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

Optimal. Leaf size=213 \[ -\frac{\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a d (a+b)^3 \left (a \cosh ^2(c+d x)+b\right )}-\frac{\sqrt{b} \left (15 a^2-10 a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{8 a^{3/2} d (a+b)^4}+\frac{b (2 a-b) \cosh (c+d x)}{4 a d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )^2}-\frac{\cosh (c+d x) \coth ^2(c+d x)}{2 d (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}+\frac{(a-5 b) \tanh ^{-1}(\cosh (c+d x))}{2 d (a+b)^4} \]

[Out]

-(Sqrt[b]*(15*a^2 - 10*a*b - b^2)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(8*a^(3/2)*(a + b)^4*d) + ((a - 5*b
)*ArcTanh[Cosh[c + d*x]])/(2*(a + b)^4*d) + ((2*a - b)*b*Cosh[c + d*x])/(4*a*(a + b)^2*d*(b + a*Cosh[c + d*x]^
2)^2) - ((4*a^2 - 9*a*b - b^2)*Cosh[c + d*x])/(8*a*(a + b)^3*d*(b + a*Cosh[c + d*x]^2)) - (Cosh[c + d*x]*Coth[
c + d*x]^2)/(2*(a + b)*d*(b + a*Cosh[c + d*x]^2)^2)

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Rubi [A]  time = 0.342228, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4133, 470, 578, 527, 522, 206, 205} \[ -\frac{\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a d (a+b)^3 \left (a \cosh ^2(c+d x)+b\right )}-\frac{\sqrt{b} \left (15 a^2-10 a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{8 a^{3/2} d (a+b)^4}+\frac{b (2 a-b) \cosh (c+d x)}{4 a d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )^2}-\frac{\cosh (c+d x) \coth ^2(c+d x)}{2 d (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}+\frac{(a-5 b) \tanh ^{-1}(\cosh (c+d x))}{2 d (a+b)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[b]*(15*a^2 - 10*a*b - b^2)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(8*a^(3/2)*(a + b)^4*d) + ((a - 5*b
)*ArcTanh[Cosh[c + d*x]])/(2*(a + b)^4*d) + ((2*a - b)*b*Cosh[c + d*x])/(4*a*(a + b)^2*d*(b + a*Cosh[c + d*x]^
2)^2) - ((4*a^2 - 9*a*b - b^2)*Cosh[c + d*x])/(8*a*(a + b)^3*d*(b + a*Cosh[c + d*x]^2)) - (Cosh[c + d*x]*Coth[
c + d*x]^2)/(2*(a + b)*d*(b + a*Cosh[c + d*x]^2)^2)

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
]

Rule 470

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

Rule 578

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

Rule 527

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

Rule 522

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

Rule 206

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

Rule 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]

Rubi steps

\begin{align*} \int \frac{\text{csch}^3(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 )^2 \left (b+a x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (3 b+(-a+2 b) x^2\right )}{\left (1-x^2\right ) \left (b+a x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{2 (a+b) d}\\ &=\frac{(2 a-b) b \cosh (c+d x)}{4 a (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac{\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{2 (2 a-b) b-2 \left (2 a^2-8 a b-b^2\right ) x^2}{\left (1-x^2\right ) \left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{8 a (a+b)^2 d}\\ &=\frac{(2 a-b) b \cosh (c+d x)}{4 a (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac{\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a (a+b)^3 d \left (b+a \cosh ^2(c+d x)\right )}-\frac{\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{-2 (11 a-b) b^2+2 b \left (4 a^2-9 a b-b^2\right ) x^2}{\left (1-x^2\right ) \left (b+a x^2\right )} \, dx,x,\cosh (c+d x)\right )}{16 a b (a+b)^3 d}\\ &=\frac{(2 a-b) b \cosh (c+d x)}{4 a (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac{\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a (a+b)^3 d \left (b+a \cosh ^2(c+d x)\right )}-\frac{\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac{(a-5 b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 (a+b)^4 d}-\frac{\left (b \left (15 a^2-10 a b-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{8 a (a+b)^4 d}\\ &=-\frac{\sqrt{b} \left (15 a^2-10 a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{8 a^{3/2} (a+b)^4 d}+\frac{(a-5 b) \tanh ^{-1}(\cosh (c+d x))}{2 (a+b)^4 d}+\frac{(2 a-b) b \cosh (c+d x)}{4 a (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac{\left (4 a^2-9 a b-b^2\right ) \cosh (c+d x)}{8 a (a+b)^3 d \left (b+a \cosh ^2(c+d x)\right )}-\frac{\cosh (c+d x) \coth ^2(c+d x)}{2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}\\ \end{align*}

