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

Optimal. Leaf size=123 \[ -\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{2 d (a+b)^{7/2}}-\frac{a b \tanh (c+d x)}{2 d (a+b)^3 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac{\coth ^3(c+d x)}{3 d (a+b)^2}+\frac{(a-b) \coth (c+d x)}{d (a+b)^3} \]

[Out]

-((3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*(a + b)^(7/2)*d) + ((a - b)*Coth[c + d*
x])/((a + b)^3*d) - Coth[c + d*x]^3/(3*(a + b)^2*d) - (a*b*Tanh[c + d*x])/(2*(a + b)^3*d*(a + b - b*Tanh[c + d
*x]^2))

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Rubi [A]  time = 0.195059, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {4132, 456, 1261, 208} \[ -\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b}}\right )}{2 d (a+b)^{7/2}}-\frac{a b \tanh (c+d x)}{2 d (a+b)^3 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac{\coth ^3(c+d x)}{3 d (a+b)^2}+\frac{(a-b) \coth (c+d x)}{d (a+b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*(a + b)^(7/2)*d) + ((a - b)*Coth[c + d*
x])/((a + b)^3*d) - Coth[c + d*x]^3/(3*(a + b)^2*d) - (a*b*Tanh[c + d*x])/(2*(a + b)^3*d*(a + b - b*Tanh[c + d
*x]^2))

Rule 4132

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

Rule 456

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1261

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 208

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

Rubi steps

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

Mathematica [B]  time = 5.93029, size = 295, normalized size = 2.4 \[ \frac{\text{sech}^4(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (-3 a b \text{sech}(2 c) \sinh (2 d x)-2 (a+b) \coth (c) \text{csch}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b)+2 (a+b) \text{csch}(c) \sinh (d x) \text{csch}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b)-4 (a-2 b) \text{csch}(c) \sinh (d x) \text{csch}(c+d x) (a \cosh (2 (c+d x))+a+2 b)-\frac{3 b (3 a-2 b) (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b) \tanh ^{-1}\left (\frac{(\cosh (2 c)-\sinh (2 c)) \text{sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}\right )}{\sqrt{a+b} \sqrt{b (\cosh (c)-\sinh (c))^4}}+3 b (a+2 b) \tanh (2 c)\right )}{24 d (a+b)^3 \left (a+b \text{sech}^2(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*(-2*(a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Coth[c]*Csch[c +
d*x]^2 - (3*(3*a - 2*b)*b*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])]*(a + 2*b + a*Cosh[2*(c + d*x)])*(Cosh[2*c] - Sinh[2*c]))/(Sqr
t[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) - 4*(a - 2*b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]*Si
nh[d*x] + 2*(a + b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]^3*Sinh[d*x] - 3*a*b*Sech[2*c]*Sinh[2
*d*x] + 3*b*(a + 2*b)*Tanh[2*c]))/(24*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^2)

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Maple [B]  time = 0.102, size = 577, normalized size = 4.7 \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)^4/(a+b*sech(d*x+c)^2)^2,x)

[Out]

