3.110 \(\int \frac {\text {csch}^3(e+f x)}{(a+b \sinh ^2(e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=139 \[ \frac {(a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{2 a^{5/2} f}-\frac {b (a-3 b) \cosh (e+f x)}{2 a^2 f (a-b) \sqrt {a+b \cosh ^2(e+f x)-b}}-\frac {\coth (e+f x) \text {csch}(e+f x)}{2 a f \sqrt {a+b \cosh ^2(e+f x)-b}} \]

[Out]

1/2*(a+3*b)*arctanh(cosh(f*x+e)*a^(1/2)/(a-b+b*cosh(f*x+e)^2)^(1/2))/a^(5/2)/f-1/2*(a-3*b)*b*cosh(f*x+e)/a^2/(
a-b)/f/(a-b+b*cosh(f*x+e)^2)^(1/2)-1/2*coth(f*x+e)*csch(f*x+e)/a/f/(a-b+b*cosh(f*x+e)^2)^(1/2)

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Rubi [A]  time = 0.18, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3186, 414, 527, 12, 377, 206} \[ -\frac {b (a-3 b) \cosh (e+f x)}{2 a^2 f (a-b) \sqrt {a+b \cosh ^2(e+f x)-b}}+\frac {(a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{2 a^{5/2} f}-\frac {\coth (e+f x) \text {csch}(e+f x)}{2 a f \sqrt {a+b \cosh ^2(e+f x)-b}} \]

Antiderivative was successfully verified.

[In]

Int[Csch[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

((a + 3*b)*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(2*a^(5/2)*f) - ((a - 3*b)*b*Cosh
[e + f*x])/(2*a^2*(a - b)*f*Sqrt[a - b + b*Cosh[e + f*x]^2]) - (Coth[e + f*x]*Csch[e + f*x])/(2*a*f*Sqrt[a - b
+ b*Cosh[e + f*x]^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, 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 3186

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

Rubi steps

\begin {align*} \int \frac {\text {csch}^3(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a-b+b x^2\right )^{3/2}} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=-\frac {\coth (e+f x) \text {csch}(e+f x)}{2 a f \sqrt {a-b+b \cosh ^2(e+f x)}}+\frac {\operatorname {Subst}\left (\int \frac {a+b+2 b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^{3/2}} \, dx,x,\cosh (e+f x)\right )}{2 a f}\\ &=-\frac {(a-3 b) b \cosh (e+f x)}{2 a^2 (a-b) f \sqrt {a-b+b \cosh ^2(e+f x)}}-\frac {\coth (e+f x) \text {csch}(e+f x)}{2 a f \sqrt {a-b+b \cosh ^2(e+f x)}}+\frac {\operatorname {Subst}\left (\int \frac {(a-b) (a+3 b)}{\left (1-x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\cosh (e+f x)\right )}{2 a^2 (a-b) f}\\ &=-\frac {(a-3 b) b \cosh (e+f x)}{2 a^2 (a-b) f \sqrt {a-b+b \cosh ^2(e+f x)}}-\frac {\coth (e+f x) \text {csch}(e+f x)}{2 a f \sqrt {a-b+b \cosh ^2(e+f x)}}+\frac {(a+3 b) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\cosh (e+f x)\right )}{2 a^2 f}\\ &=-\frac {(a-3 b) b \cosh (e+f x)}{2 a^2 (a-b) f \sqrt {a-b+b \cosh ^2(e+f x)}}-\frac {\coth (e+f x) \text {csch}(e+f x)}{2 a f \sqrt {a-b+b \cosh ^2(e+f x)}}+\frac {(a+3 b) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{2 a^2 f}\\ &=\frac {(a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{2 a^{5/2} f}-\frac {(a-3 b) b \cosh (e+f x)}{2 a^2 (a-b) f \sqrt {a-b+b \cosh ^2(e+f x)}}-\frac {\coth (e+f x) \text {csch}(e+f x)}{2 a f \sqrt {a-b+b \cosh ^2(e+f x)}}\\ \end {align*}