Mathematica [C]  time = 3.73653, size = 524, normalized size = 2.46 \[ \frac{\text{sech}^5(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\frac{\sqrt{b} \left (-15 a^2+10 a b+b^2\right ) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left (\frac{\sinh (c) \tanh \left (\frac{d x}{2}\right ) \left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )\right )}{\sqrt{b}}\right )}{a^{3/2}}+\frac{\sqrt{b} \left (-15 a^2+10 a b+b^2\right ) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left (\frac{\sinh (c) \tanh \left (\frac{d x}{2}\right ) \left (\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )\right )}{\sqrt{b}}\right )}{a^{3/2}}-\frac{8 b^2 (a+b)^2}{a}+\frac{2 b (9 a+b) (a+b) (a \cosh (2 (c+d x))+a+2 b)}{a}-(a+b) \text{sech}^2\left (\frac{1}{2} (c+d x)\right ) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2+4 (a-5 b) \text{sech}(c+d x) \log \left (\cosh \left (\frac{1}{2} (c+d x)\right )\right ) (a \cosh (2 (c+d x))+a+2 b)^2-(a+b) \text{csch}^2\left (\frac{1}{2} (c+d x)\right ) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2-4 (a-5 b) \text{sech}(c+d x) \log \left (\sinh \left (\frac{1}{2} (c+d x)\right )\right ) (a \cosh (2 (c+d x))+a+2 b)^2\right )}{64 d (a+b)^4 \left (a+b \text{sech}^2(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^3/(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*b*(a + b)*(9*a + b)*(a + 2*b + a*C
osh[2*(c + d*x)]))/a + (Sqrt[b]*(-15*a^2 + 10*a*b + 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^(3/2) + (Sqrt[b]*(-15*a^2 + 10*a*b + 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^(3/2)
- (a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Csch[(c + d*x)/2]^2*Sech[c + d*x] + 4*(a - 5*b)*(a + 2*b + a*Cosh
[2*(c + d*x)])^2*Log[Cosh[(c + d*x)/2]]*Sech[c + d*x] - 4*(a - 5*b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Log[Sinh
[(c + d*x)/2]]*Sech[c + d*x] - (a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[(c + d*x)/2]^2*Sech[c + d*x]))/(
64*(a + b)^4*d*(a + b*Sech[c + d*x]^2)^3)