-1/24/d/(a+b)/(a^2+2*a*b+b^2)*a*tanh(1/2*d*x+1/2*c)^3-1/24/d/(a+b)/(a^2+2*a*b+b^2)*b*tanh(1/2*d*x+1/2*c)^3+3/8
/d/(a+b)/(a^2+2*a*b+b^2)*a*tanh(1/2*d*x+1/2*c)-5/8/d/(a+b)/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)*b-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*t
anh(1/2*d*x+1/2*c)^3-1/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*tanh(1/2*d*x+1/2*c)-3/4/d*b^(1/2)/(a+b)^(7/2)*a*ln((a+b)^(1/2)*tanh(1/2*d*x+1
/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))+3/4/d*b^(1/2)/(a+b)^(7/2)*a*ln(-(a+b)^(1/2)*tanh(1/2*d*x+1/
2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)-(a+b)^(1/2))+1/2/d*b^(3/2)/(a+b)^(7/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)
^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))-1/2/d*b^(3/2)/(a+b)^(7/2)*ln(-(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+
2*tanh(1/2*d*x+1/2*c)*b^(1/2)-(a+b)^(1/2))-1/24/d/(a+b)^2/tanh(1/2*d*x+1/2*c)^3+3/8/d/(a+b)^3/tanh(1/2*d*x+1/2
*c)*a-5/8/d/(a+b)^3/tanh(1/2*d*x+1/2*c)*b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.99228, size = 14893, normalized size = 121.08 \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)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[-1/12*(12*(3*a*b - 2*b^2)*cosh(d*x + c)^8 + 96*(3*a*b - 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + 12*(3*a*b - 2*
b^2)*sinh(d*x + c)^8 + 24*(2*a^2 + 3*a*b + 11*b^2)*cosh(d*x + c)^6 + 24*(14*(3*a*b - 2*b^2)*cosh(d*x + c)^2 +
2*a^2 + 3*a*b + 11*b^2)*sinh(d*x + c)^6 + 48*(14*(3*a*b - 2*b^2)*cosh(d*x + c)^3 + 3*(2*a^2 + 3*a*b + 11*b^2)*
cosh(d*x + c))*sinh(d*x + c)^5 + 8*(10*a^2 + 22*a*b - 33*b^2)*cosh(d*x + c)^4 + 8*(105*(3*a*b - 2*b^2)*cosh(d*
x + c)^4 + 45*(2*a^2 + 3*a*b + 11*b^2)*cosh(d*x + c)^2 + 10*a^2 + 22*a*b - 33*b^2)*sinh(d*x + c)^4 + 32*(21*(3
*a*b - 2*b^2)*cosh(d*x + c)^5 + 15*(2*a^2 + 3*a*b + 11*b^2)*cosh(d*x + c)^3 + (10*a^2 + 22*a*b - 33*b^2)*cosh(
d*x + c))*sinh(d*x + c)^3 + 8*(2*a^2 - 9*a*b + 19*b^2)*cosh(d*x + c)^2 + 8*(42*(3*a*b - 2*b^2)*cosh(d*x + c)^6
+ 45*(2*a^2 + 3*a*b + 11*b^2)*cosh(d*x + c)^4 + 6*(10*a^2 + 22*a*b - 33*b^2)*cosh(d*x + c)^2 + 2*a^2 - 9*a*b
+ 19*b^2)*sinh(d*x + c)^2 + 3*((3*a^2 - 2*a*b)*cosh(d*x + c)^10 + 10*(3*a^2 - 2*a*b)*cosh(d*x + c)*sinh(d*x +
c)^9 + (3*a^2 - 2*a*b)*sinh(d*x + c)^10 - (3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^8 + (45*(3*a^2 - 2*a*b)*cosh(
d*x + c)^2 - 3*a^2 + 14*a*b - 8*b^2)*sinh(d*x + c)^8 + 8*(15*(3*a^2 - 2*a*b)*cosh(d*x + c)^3 - (3*a^2 - 14*a*b
+ 8*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c)^6 + 2*(105*(3*a^2 - 2*a*b
)*cosh(d*x + c)^4 - 14*(3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^2 - 3*a^2 - 16*a*b + 12*b^2)*sinh(d*x + c)^6 + 4
*(63*(3*a^2 - 2*a*b)*cosh(d*x + c)^5 - 14*(3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^3 - 3*(3*a^2 + 16*a*b - 12*b^
2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c)^4 + 2*(105*(3*a^2 - 2*a*b)*cosh(
d*x + c)^6 - 35*(3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^4 - 15*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c)^2 + 3*a^
2 + 16*a*b - 12*b^2)*sinh(d*x + c)^4 + 8*(15*(3*a^2 - 2*a*b)*cosh(d*x + c)^7 - 7*(3*a^2 - 14*a*b + 8*b^2)*cosh
(d*x + c)^5 - 5*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c)^3 + (3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c))*sinh(d*x
+ c)^3 + (3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^2 + (45*(3*a^2 - 2*a*b)*cosh(d*x + c)^8 - 28*(3*a^2 - 14*a*b +
8*b^2)*cosh(d*x + c)^6 - 30*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c)^4 + 12*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x
+ c)^2 + 3*a^2 - 14*a*b + 8*b^2)*sinh(d*x + c)^2 - 3*a^2 + 2*a*b + 2*(5*(3*a^2 - 2*a*b)*cosh(d*x + c)^9 - 4*(
3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^7 - 6*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c)^5 + 4*(3*a^2 + 16*a*b - 12
*b^2)*cosh(d*x + c)^3 + (3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/(a + b))*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^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c)
+ (a^2 + a*b)*sinh(d*x + c)^2 + a^2 + 3*a*b + 2*b^2)*sqrt(b/(a + b)))/(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)) - 16*a^2 + 44*a*b + 16*(6*(3*a*b
