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

Optimal. Leaf size=147 \[ -\frac {(a-b) \cosh (c+d x)}{2 d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )}-\frac {\sqrt {b} (3 a-b) \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{2 \sqrt {a} d (a+b)^3}+\frac {(a-3 b) \tanh ^{-1}(\cosh (c+d x))}{2 d (a+b)^3}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 d (a+b) \left (a \cosh ^2(c+d x)+b\right )} \]

[Out]

1/2*(a-3*b)*arctanh(cosh(d*x+c))/d/(a+b)^3-1/2*(a-b)*cosh(d*x+c)/(a+b)^2/d/(b+a*cosh(d*x+c)^2)-1/2*coth(d*x+c)
*csch(d*x+c)/(a+b)/d/(b+a*cosh(d*x+c)^2)-1/2*(3*a-b)*arctan(cosh(d*x+c)*a^(1/2)/b^(1/2))*b^(1/2)/(a+b)^3/d/a^(
1/2)

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Rubi [A]  time = 0.21, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4133, 470, 527, 522, 206, 205} \[ -\frac {(a-b) \cosh (c+d x)}{2 d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )}-\frac {\sqrt {b} (3 a-b) \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{2 \sqrt {a} d (a+b)^3}+\frac {(a-3 b) \tanh ^{-1}(\cosh (c+d x))}{2 d (a+b)^3}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 d (a+b) \left (a \cosh ^2(c+d x)+b\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(2*Sqrt[a]*(a + b)^3*d) + ((a - 3*b)*ArcTanh[Cosh
[c + d*x]])/(2*(a + b)^3*d) - ((a - b)*Cosh[c + d*x])/(2*(a + b)^2*d*(b + a*Cosh[c + d*x]^2)) - (Coth[c + d*x]
*Csch[c + d*x])/(2*(a + b)*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 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 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}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2 \left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {b+(-a+2 b) x^2}{\left (1-x^2\right ) \left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{2 (a+b) d}\\ &=-\frac {(a-b) \cosh (c+d x)}{2 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {-4 b^2+2 (a-b) b x^2}{\left (1-x^2\right ) \left (b+a x^2\right )} \, dx,x,\cosh (c+d x)\right )}{4 b (a+b)^2 d}\\ &=-\frac {(a-b) \cosh (c+d x)}{2 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )}+\frac {(a-3 b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 (a+b)^3 d}-\frac {((3 a-b) b) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{2 (a+b)^3 d}\\ &=-\frac {(3 a-b) \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{2 \sqrt {a} (a+b)^3 d}+\frac {(a-3 b) \tanh ^{-1}(\cosh (c+d x))}{2 (a+b)^3 d}-\frac {(a-b) \cosh (c+d x)}{2 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [C]  time = 2.23, size = 462, normalized size = 3.14 \[ \frac {\text {sech}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (-(a+b) \text {sech}^2\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)+4 (a-3 b) \text {sech}(c+d x) \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right ) (a \cosh (2 (c+d x))+a+2 b)-\left ((a+b) \text {csch}^2\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)\right )-4 (a-3 b) \text {sech}(c+d x) \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right ) (a \cosh (2 (c+d x))+a+2 b)+\frac {4 \sqrt {b} (b-3 a) \text {sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \tan ^{-1}\left (\frac {\sinh (c) \tanh \left (\frac {d x}{2}\right ) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )}{\sqrt {a}}+\frac {4 \sqrt {b} (b-3 a) \text {sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \tan ^{-1}\left (\frac {\sinh (c) \tanh \left (\frac {d x}{2}\right ) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )}{\sqrt {a}}+8 b (a+b)\right )}{32 d (a+b)^3 \left (a+b \text {sech}^2(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^3*(8*b*(a + b) + (4*Sqrt[b]*(-3*a + b)*ArcTan[((Sqrt[a] - I*Sqr
t[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)])*Sech[c + d*x])/Sqrt[a] + (4*Sqrt[b]*(-3*
a + b)*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)])*Sech[c +
d*x])/Sqrt[a] - (a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[(c + d*x)/2]^2*Sech[c + d*x] + 4*(a - 3*b)*(a + 2
*b + a*Cosh[2*(c + d*x)])*Log[Cosh[(c + d*x)/2]]*Sech[c + d*x] - 4*(a - 3*b)*(a + 2*b + a*Cosh[2*(c + d*x)])*L
og[Sinh[(c + d*x)/2]]*Sech[c + d*x] - (a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[(c + d*x)/2]^2*Sech[c + d*x
]))/(32*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^2)

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fricas [B]  time = 0.73, size = 6878, normalized size = 46.79 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(4*(a^2 - b^2)*cosh(d*x + c)^7 + 28*(a^2 - b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + 4*(a^2 - b^2)*sinh(d*x +
c)^7 + 4*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^5 + 4*(21*(a^2 - b^2)*cosh(d*x + c)^2 + 3*a^2 + 8*a*b + 5*b^2)
*sinh(d*x + c)^5 + 20*(7*(a^2 - b^2)*cosh(d*x + c)^3 + (3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^4
+ 4*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^3 + 4*(35*(a^2 - b^2)*cosh(d*x + c)^4 + 10*(3*a^2 + 8*a*b + 5*b^2)*c
osh(d*x + c)^2 + 3*a^2 + 8*a*b + 5*b^2)*sinh(d*x + c)^3 + 4*(21*(a^2 - b^2)*cosh(d*x + c)^5 + 10*(3*a^2 + 8*a*
b + 5*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + ((3*a^2 - a*b)*cosh(d*
x + c)^8 + 8*(3*a^2 - a*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^2 - a*b)*sinh(d*x + c)^8 + 4*(3*a*b - b^2)*cos
h(d*x + c)^6 + 4*(7*(3*a^2 - a*b)*cosh(d*x + c)^2 + 3*a*b - b^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2 - a*b)*cosh(d*x
+ c)^3 + 3*(3*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(3*a^2 + 11*a*b - 4*b^2)*cosh(d*x + c)^4 + 2*(35*
(3*a^2 - a*b)*cosh(d*x + c)^4 + 30*(3*a*b - b^2)*cosh(d*x + c)^2 - 3*a^2 - 11*a*b + 4*b^2)*sinh(d*x + c)^4 + 8
*(7*(3*a^2 - a*b)*cosh(d*x + c)^5 + 10*(3*a*b - b^2)*cosh(d*x + c)^3 - (3*a^2 + 11*a*b - 4*b^2)*cosh(d*x + c))
*sinh(d*x + c)^3 + 4*(3*a*b - b^2)*cosh(d*x + c)^2 + 4*(7*(3*a^2 - a*b)*cosh(d*x + c)^6 + 15*(3*a*b - b^2)*cos
h(d*x + c)^4 - 3*(3*a^2 + 11*a*b - 4*b^2)*cosh(d*x + c)^2 + 3*a*b - b^2)*sinh(d*x + c)^2 + 3*a^2 - a*b + 8*((3
*a^2 - a*b)*cosh(d*x + c)^7 + 3*(3*a*b - b^2)*cosh(d*x + c)^5 - (3*a^2 + 11*a*b - 4*b^2)*cosh(d*x + c)^3 + (3*
a*b - b^2)*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*cos
h(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*(a^2 - b^2)*cosh(d*x + c) - 2*((a^2 - 3*a*b)*cosh(d*x + c)^8 + 8*(a^2 - 3*a*b)*cosh(d*x + c)*sinh(d*x + c)
^7 + (a^2 - 3*a*b)*sinh(d*x + c)^8 + 4*(a*b - 3*b^2)*cosh(d*x + c)^6 + 4*(7*(a^2 - 3*a*b)*cosh(d*x + c)^2 + a*