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Mathematica [A]  time = 0.72, size = 134, normalized size = 0.96 \[ \frac {\frac {(a+3 b) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \cosh (e+f x)}{\sqrt {2 a+b \cosh (2 (e+f x))-b}}\right )}{a^{5/2}}-\frac {\coth (e+f x) \text {csch}(e+f x) \left (2 a^2+b (a-3 b) \cosh (2 (e+f x))-3 a b+3 b^2\right )}{a^2 (a-b) \sqrt {4 a+2 b \cosh (2 (e+f x))-2 b}}}{2 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

(((a + 3*b)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]])/a^(5/2) - ((2*a^2 -
3*a*b + 3*b^2 + (a - 3*b)*b*Cosh[2*(e + f*x)])*Coth[e + f*x]*Csch[e + f*x])/(a^2*(a - b)*Sqrt[4*a - 2*b + 2*b*
Cosh[2*(e + f*x)]]))/(2*f)

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fricas [B]  time = 3.00, size = 4441, normalized size = 31.95 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(((a^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)^8 + 8*(a^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)*sinh(f*x + e)^7 +
(a^2*b + 2*a*b^2 - 3*b^3)*sinh(f*x + e)^8 + 4*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e)^6 + 4*(a^3 + a^2*
b - 5*a*b^2 + 3*b^3 + 7*(a^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 8*(7*(a^2*b + 2*a*b^2 - 3
*b^3)*cosh(f*x + e)^3 + 3*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e))*sinh(f*x + e)^5 - 2*(4*a^3 + 5*a^2*b
- 18*a*b^2 + 9*b^3)*cosh(f*x + e)^4 + 2*(35*(a^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)^4 - 4*a^3 - 5*a^2*b + 18*a
*b^2 - 9*b^3 + 30*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 8*(7*(a^2*b + 2*a*b^2 - 3
*b^3)*cosh(f*x + e)^5 + 10*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e)^3 - (4*a^3 + 5*a^2*b - 18*a*b^2 + 9*b
^3)*cosh(f*x + e))*sinh(f*x + e)^3 + a^2*b + 2*a*b^2 - 3*b^3 + 4*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e)
^2 + 4*(7*(a^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)^6 + 15*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e)^4 + a^3
+ a^2*b - 5*a*b^2 + 3*b^3 - 3*(4*a^3 + 5*a^2*b - 18*a*b^2 + 9*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*((a^2
*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)^7 + 3*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e)^5 - (4*a^3 + 5*a^2*b -
18*a*b^2 + 9*b^3)*cosh(f*x + e)^3 + (a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e))*sinh(f*x + e))*sqrt(a)*log
(-((a + b)*cosh(f*x + e)^4 + 4*(a + b)*cosh(f*x + e)*sinh(f*x + e)^3 + (a + b)*sinh(f*x + e)^4 + 2*(3*a - b)*c
osh(f*x + e)^2 + 2*(3*(a + b)*cosh(f*x + e)^2 + 3*a - b)*sinh(f*x + e)^2 + 2*sqrt(2)*(cosh(f*x + e)^2 + 2*cosh
(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 + 1)*sqrt(a)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/
(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)) + 4*((a + b)*cosh(f*x + e)^3 + (3*a - b)*
cosh(f*x + e))*sinh(f*x + e) + a + b)/(cosh(f*x + e)^4 + 4*cosh(f*x + e)*sinh(f*x + e)^3 + sinh(f*x + e)^4 + 2
*(3*cosh(f*x + e)^2 - 1)*sinh(f*x + e)^2 - 2*cosh(f*x + e)^2 + 4*(cosh(f*x + e)^3 - cosh(f*x + e))*sinh(f*x +