- 2*b^2)*cosh(d*x + c)^7 + 9*(2*a^2 + 3*a*b + 11*b^2)*cosh(d*x + c)^5 + 2*(10*a^2 + 22*a*b - 33*b^2)*cosh(d*x
+ c)^3 + (2*a^2 - 9*a*b + 19*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d
*x + c)^10 + 10*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^4 + 3*a^3*b + 3*a^2*b
^2 + a*b^3)*d*sinh(d*x + c)^10 - (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c)^8 + (45*(a^4 + 3
*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^2 - (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d)*sinh(d*x + c)^
8 - 2*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c)^6 + 8*(15*(a^4 + 3*a^3*b + 3*a^2*b^2 + a
*b^3)*d*cosh(d*x + c)^3 - (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(1
05*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^4 - 14*(a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*c
osh(d*x + c)^2 - (a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d)*sinh(d*x + c)^6 + 2*(a^4 + 9*a^3*b + 21*a^
2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c)^4 + 4*(63*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^5 - 14
*(a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c)^3 - 3*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6
*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^6 - 35*(a^
4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c)^4 - 15*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^
4)*d*cosh(d*x + c)^2 + (a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d)*sinh(d*x + c)^4 + (a^4 - a^3*b - 9*a
^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c)^2 + 8*(15*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^7 - 7
*(a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c)^5 - 5*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6
*b^4)*d*cosh(d*x + c)^3 + (a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (
45*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^8 - 28*(a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*c
osh(d*x + c)^6 - 30*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c)^4 + 12*(a^4 + 9*a^3*b + 21
*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c)^2 + (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d)*sinh(d*x + c)
^2 - (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d + 2*(5*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^9 - 4*(a
^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c)^7 - 6*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^
4)*d*cosh(d*x + c)^5 + 4*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c)^3 + (a^4 - a^3*b - 9*
a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c))*sinh(d*x + c)), -1/6*(6*(3*a*b - 2*b^2)*cosh(d*x + c)^8 + 48*(3*a
*b - 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + 6*(3*a*b - 2*b^2)*sinh(d*x + c)^8 + 12*(2*a^2 + 3*a*b + 11*b^2)*co
sh(d*x + c)^6 + 12*(14*(3*a*b - 2*b^2)*cosh(d*x + c)^2 + 2*a^2 + 3*a*b + 11*b^2)*sinh(d*x + c)^6 + 24*(14*(3*a
*b - 2*b^2)*cosh(d*x + c)^3 + 3*(2*a^2 + 3*a*b + 11*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 4*(10*a^2 + 22*a*b -
33*b^2)*cosh(d*x + c)^4 + 4*(105*(3*a*b - 2*b^2)*cosh(d*x + c)^4 + 45*(2*a^2 + 3*a*b + 11*b^2)*cosh(d*x + c)^
2 + 10*a^2 + 22*a*b - 33*b^2)*sinh(d*x + c)^4 + 16*(21*(3*a*b - 2*b^2)*cosh(d*x + c)^5 + 15*(2*a^2 + 3*a*b + 1
1*b^2)*cosh(d*x + c)^3 + (10*a^2 + 22*a*b - 33*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(2*a^2 - 9*a*b + 19*b^2
)*cosh(d*x + c)^2 + 4*(42*(3*a*b - 2*b^2)*cosh(d*x + c)^6 + 45*(2*a^2 + 3*a*b + 11*b^2)*cosh(d*x + c)^4 + 6*(1
0*a^2 + 22*a*b - 33*b^2)*cosh(d*x + c)^2 + 2*a^2 - 9*a*b + 19*b^2)*sinh(d*x + c)^2 + 3*((3*a^2 - 2*a*b)*cosh(d
*x + c)^10 + 10*(3*a^2 - 2*a*b)*cosh(d*x + c)*sinh(d*x + c)^9 + (3*a^2 - 2*a*b)*sinh(d*x + c)^10 - (3*a^2 - 14
*a*b + 8*b^2)*cosh(d*x + c)^8 + (45*(3*a^2 - 2*a*b)*cosh(d*x + c)^2 - 3*a^2 + 14*a*b - 8*b^2)*sinh(d*x + c)^8
+ 8*(15*(3*a^2 - 2*a*b)*cosh(d*x + c)^3 - (3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(3*a^2 +
16*a*b - 12*b^2)*cosh(d*x + c)^6 + 2*(105*(3*a^2 - 2*a*b)*cosh(d*x + c)^4 - 14*(3*a^2 - 14*a*b + 8*b^2)*cosh(
d*x + c)^2 - 3*a^2 - 16*a*b + 12*b^2)*sinh(d*x + c)^6 + 4*(63*(3*a^2 - 2*a*b)*cosh(d*x + c)^5 - 14*(3*a^2 - 14