b - 3*b^2)*sinh(d*x + c)^6 + 8*(7*(a^2 - 3*a*b)*cosh(d*x + c)^3 + 3*(a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c)
^5 - 2*(a^2 + a*b - 12*b^2)*cosh(d*x + c)^4 + 2*(35*(a^2 - 3*a*b)*cosh(d*x + c)^4 + 30*(a*b - 3*b^2)*cosh(d*x
+ c)^2 - a^2 - a*b + 12*b^2)*sinh(d*x + c)^4 + 8*(7*(a^2 - 3*a*b)*cosh(d*x + c)^5 + 10*(a*b - 3*b^2)*cosh(d*x
+ c)^3 - (a^2 + a*b - 12*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a*b - 3*b^2)*cosh(d*x + c)^2 + 4*(7*(a^2 - 3
*a*b)*cosh(d*x + c)^6 + 15*(a*b - 3*b^2)*cosh(d*x + c)^4 - 3*(a^2 + a*b - 12*b^2)*cosh(d*x + c)^2 + a*b - 3*b^
2)*sinh(d*x + c)^2 + a^2 - 3*a*b + 8*((a^2 - 3*a*b)*cosh(d*x + c)^7 + 3*(a*b - 3*b^2)*cosh(d*x + c)^5 - (a^2 +
a*b - 12*b^2)*cosh(d*x + c)^3 + (a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c)
+ 1) + 2*((a^2 - 3*a*b)*cosh(d*x + c)^8 + 8*(a^2 - 3*a*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 - 3*a*b)*sinh(
d*x + c)^8 + 4*(a*b - 3*b^2)*cosh(d*x + c)^6 + 4*(7*(a^2 - 3*a*b)*cosh(d*x + c)^2 + a*b - 3*b^2)*sinh(d*x + c)
^6 + 8*(7*(a^2 - 3*a*b)*cosh(d*x + c)^3 + 3*(a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(a^2 + a*b - 12*b
^2)*cosh(d*x + c)^4 + 2*(35*(a^2 - 3*a*b)*cosh(d*x + c)^4 + 30*(a*b - 3*b^2)*cosh(d*x + c)^2 - a^2 - a*b + 12*
b^2)*sinh(d*x + c)^4 + 8*(7*(a^2 - 3*a*b)*cosh(d*x + c)^5 + 10*(a*b - 3*b^2)*cosh(d*x + c)^3 - (a^2 + a*b - 12
*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a*b - 3*b^2)*cosh(d*x + c)^2 + 4*(7*(a^2 - 3*a*b)*cosh(d*x + c)^6 +
15*(a*b - 3*b^2)*cosh(d*x + c)^4 - 3*(a^2 + a*b - 12*b^2)*cosh(d*x + c)^2 + a*b - 3*b^2)*sinh(d*x + c)^2 + a^2
- 3*a*b + 8*((a^2 - 3*a*b)*cosh(d*x + c)^7 + 3*(a*b - 3*b^2)*cosh(d*x + c)^5 - (a^2 + a*b - 12*b^2)*cosh(d*x
+ c)^3 + (a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 4*(7*(a^2 - b^2)
*cosh(d*x + c)^6 + 5*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^4 + 3*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^2 + a^2
- b^2)*sinh(d*x + c))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^8 + 8*(a^4 + 3*a^3*b + 3*a^2*b^2 +
a*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*sinh(d*x + c)^8 + 4*(a^3*b + 3
*a^2*b^2 + 3*a*b^3 + b^4)*d*cosh(d*x + c)^6 + 4*(7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^2 + (a^
3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d)*sinh(d*x + c)^6 - 2*(a^4 + 7*a^3*b + 15*a^2*b^2 + 13*a*b^3 + 4*b^4)*d*cosh
(d*x + c)^4 + 8*(7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^3 + 3*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^
4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^4 + 30*(a^3*b
+ 3*a^2*b^2 + 3*a*b^3 + b^4)*d*cosh(d*x + c)^2 - (a^4 + 7*a^3*b + 15*a^2*b^2 + 13*a*b^3 + 4*b^4)*d)*sinh(d*x +
c)^4 + 4*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*cosh(d*x + c)^2 + 8*(7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*c
osh(d*x + c)^5 + 10*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*cosh(d*x + c)^3 - (a^4 + 7*a^3*b + 15*a^2*b^2 + 13*a
*b^3 + 4*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^6 +
15*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*cosh(d*x + c)^4 - 3*(a^4 + 7*a^3*b + 15*a^2*b^2 + 13*a*b^3 + 4*b^4)*d