e) + 1)) - 2*sqrt(2)*((a^2*b - 3*a*b^2)*cosh(f*x + e)^6 + 6*(a^2*b - 3*a*b^2)*cosh(f*x + e)*sinh(f*x + e)^5 +
(a^2*b - 3*a*b^2)*sinh(f*x + e)^6 + (4*a^3 - 5*a^2*b + 3*a*b^2)*cosh(f*x + e)^4 + (4*a^3 - 5*a^2*b + 3*a*b^2 +
15*(a^2*b - 3*a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(5*(a^2*b - 3*a*b^2)*cosh(f*x + e)^3 + (4*a^3 - 5*a
^2*b + 3*a*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + a^2*b - 3*a*b^2 + (4*a^3 - 5*a^2*b + 3*a*b^2)*cosh(f*x + e)^2
+ (15*(a^2*b - 3*a*b^2)*cosh(f*x + e)^4 + 4*a^3 - 5*a^2*b + 3*a*b^2 + 6*(4*a^3 - 5*a^2*b + 3*a*b^2)*cosh(f*x
+ e)^2)*sinh(f*x + e)^2 + 2*(3*(a^2*b - 3*a*b^2)*cosh(f*x + e)^5 + 2*(4*a^3 - 5*a^2*b + 3*a*b^2)*cosh(f*x + e)
^3 + (4*a^3 - 5*a^2*b + 3*a*b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2
*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/((a^4*b - a^3*b^2)*f*cosh(f*x +
e)^8 + 8*(a^4*b - a^3*b^2)*f*cosh(f*x + e)*sinh(f*x + e)^7 + (a^4*b - a^3*b^2)*f*sinh(f*x + e)^8 + 4*(a^5 - 2*
a^4*b + a^3*b^2)*f*cosh(f*x + e)^6 + 4*(7*(a^4*b - a^3*b^2)*f*cosh(f*x + e)^2 + (a^5 - 2*a^4*b + a^3*b^2)*f)*s
inh(f*x + e)^6 - 2*(4*a^5 - 7*a^4*b + 3*a^3*b^2)*f*cosh(f*x + e)^4 + 8*(7*(a^4*b - a^3*b^2)*f*cosh(f*x + e)^3
+ 3*(a^5 - 2*a^4*b + a^3*b^2)*f*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35*(a^4*b - a^3*b^2)*f*cosh(f*x + e)^4 + 3
0*(a^5 - 2*a^4*b + a^3*b^2)*f*cosh(f*x + e)^2 - (4*a^5 - 7*a^4*b + 3*a^3*b^2)*f)*sinh(f*x + e)^4 + 4*(a^5 - 2*
a^4*b + a^3*b^2)*f*cosh(f*x + e)^2 + 8*(7*(a^4*b - a^3*b^2)*f*cosh(f*x + e)^5 + 10*(a^5 - 2*a^4*b + a^3*b^2)*f
*cosh(f*x + e)^3 - (4*a^5 - 7*a^4*b + 3*a^3*b^2)*f*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(7*(a^4*b - a^3*b^2)*f*c
osh(f*x + e)^6 + 15*(a^5 - 2*a^4*b + a^3*b^2)*f*cosh(f*x + e)^4 - 3*(4*a^5 - 7*a^4*b + 3*a^3*b^2)*f*cosh(f*x +
e)^2 + (a^5 - 2*a^4*b + a^3*b^2)*f)*sinh(f*x + e)^2 + (a^4*b - a^3*b^2)*f + 8*((a^4*b - a^3*b^2)*f*cosh(f*x +
e)^7 + 3*(a^5 - 2*a^4*b + a^3*b^2)*f*cosh(f*x + e)^5 - (4*a^5 - 7*a^4*b + 3*a^3*b^2)*f*cosh(f*x + e)^3 + (a^5
- 2*a^4*b + a^3*b^2)*f*cosh(f*x + e))*sinh(f*x + e)), -1/2*(((a^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)^8 + 8*(a
^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)*sinh(f*x + e)^7 + (a^2*b + 2*a*b^2 - 3*b^3)*sinh(f*x + e)^8 + 4*(a^3 + a
^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e)^6 + 4*(a^3 + a^2*b - 5*a*b^2 + 3*b^3 + 7*(a^2*b + 2*a*b^2 - 3*b^3)*cosh(
f*x + e)^2)*sinh(f*x + e)^6 + 8*(7*(a^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)^3 + 3*(a^3 + a^2*b - 5*a*b^2 + 3*b^
3)*cosh(f*x + e))*sinh(f*x + e)^5 - 2*(4*a^3 + 5*a^2*b - 18*a*b^2 + 9*b^3)*cosh(f*x + e)^4 + 2*(35*(a^2*b + 2*
a*b^2 - 3*b^3)*cosh(f*x + e)^4 - 4*a^3 - 5*a^2*b + 18*a*b^2 - 9*b^3 + 30*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(
f*x + e)^2)*sinh(f*x + e)^4 + 8*(7*(a^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)^5 + 10*(a^3 + a^2*b - 5*a*b^2 + 3*b
^3)*cosh(f*x + e)^3 - (4*a^3 + 5*a^2*b - 18*a*b^2 + 9*b^3)*cosh(f*x + e))*sinh(f*x + e)^3 + a^2*b + 2*a*b^2 -