*a*b + 8*b^2)*cosh(d*x + c)^3 - 3*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^2 + 16*a*b
- 12*b^2)*cosh(d*x + c)^4 + 2*(105*(3*a^2 - 2*a*b)*cosh(d*x + c)^6 - 35*(3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c
)^4 - 15*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c)^2 + 3*a^2 + 16*a*b - 12*b^2)*sinh(d*x + c)^4 + 8*(15*(3*a^2 -
2*a*b)*cosh(d*x + c)^7 - 7*(3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^5 - 5*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x +
c)^3 + (3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + (3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^2 + (
45*(3*a^2 - 2*a*b)*cosh(d*x + c)^8 - 28*(3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^6 - 30*(3*a^2 + 16*a*b - 12*b^2
)*cosh(d*x + c)^4 + 12*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c)^2 + 3*a^2 - 14*a*b + 8*b^2)*sinh(d*x + c)^2 - 3
*a^2 + 2*a*b + 2*(5*(3*a^2 - 2*a*b)*cosh(d*x + c)^9 - 4*(3*a^2 - 14*a*b + 8*b^2)*cosh(d*x + c)^7 - 6*(3*a^2 +
16*a*b - 12*b^2)*cosh(d*x + c)^5 + 4*(3*a^2 + 16*a*b - 12*b^2)*cosh(d*x + c)^3 + (3*a^2 - 14*a*b + 8*b^2)*cosh
(d*x + c))*sinh(d*x + c))*sqrt(-b/(a + b))*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(-b/(a + b))/b) - 8*a^2 + 22*a*b + 8*(6*(3*a*b - 2*b^2)*cosh(d*x + c)^7 + 9*(2
*a^2 + 3*a*b + 11*b^2)*cosh(d*x + c)^5 + 2*(10*a^2 + 22*a*b - 33*b^2)*cosh(d*x + c)^3 + (2*a^2 - 9*a*b + 19*b^
2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^10 + 10*(a^4 + 3*a^3*b +
3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*sinh(d*x + c)^10 -
(a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c)^8 + (45*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*co
sh(d*x + c)^2 - (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d)*sinh(d*x + c)^8 - 2*(a^4 + 9*a^3*b + 21*a^2*b^
2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c)^6 + 8*(15*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^3 - (a^4 -
a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^4 + 3*a^3*b + 3*a^2*b^2 +
a*b^3)*d*cosh(d*x + c)^4 - 14*(a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c)^2 - (a^4 + 9*a^3*b
+ 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d)*sinh(d*x + c)^6 + 2*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh
(d*x + c)^4 + 4*(63*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^5 - 14*(a^4 - a^3*b - 9*a^2*b^2 - 11*a
*b^3 - 4*b^4)*d*cosh(d*x + c)^3 - 3*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c))*sinh(d*x
+ c)^5 + 2*(105*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^6 - 35*(a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3
- 4*b^4)*d*cosh(d*x + c)^4 - 15*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c)^2 + (a^4 + 9*
a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d)*sinh(d*x + c)^4 + (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cos
h(d*x + c)^2 + 8*(15*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^7 - 7*(a^4 - a^3*b - 9*a^2*b^2 - 11*a
*b^3 - 4*b^4)*d*cosh(d*x + c)^5 - 5*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c)^3 + (a^4 +
9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^4 + 3*a^3*b + 3*a^2*b^2 +
a*b^3)*d*cosh(d*x + c)^8 - 28*(a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*cosh(d*x + c)^6 - 30*(a^4 + 9*a^3
*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c)^4 + 12*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*c
osh(d*x + c)^2 + (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d)*sinh(d*x + c)^2 - (a^4 + 3*a^3*b + 3*a^2*b^2
+ a*b^3)*d + 2*(5*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^9 - 4*(a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^
3 - 4*b^4)*d*cosh(d*x + c)^7 - 6*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c)^5 + 4*(a^4 +
9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*cosh(d*x + c)^3 + (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*co
sh(d*x + c))*sinh(d*x + c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{4}{\left (c + d x \right )}}{\left (a + b \operatorname{sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(c + d*x)**4/(a + b*sech(c + d*x)**2)**2, x)

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Giac [B]  time = 1.37426, size = 358, normalized size = 2.91 \begin{align*} -\frac{{\left (3 \, a b - 2 \, b^{2}\right )} \arctan \left (\frac{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt{-a b - b^{2}}}\right )}{2 \,{\left (a^{3} d + 3 \, a^{2} b d + 3 \, a b^{2} d + b^{3} d\right )} \sqrt{-a b - b^{2}}} + \frac{a b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a b}{{\left (a^{3} d + 3 \, a^{2} b d + 3 \, a b^{2} d + b^{3} d\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 )}} - \frac{4 \,{\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b e^{\left (2 \, d x + 2 \, c\right )} - a + 2 \, b\right )}}{3 \,{\left (a^{3} d + 3 \, a^{2} b d + 3 \, a b^{2} d + b^{3} d\right )}{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*a*b - 2*b^2)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^3*d + 3*a^2*b*d + 3*a*b^2*
d + b^3*d)*sqrt(-a*b - b^2)) + (a*b*e^(2*d*x + 2*c) + 2*b^2*e^(2*d*x + 2*c) + a*b)/((a^3*d + 3*a^2*b*d + 3*a*b
^2*d + b^3*d)*(a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)) - 4/3*(3*b*e^(4*d*x + 4*c)
+ 3*a*e^(2*d*x + 2*c) - 3*b*e^(2*d*x + 2*c) - a + 2*b)/((a^3*d + 3*a^2*b*d + 3*a*b^2*d + b^3*d)*(e^(2*d*x + 2*
c) - 1)^3)