*cosh(d*x + c)^2 + (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d)*sinh(d*x + c)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3
)*d + 8*((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^7 + 3*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*cosh(
d*x + c)^5 - (a^4 + 7*a^3*b + 15*a^2*b^2 + 13*a*b^3 + 4*b^4)*d*cosh(d*x + c)^3 + (a^3*b + 3*a^2*b^2 + 3*a*b^3
+ b^4)*d*cosh(d*x + c))*sinh(d*x + c)), -1/2*(2*(a^2 - b^2)*cosh(d*x + c)^7 + 14*(a^2 - b^2)*cosh(d*x + c)*sin
h(d*x + c)^6 + 2*(a^2 - b^2)*sinh(d*x + c)^7 + 2*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^5 + 2*(21*(a^2 - b^2)*c
osh(d*x + c)^2 + 3*a^2 + 8*a*b + 5*b^2)*sinh(d*x + c)^5 + 10*(7*(a^2 - b^2)*cosh(d*x + c)^3 + (3*a^2 + 8*a*b +
5*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + 2*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^3 + 2*(35*(a^2 - b^2)*cosh(d*
x + c)^4 + 10*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^2 + 3*a^2 + 8*a*b + 5*b^2)*sinh(d*x + c)^3 + 2*(21*(a^2 -
b^2)*cosh(d*x + c)^5 + 10*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c))*s
inh(d*x + c)^2 - ((3*a^2 - a*b)*cosh(d*x + c)^8 + 8*(3*a^2 - a*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^2 - a*b
)*sinh(d*x + c)^8 + 4*(3*a*b - b^2)*cosh(d*x + c)^6 + 4*(7*(3*a^2 - a*b)*cosh(d*x + c)^2 + 3*a*b - b^2)*sinh(d
*x + c)^6 + 8*(7*(3*a^2 - a*b)*cosh(d*x + c)^3 + 3*(3*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(3*a^2 + 1
1*a*b - 4*b^2)*cosh(d*x + c)^4 + 2*(35*(3*a^2 - a*b)*cosh(d*x + c)^4 + 30*(3*a*b - b^2)*cosh(d*x + c)^2 - 3*a^
2 - 11*a*b + 4*b^2)*sinh(d*x + c)^4 + 8*(7*(3*a^2 - a*b)*cosh(d*x + c)^5 + 10*(3*a*b - b^2)*cosh(d*x + c)^3 -
(3*a^2 + 11*a*b - 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(3*a*b - b^2)*cosh(d*x + c)^2 + 4*(7*(3*a^2 - a*b)
*cosh(d*x + c)^6 + 15*(3*a*b - b^2)*cosh(d*x + c)^4 - 3*(3*a^2 + 11*a*b - 4*b^2)*cosh(d*x + c)^2 + 3*a*b - b^2
)*sinh(d*x + c)^2 + 3*a^2 - a*b + 8*((3*a^2 - a*b)*cosh(d*x + c)^7 + 3*(3*a*b - b^2)*cosh(d*x + c)^5 - (3*a^2
+ 11*a*b - 4*b^2)*cosh(d*x + c)^3 + (3*a*b - b^2)*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) + ((3*a^2 - a*b)*cosh(d*x + c)^8 + 8*(3*a^2 - a*b)*cosh(d*x + c)*s
inh(d*x + c)^7 + (3*a^2 - a*b)*sinh(d*x + c)^8 + 4*(3*a*b - b^2)*cosh(d*x + c)^6 + 4*(7*(3*a^2 - a*b)*cosh(d*x
+ c)^2 + 3*a*b - b^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2 - a*b)*cosh(d*x + c)^3 + 3*(3*a*b - b^2)*cosh(d*x + c))*s
inh(d*x + c)^5 - 2*(3*a^2 + 11*a*b - 4*b^2)*cosh(d*x + c)^4 + 2*(35*(3*a^2 - a*b)*cosh(d*x + c)^4 + 30*(3*a*b
- b^2)*cosh(d*x + c)^2 - 3*a^2 - 11*a*b + 4*b^2)*sinh(d*x + c)^4 + 8*(7*(3*a^2 - a*b)*cosh(d*x + c)^5 + 10*(3*
a*b - b^2)*cosh(d*x + c)^3 - (3*a^2 + 11*a*b - 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(3*a*b - b^2)*cosh(d*
x + c)^2 + 4*(7*(3*a^2 - a*b)*cosh(d*x + c)^6 + 15*(3*a*b - b^2)*cosh(d*x + c)^4 - 3*(3*a^2 + 11*a*b - 4*b^2)*
cosh(d*x + c)^2 + 3*a*b - b^2)*sinh(d*x + c)^2 + 3*a^2 - a*b + 8*((3*a^2 - a*b)*cosh(d*x + c)^7 + 3*(3*a*b - b
^2)*cosh(d*x + c)^5 - (3*a^2 + 11*a*b - 4*b^2)*cosh(d*x + c)^3 + (3*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c))*s