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Maple [B]  time = 0.101, size = 1555, normalized size = 7.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/d*tanh(1/2*d*x+1/2*c)^2/(a^3+3*a^2*b+3*a*b^2+b^3)+9/4/d*b/(a+b)^4/(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)^6-5/4/d*b^2/(a+b)^
4/(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)^6-13/4/d*b^3/(a+b)^4/(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)^6+1/4/d*b^4/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*ta
nh(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+27/4/d*
b/(a+b)^4/(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)^4-21/4/d*b^2/(a+b)^4/(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+29/4/d*b^3/(a+b)^4/(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
)^4-3/4/d*b^4/(a+b)^4/(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+27/4/d*b/(a+b)^4/(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+1/4/d*b^2/(a+b)^4/(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)^2-23/4/d*b^3/(a+b)^4/(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+3/4/d*b^4/(a+b)^4/(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/(a+b)^4/
(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+17/4/d*b^2/(a+b)^4/(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+7/4/d*b^3/(a+b)^4/(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)^2-1/4/d*b^4/(a+b)^4/(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-15/8/d*b/(a+b)^4*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))+5/4/d*b^2/(a+b)^4/(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*b^3/(a+b)^4/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))-1/8/d/(a+b)^3/tanh(1/2*d*x+1/2*c)^2-1/2/d/(a+b)^4*ln(tanh(1/2*d*x+1/2*c))*a+5/2/d/(a+b)^4*ln(tanh(1/
2*d*x+1/2*c))*b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a - 5*b)*log((e^(d*x + c) + 1)*e^(-c))/(a^4*d + 4*a^3*b*d + 6*a^2*b^2*d + 4*a*b^3*d + b^4*d) - 1/2*(a - 5
*b)*log((e^(d*x + c) - 1)*e^(-c))/(a^4*d + 4*a^3*b*d + 6*a^2*b^2*d + 4*a*b^3*d + b^4*d) - 1/4*((4*a^3*e^(11*c)
- 9*a^2*b*e^(11*c) - a*b^2*e^(11*c))*e^(11*d*x) + (20*a^3*e^(9*c) + 23*a^2*b*e^(9*c) - 29*a*b^2*e^(9*c) + 4*b
^3*e^(9*c))*e^(9*d*x) + 2*(20*a^3*e^(7*c) + 57*a^2*b*e^(7*c) + 47*a*b^2*e^(7*c) - 2*b^3*e^(7*c))*e^(7*d*x) + 2
*(20*a^3*e^(5*c) + 57*a^2*b*e^(5*c) + 47*a*b^2*e^(5*c) - 2*b^3*e^(5*c))*e^(5*d*x) + (20*a^3*e^(3*c) + 23*a^2*b
*e^(3*c) - 29*a*b^2*e^(3*c) + 4*b^3*e^(3*c))*e^(3*d*x) + (4*a^3*e^c - 9*a^2*b*e^c - a*b^2*e^c)*e^(d*x))/(a^6*d
+ 3*a^5*b*d + 3*a^4*b^2*d + a^3*b^3*d + (a^6*d*e^(12*c) + 3*a^5*b*d*e^(12*c) + 3*a^4*b^2*d*e^(12*c) + a^3*b^3
*d*e^(12*c))*e^(12*d*x) + 2*(a^6*d*e^(10*c) + 7*a^5*b*d*e^(10*c) + 15*a^4*b^2*d*e^(10*c) + 13*a^3*b^3*d*e^(10*
c) + 4*a^2*b^4*d*e^(10*c))*e^(10*d*x) - (a^6*d*e^(8*c) + 3*a^5*b*d*e^(8*c) - 13*a^4*b^2*d*e^(8*c) - 47*a^3*b^3
*d*e^(8*c) - 48*a^2*b^4*d*e^(8*c) - 16*a*b^5*d*e^(8*c))*e^(8*d*x) - 4*(a^6*d*e^(6*c) + 7*a^5*b*d*e^(6*c) + 23*
a^4*b^2*d*e^(6*c) + 37*a^3*b^3*d*e^(6*c) + 28*a^2*b^4*d*e^(6*c) + 8*a*b^5*d*e^(6*c))*e^(6*d*x) - (a^6*d*e^(4*c
) + 3*a^5*b*d*e^(4*c) - 13*a^4*b^2*d*e^(4*c) - 47*a^3*b^3*d*e^(4*c) - 48*a^2*b^4*d*e^(4*c) - 16*a*b^5*d*e^(4*c
))*e^(4*d*x) + 2*(a^6*d*e^(2*c) + 7*a^5*b*d*e^(2*c) + 15*a^4*b^2*d*e^(2*c) + 13*a^3*b^3*d*e^(2*c) + 4*a^2*b^4*
d*e^(2*c))*e^(2*d*x)) - 8*integrate(1/32*((15*a^2*b*e^(3*c) - 10*a*b^2*e^(3*c) - b^3*e^(3*c))*e^(3*d*x) - (15*
a^2*b*e^c - 10*a*b^2*e^c - b^3*e^c)*e^(d*x))/(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4 + (a^6*e^(4*c) +
4*a^5*b*e^(4*c) + 6*a^4*b^2*e^(4*c) + 4*a^3*b^3*e^(4*c) + a^2*b^4*e^(4*c))*e^(4*d*x) + 2*(a^6*e^(2*c) + 6*a^5
*b*e^(2*c) + 14*a^4*b^2*e^(2*c) + 16*a^3*b^3*e^(2*c) + 9*a^2*b^4*e^(2*c) + 2*a*b^5*e^(2*c))*e^(2*d*x)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{3}{\left (c + d x \right )}}{\left (a + b \operatorname{sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError