 9*a^5*b^
2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^2 + 8*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^5 +
10*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^3 + (3*a^7 + 17*a^6*b + 41*a^5*b^2 + 51
*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^
3)*d*cosh(d*x + c)^6 + 15*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^4 + 3*(3*a^7 + 1
7*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c)^2 + (a^7 + 5*a^6*b + 9*a^5*b^2 + 7
*a^4*b^3 + 2*a^3*b^4)*d)*sinh(d*x + c)^2 + (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d + 8*((a^7 + 3*a^6*b + 3*a^5
*b^2 + a^4*b^3)*d*cosh(d*x + c)^7 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c)^5 +
(3*a^7 + 17*a^6*b + 41*a^5*b^2 + 51*a^4*b^3 + 32*a^3*b^4 + 8*a^2*b^5)*d*cosh(d*x + c)^3 + (a^7 + 5*a^6*b + 9*a
^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c))]

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[84,-86]Warning, need to choose a branch for the root of a polynomial with parameters. Thi
s might be wrong.The choice was done assuming [a,b]=[-42,-12]Warning, need to choose a branch for the root of
a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-43,-99]Warning, need to
choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assumin
g [a,b]=[-28,94]Warning, need to choose a branch for the root of a polynomial with parameters. This might be w
rong.The choice was done assuming [a,b]=[-7,46]Warning, need to choose a branch for the root of a polynomial w
ith parameters. This might be wrong.The choice was done assuming [a,b]=[-35,-99]Undef/Unsigned Inf encountered
in limitEvaluation time: 1.19Limit: Max order reached or unable to make series expansion Error: Bad Argument
Value