qrt(b/a)*arctan(1/2*(a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt(b/a)/b) + 2*(a^2 - b^2)*cosh(d*x + c) - ((a^2 - 3
*a*b)*cosh(d*x + c)^8 + 8*(a^2 - 3*a*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 - 3*a*b)*sinh(d*x + c)^8 + 4*(a*b
- 3*b^2)*cosh(d*x + c)^6 + 4*(7*(a^2 - 3*a*b)*cosh(d*x + c)^2 + a*b - 3*b^2)*sinh(d*x + c)^6 + 8*(7*(a^2 - 3*
a*b)*cosh(d*x + c)^3 + 3*(a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(a^2 + a*b - 12*b^2)*cosh(d*x + c)^4
+ 2*(35*(a^2 - 3*a*b)*cosh(d*x + c)^4 + 30*(a*b - 3*b^2)*cosh(d*x + c)^2 - a^2 - a*b + 12*b^2)*sinh(d*x + c)^
4 + 8*(7*(a^2 - 3*a*b)*cosh(d*x + c)^5 + 10*(a*b - 3*b^2)*cosh(d*x + c)^3 - (a^2 + a*b - 12*b^2)*cosh(d*x + c)
)*sinh(d*x + c)^3 + 4*(a*b - 3*b^2)*cosh(d*x + c)^2 + 4*(7*(a^2 - 3*a*b)*cosh(d*x + c)^6 + 15*(a*b - 3*b^2)*co
sh(d*x + c)^4 - 3*(a^2 + a*b - 12*b^2)*cosh(d*x + c)^2 + a*b - 3*b^2)*sinh(d*x + c)^2 + a^2 - 3*a*b + 8*((a^2
- 3*a*b)*cosh(d*x + c)^7 + 3*(a*b - 3*b^2)*cosh(d*x + c)^5 - (a^2 + a*b - 12*b^2)*cosh(d*x + c)^3 + (a*b - 3*b
^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + ((a^2 - 3*a*b)*cosh(d*x + c)^8 + 8*
(a^2 - 3*a*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 - 3*a*b)*sinh(d*x + c)^8 + 4*(a*b - 3*b^2)*cosh(d*x + c)^6
+ 4*(7*(a^2 - 3*a*b)*cosh(d*x + c)^2 + a*b - 3*b^2)*sinh(d*x + c)^6 + 8*(7*(a^2 - 3*a*b)*cosh(d*x + c)^3 + 3*(
a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(a^2 + a*b - 12*b^2)*cosh(d*x + c)^4 + 2*(35*(a^2 - 3*a*b)*cos
h(d*x + c)^4 + 30*(a*b - 3*b^2)*cosh(d*x + c)^2 - a^2 - a*b + 12*b^2)*sinh(d*x + c)^4 + 8*(7*(a^2 - 3*a*b)*cos
h(d*x + c)^5 + 10*(a*b - 3*b^2)*cosh(d*x + c)^3 - (a^2 + a*b - 12*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a*b
- 3*b^2)*cosh(d*x + c)^2 + 4*(7*(a^2 - 3*a*b)*cosh(d*x + c)^6 + 15*(a*b - 3*b^2)*cosh(d*x + c)^4 - 3*(a^2 + a
*b - 12*b^2)*cosh(d*x + c)^2 + a*b - 3*b^2)*sinh(d*x + c)^2 + a^2 - 3*a*b + 8*((a^2 - 3*a*b)*cosh(d*x + c)^7 +
3*(a*b - 3*b^2)*cosh(d*x + c)^5 - (a^2 + a*b - 12*b^2)*cosh(d*x + c)^3 + (a*b - 3*b^2)*cosh(d*x + c))*sinh(d*
x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(7*(a^2 - b^2)*cosh(d*x + c)^6 + 5*(3*a^2 + 8*a*b + 5*b^2)*
cosh(d*x + c)^4 + 3*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c))/((a^4 + 3*a^3*b + 3*a^
2*b^2 + a*b^3)*d*cosh(d*x + c)^8 + 8*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^
4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*sinh(d*x + c)^8 + 4*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*cosh(d*x + c)^6 +
4*(7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^2 + (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d)*sinh(d*x
+ c)^6 - 2*(a^4 + 7*a^3*b + 15*a^2*b^2 + 13*a*b^3 + 4*b^4)*d*cosh(d*x + c)^4 + 8*(7*(a^4 + 3*a^3*b + 3*a^2*b^2
+ a*b^3)*d*cosh(d*x + c)^3 + 3*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(
a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^4 + 30*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*cosh(d*x + c)^
2 - (a^4 + 7*a^3*b + 15*a^2*b^2 + 13*a*b^3 + 4*b^4)*d)*sinh(d*x + c)^4 + 4*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)
*d*cosh(d*x + c)^2 + 8*(7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^5 + 10*(a^3*b + 3*a^2*b^2 + 3*a*
b^3 + b^4)*d*cosh(d*x + c)^3 - (a^4 + 7*a^3*b + 15*a^2*b^2 + 13*a*b^3 + 4*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^
3 + 4*(7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^6 + 15*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*cosh
(d*x + c)^4 - 3*(a^4 + 7*a^3*b + 15*a^2*b^2 + 13*a*b^3 + 4*b^4)*d*cosh(d*x + c)^2 + (a^3*b + 3*a^2*b^2 + 3*a*b
^3 + b^4)*d)*sinh(d*x + c)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d + 8*((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*
d*cosh(d*x + c)^7 + 3*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*cosh(d*x + c)^5 - (a^4 + 7*a^3*b + 15*a^2*b^2 + 13
*a*b^3 + 4*b^4)*d*cosh(d*x + c)^3 + (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*cosh(d*x + c))*sinh(d*x + c))]

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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)^3/(a+b*sech(d*x+c)^2)^2,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]=[6,-20]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]=[89,-63]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]=[12,-32]Warning, need to ch
oose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [
a,b]=[2,72]Undef/Unsigned Inf encountered in limitEvaluation time: 0.7Limit: Max order reached or unable to ma
ke series expansion Error: Bad Argument Value

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maple [B]  time = 0.42, size = 496, normalized size = 3.37 \[ \frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a +b \right )^{3} \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}-\frac {b^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a +b \right )^{3} \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}+\frac {b a}{d \left (a +b \right )^{3} \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}+\frac {b^{2}}{d \left (a +b \right )^{3} \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}-\frac {3 b \arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a -2 b}{4 \sqrt {a b}}\right ) a}{2 d \left (a +b \right )^{3} \sqrt {a b}}+\frac {b^{2} \arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a -2 b}{4 \sqrt {a b}}\right )}{2 d \left (a +b \right )^{3} \sqrt {a b}}-\frac {1}{8 d \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2 d \left (a +b \right )^{3}}+\frac {3 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{2 d \left (a +b \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/d*tanh(1/2*d*x+1/2*c)^2/(a^2+2*a*b+b^2)+1/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*t
anh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)*a*tanh(1/2*d*x+1/2*c)^2-1/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)*tanh(1/2*d*x+1/2*c)^
2+1/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)*a+1/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*tan
h(1/2*d*x+1/2*c)^2*b+a+b)-3/2/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))*a+1/2/d*b^2/(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))-1/8/d/