3*b^3 + 4*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e)^2 + 4*(7*(a^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)^6 + 1
5*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e)^4 + a^3 + a^2*b - 5*a*b^2 + 3*b^3 - 3*(4*a^3 + 5*a^2*b - 18*a*
b^2 + 9*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*((a^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)^7 + 3*(a^3 + a^2*b
- 5*a*b^2 + 3*b^3)*cosh(f*x + e)^5 - (4*a^3 + 5*a^2*b - 18*a*b^2 + 9*b^3)*cosh(f*x + e)^3 + (a^3 + a^2*b - 5*a
*b^2 + 3*b^3)*cosh(f*x + e))*sinh(f*x + e))*sqrt(-a)*arctan(sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*
x + e) + sinh(f*x + e)^2 + 1)*sqrt(-a)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2
- 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2))/(b*cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*sinh(f*x + e)^3 +
b*sinh(f*x + e)^4 + 2*(2*a - b)*cosh(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2 + 2*a - b)*sinh(f*x + e)^2 + 4*(b*cos
h(f*x + e)^3 + (2*a - b)*cosh(f*x + e))*sinh(f*x + e) + b)) + sqrt(2)*((a^2*b - 3*a*b^2)*cosh(f*x + e)^6 + 6*(
a^2*b - 3*a*b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (a^2*b - 3*a*b^2)*sinh(f*x + e)^6 + (4*a^3 - 5*a^2*b + 3*a*b^
2)*cosh(f*x + e)^4 + (4*a^3 - 5*a^2*b + 3*a*b^2 + 15*(a^2*b - 3*a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(5
*(a^2*b - 3*a*b^2)*cosh(f*x + e)^3 + (4*a^3 - 5*a^2*b + 3*a*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + a^2*b - 3*a*
b^2 + (4*a^3 - 5*a^2*b + 3*a*b^2)*cosh(f*x + e)^2 + (15*(a^2*b - 3*a*b^2)*cosh(f*x + e)^4 + 4*a^3 - 5*a^2*b +
3*a*b^2 + 6*(4*a^3 - 5*a^2*b + 3*a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(3*(a^2*b - 3*a*b^2)*cosh(f*x + e
)^5 + 2*(4*a^3 - 5*a^2*b + 3*a*b^2)*cosh(f*x + e)^3 + (4*a^3 - 5*a^2*b + 3*a*b^2)*cosh(f*x + e))*sinh(f*x + e)
)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + si
nh(f*x + e)^2)))/((a^4*b - a^3*b^2)*f*cosh(f*x + e)^8 + 8*(a^4*b - a^3*b^2)*f*cosh(f*x + e)*sinh(f*x + e)^7 +
(a^4*b - a^3*b^2)*f*sinh(f*x + e)^8 + 4*(a^5 - 2*a^4*b + a^3*b^2)*f*cosh(f*x + e)^6 + 4*(7*(a^4*b - a^3*b^2)*f
*cosh(f*x + e)^2 + (a^5 - 2*a^4*b + a^3*b^2)*f)*sinh(f*x + e)^6 - 2*(4*a^5 - 7*a^4*b + 3*a^3*b^2)*f*cosh(f*x +
e)^4 + 8*(7*(a^4*b - a^3*b^2)*f*cosh(f*x + e)^3 + 3*(a^5 - 2*a^4*b + a^3*b^2)*f*cosh(f*x + e))*sinh(f*x + e)^
5 + 2*(35*(a^4*b - a^3*b^2)*f*cosh(f*x + e)^4 + 30*(a^5 - 2*a^4*b + a^3*b^2)*f*cosh(f*x + e)^2 - (4*a^5 - 7*a^
4*b + 3*a^3*b^2)*f)*sinh(f*x + e)^4 + 4*(a^5 - 2*a^4*b + a^3*b^2)*f*cosh(f*x + e)^2 + 8*(7*(a^4*b - a^3*b^2)*f
*cosh(f*x + e)^5 + 10*(a^5 - 2*a^4*b + a^3*b^2)*f*cosh(f*x + e)^3 - (4*a^5 - 7*a^4*b + 3*a^3*b^2)*f*cosh(f*x +
e))*sinh(f*x + e)^3 + 4*(7*(a^4*b - a^3*b^2)*f*cosh(f*x + e)^6 + 15*(a^5 - 2*a^4*b + a^3*b^2)*f*cosh(f*x + e)
^4 - 3*(4*a^5 - 7*a^4*b + 3*a^3*b^2)*f*cosh(f*x + e)^2 + (a^5 - 2*a^4*b + a^3*b^2)*f)*sinh(f*x + e)^2 + (a^4*b
- a^3*b^2)*f + 8*((a^4*b - a^3*b^2)*f*cosh(f*x + e)^7 + 3*(a^5 - 2*a^4*b + a^3*b^2)*f*cosh(f*x + e)^5 - (4*a^
5 - 7*a^4*b + 3*a^3*b^2)*f*cosh(f*x + e)^3 + (a^5 - 2*a^4*b + a^3*b^2)*f*cosh(f*x + e))*sinh(f*x + e))]