________________________________________________________________________________________

maple [B]  time = 0.36, size = 1476, normalized size = 9.58 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/4/d*b/(a+b)^3/(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*tanh(1/2*d*x+1/2*c)^6+1/4/d*b^2/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*ta
nh(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)^6+13/4/d*b^3/(a+b)^3/(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*tanh(1/2*d*x+1
/2*c)^6+3/4/d*b^4/(a+b)^3/(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^2*tanh(1/2*d*x+1/2*c)^6-27/4/d*b/(a+b)^3/(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)^4*a+9/4/d*b^2/(a+b)^3/(ta
nh(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)^4-21/4/d*b^3/(a+b)^3/(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*tanh(1/2*d*x+1/2*c)^4-9/4/d*b^4/(a+b)^3/(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^2*tanh(1/2*d*x+1/2*c)^4-27/4/d*b/(
a+b)^3/(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*tanh(1/2*d*x+1/2*c)^2-13/4/d*b^2/(a+b)^3/(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)^2+23/4/d*b^3/(a+b)^3/(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*tanh(1/2*d*x+1/2*c)^2+9
/4/d*b^4/(a+b)^3/(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^2*tanh(1/2*d*x+1/2*c)^2-9/4/d*b/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*ta
nh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*a-21/4/d*b^2/(a+b)^3/(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)^2-15/4/d*b^3/(a+b)^3/(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-3/4/d*b^4/(a+b)^3
/(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
^2+15/8/d*b/(a+b)^3/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b)^(1/2))+5/4/d*b^2/(a+b
)^3/a/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b)^(1/2))+3/8/d*b^3/(a+b)^3/a^2/(a*b)^
(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b)^(1/2))+1/d/(a+b)^3*ln(tanh(1/2*d*x+1/2*c))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (9 \, a^{2} b e^{\left (7 \, c\right )} + 5 \, a b^{2} e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} + {\left (27 \, a^{2} b e^{\left (5 \, c\right )} + 43 \, a b^{2} e^{\left (5 \, c\right )} + 12 \, b^{3} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} + {\left (27 \, a^{2} b e^{\left (3 \, c\right )} + 43 \, a b^{2} e^{\left (3 \, c\right )} + 12 \, b^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (9 \, a^{2} b e^{c} + 5 \, a b^{2} e^{c}\right )} e^{\left (d x\right )}}{4 \, {\left (a^{6} d + 2 \, a^{5} b d + a^{4} b^{2} d + {\left (a^{6} d e^{\left (8 \, c\right )} + 2 \, a^{5} b d e^{\left (8 \, c\right )} + a^{4} b^{2} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 4 \, {\left (a^{6} d e^{\left (6 \, c\right )} + 4 \, a^{5} b d e^{\left (6 \, c\right )} + 5 \, a^{4} b^{2} d e^{\left (6 \, c\right )} + 2 \, a^{3} b^{3} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 2 \, {\left (3 \, a^{6} d e^{\left (4 \, c\right )} + 14 \, a^{5} b d e^{\left (4 \, c\right )} + 27 \, a^{4} b^{2} d e^{\left (4 \, c\right )} + 24 \, a^{3} b^{3} d e^{\left (4 \, c\right )} + 8 \, a^{2} b^{4} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \, {\left (a^{6} d e^{\left (2 \, c\right )} + 4 \, a^{5} b d e^{\left (2 \, c\right )} + 5 \, a^{4} b^{2} d e^{\left (2 \, c\right )} + 2 \, a^{3} b^{3} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}} - \frac {\log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{a^{3} d + 3 \, a^{2} b d + 3 \, a b^{2} d + b^{3} d} + \frac {\log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{a^{3} d + 3 \, a^{2} b d + 3 \, a b^{2} d + b^{3} d} + 2 \, \int \frac {{\left (15 \, a^{2} b e^{\left (3 \, c\right )} + 10 \, a b^{2} e^{\left (3 \, c\right )} + 3 \, b^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - {\left (15 \, a^{2} b e^{c} + 10 \, a b^{2} e^{c} + 3 \, b^{3} e^{c}\right )} e^{\left (d x\right )}}{8 \, {\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3} + {\left (a^{6} e^{\left (4 \, c\right )} + 3 \, a^{5} b e^{\left (4 \, c\right )} + 3 \, a^{4} b^{2} e^{\left (4 \, c\right )} + a^{3} b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a^{6} e^{\left (2 \, c\right )} + 5 \, a^{5} b e^{\left (2 \, c\right )} + 9 \, a^{4} b^{2} e^{\left (2 \, c\right )} + 7 \, a^{3} b^{3} e^{\left (2 \, c\right )} + 2 \, a^{2} b^{4} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((9*a^2*b*e^(7*c) + 5*a*b^2*e^(7*c))*e^(7*d*x) + (27*a^2*b*e^(5*c) + 43*a*b^2*e^(5*c) + 12*b^3*e^(5*c))*e
^(5*d*x) + (27*a^2*b*e^(3*c) + 43*a*b^2*e^(3*c) + 12*b^3*e^(3*c))*e^(3*d*x) + (9*a^2*b*e^c + 5*a*b^2*e^c)*e^(d
*x))/(a^6*d + 2*a^5*b*d + a^4*b^2*d + (a^6*d*e^(8*c) + 2*a^5*b*d*e^(8*c) + a^4*b^2*d*e^(8*c))*e^(8*d*x) + 4*(a
^6*d*e^(6*c) + 4*a^5*b*d*e^(6*c) + 5*a^4*b^2*d*e^(6*c) + 2*a^3*b^3*d*e^(6*c))*e^(6*d*x) + 2*(3*a^6*d*e^(4*c) +
14*a^5*b*d*e^(4*c) + 27*a^4*b^2*d*e^(4*c) + 24*a^3*b^3*d*e^(4*c) + 8*a^2*b^4*d*e^(4*c))*e^(4*d*x) + 4*(a^6*d*
e^(2*c) + 4*a^5*b*d*e^(2*c) + 5*a^4*b^2*d*e^(2*c) + 2*a^3*b^3*d*e^(2*c))*e^(2*d*x)) - log((e^(d*x + c) + 1)*e^
(-c))/(a^3*d + 3*a^2*b*d + 3*a*b^2*d + b^3*d) + log((e^(d*x + c) - 1)*e^(-c))/(a^3*d + 3*a^2*b*d + 3*a*b^2*d +
b^3*d) + 2*integrate(1/8*((15*a^2*b*e^(3*c) + 10*a*b^2*e^(3*c) + 3*b^3*e^(3*c))*e^(3*d*x) - (15*a^2*b*e^c + 1
0*a*b^2*e^c + 3*b^3*e^c)*e^(d*x))/(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3 + (a^6*e^(4*c) + 3*a^5*b*e^(4*c) + 3*a^
4*b^2*e^(4*c) + a^3*b^3*e^(4*c))*e^(4*d*x) + 2*(a^6*e^(2*c) + 5*a^5*b*e^(2*c) + 9*a^4*b^2*e^(2*c) + 7*a^3*b^3*
e^(2*c) + 2*a^2*b^4*e^(2*c))*e^(2*d*x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)*(a + b/cosh(c + d*x)^2)^3),x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(c + d*x)/(a + b*sech(c + d*x)**2)**3, x)

________________________________________________________________________________________