(a+b)^2/tanh(1/2*d*x+1/2*c)^2-1/2/d/(a+b)^3*ln(tanh(1/2*d*x+1/2*c))*a+3/2/d/(a+b)^3*ln(tanh(1/2*d*x+1/2*c))*b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (a - 3 \, b\right )} \log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{2 \, {\left (a^{3} d + 3 \, a^{2} b d + 3 \, a b^{2} d + b^{3} d\right )}} - \frac {{\left (a - 3 \, b\right )} \log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{2 \, {\left (a^{3} d + 3 \, a^{2} b d + 3 \, a b^{2} d + b^{3} d\right )}} - \frac {{\left (a e^{\left (7 \, c\right )} - b e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} + {\left (3 \, a e^{\left (5 \, c\right )} + 5 \, b e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} + {\left (3 \, a e^{\left (3 \, c\right )} + 5 \, b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (a e^{c} - b e^{c}\right )} e^{\left (d x\right )}}{a^{3} d + 2 \, a^{2} b d + a b^{2} d + {\left (a^{3} d e^{\left (8 \, c\right )} + 2 \, a^{2} b d e^{\left (8 \, c\right )} + a b^{2} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 4 \, {\left (a^{2} b d e^{\left (6 \, c\right )} + 2 \, a b^{2} d e^{\left (6 \, c\right )} + b^{3} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} - 2 \, {\left (a^{3} d e^{\left (4 \, c\right )} + 6 \, a^{2} b d e^{\left (4 \, c\right )} + 9 \, a b^{2} d e^{\left (4 \, c\right )} + 4 \, b^{3} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \, {\left (a^{2} b d e^{\left (2 \, c\right )} + 2 \, a b^{2} d e^{\left (2 \, c\right )} + b^{3} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} - 8 \, \int \frac {{\left (3 \, a b e^{\left (3 \, c\right )} - b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - {\left (3 \, a b e^{c} - b^{2} e^{c}\right )} e^{\left (d x\right )}}{8 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} + {\left (a^{4} e^{\left (4 \, c\right )} + 3 \, a^{3} b e^{\left (4 \, c\right )} + 3 \, a^{2} b^{2} e^{\left (4 \, c\right )} + a b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a^{4} e^{\left (2 \, c\right )} + 5 \, a^{3} b e^{\left (2 \, c\right )} + 9 \, a^{2} b^{2} e^{\left (2 \, c\right )} + 7 \, a b^{3} e^{\left (2 \, c\right )} + 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)^3/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/2*(a - 3*b)*log((e^(d*x + c) + 1)*e^(-c))/(a^3*d + 3*a^2*b*d + 3*a*b^2*d + b^3*d) - 1/2*(a - 3*b)*log((e^(d*
x + c) - 1)*e^(-c))/(a^3*d + 3*a^2*b*d + 3*a*b^2*d + b^3*d) - ((a*e^(7*c) - b*e^(7*c))*e^(7*d*x) + (3*a*e^(5*c
) + 5*b*e^(5*c))*e^(5*d*x) + (3*a*e^(3*c) + 5*b*e^(3*c))*e^(3*d*x) + (a*e^c - b*e^c)*e^(d*x))/(a^3*d + 2*a^2*b
*d + a*b^2*d + (a^3*d*e^(8*c) + 2*a^2*b*d*e^(8*c) + a*b^2*d*e^(8*c))*e^(8*d*x) + 4*(a^2*b*d*e^(6*c) + 2*a*b^2*
d*e^(6*c) + b^3*d*e^(6*c))*e^(6*d*x) - 2*(a^3*d*e^(4*c) + 6*a^2*b*d*e^(4*c) + 9*a*b^2*d*e^(4*c) + 4*b^3*d*e^(4
*c))*e^(4*d*x) + 4*(a^2*b*d*e^(2*c) + 2*a*b^2*d*e^(2*c) + b^3*d*e^(2*c))*e^(2*d*x)) - 8*integrate(1/8*((3*a*b*
e^(3*c) - b^2*e^(3*c))*e^(3*d*x) - (3*a*b*e^c - b^2*e^c)*e^(d*x))/(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 + (a^4*e^
(4*c) + 3*a^3*b*e^(4*c) + 3*a^2*b^2*e^(4*c) + a*b^3*e^(4*c))*e^(4*d*x) + 2*(a^4*e^(2*c) + 5*a^3*b*e^(2*c) + 9*
a^2*b^2*e^(2*c) + 7*a*b^3*e^(2*c) + 2*b^4*e^(2*c))*e^(2*d*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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