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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(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.47Error: Bad Argument Type

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maple [B]  time = 0.21, size = 251, normalized size = 1.81 \[ \frac {\sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (\frac {b^{2} \left (\cosh ^{2}\left (f x +e \right )\right )}{a^{2} \left (a -b \right ) \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}-\frac {\sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}{2 a^{2} \sinh \left (f x +e \right )^{2}}+\frac {\ln \left (\frac {2 a +\left (a +b \right ) \left (\sinh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}{\sinh \left (f x +e \right )^{2}}\right )}{4 a^{\frac {3}{2}}}+\frac {3 b \ln \left (\frac {2 a +\left (a +b \right ) \left (\sinh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}{\sinh \left (f x +e \right )^{2}}\right )}{4 a^{\frac {5}{2}}}\right )}{\cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x)

[Out]

((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(b^2/a^2*cosh(f*x+e)^2/(a-b)/((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/
2)-1/2/a^2/sinh(f*x+e)^2*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)+1/4/a^(3/2)*ln((2*a+(a+b)*sinh(f*x+e)^2+2*a
^(1/2)*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2))/sinh(f*x+e)^2)+3/4/a^(5/2)*b*ln((2*a+(a+b)*sinh(f*x+e)^2+2*a
^(1/2)*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2))/sinh(f*x+e)^2))/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}\left (f x + e\right )^{3}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(csch(f*x + e)^3/(b*sinh(f*x + e)^2 + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(e + f*x)^3*(a + b*sinh(e + f*x)^2)^(3/2)),x)

[Out]

int(1/(sinh(e + f*x)^3*(a + b*sinh(e + f*x)^2)^(3/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)**3/(a+b*sinh(f*x+e)**2)**(3/2),x)

[Out]

Integral(csch(e + f*x)**3/(a + b*sinh(e + f*x)**2)**(3/2), x)

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