3.119 \(\int \frac{\text{csch}(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=136 \[ -\frac{b (5 a-3 b) \cosh (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \cosh ^2(e+f x)-b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right )}{a^{5/2} f}-\frac{b \cosh (e+f x)}{3 a f (a-b) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.171386, antiderivative size = 136, normalized size of antiderivative = 1., 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 = {3186, 414, 527, 12, 377, 206} \[ -\frac{b (5 a-3 b) \cosh (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \cosh ^2(e+f x)-b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right )}{a^{5/2} f}-\frac{b \cosh (e+f x)}{3 a f (a-b) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csch[e + f*x]/(a + b*Sinh[e + f*x]^2)^(5/2),x]

[Out]

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

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]

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 12

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

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 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])

Rubi steps

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

Mathematica [A]  time = 0.797738, size = 130, normalized size = 0.96 \[ \frac{\frac{\sqrt{2} b \cosh (e+f x) \left (-12 a^2+b (3 b-5 a) \cosh (2 (e+f x))+13 a b-3 b^2\right )}{3 a^2 (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right )}{a^{5/2}}}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[e + f*x]/(a + b*Sinh[e + f*x]^2)^(5/2),x]

[Out]

(-(ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]]/a^(5/2)) + (Sqrt[2]*b*Cosh[e +
f*x]*(-12*a^2 + 13*a*b - 3*b^2 + b*(-5*a + 3*b)*Cosh[2*(e + f*x)]))/(3*a^2*(a - b)^2*(2*a - b + b*Cosh[2*(e +
f*x)])^(3/2)))/f

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Maple [A]  time = 0.176, size = 236, normalized size = 1.7 \begin{align*}{\frac{1}{f\cosh \left ( fx+e \right ) }\sqrt{ \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}} \left ( -{\frac{b \left ( 2\,b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}+3\,a-b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{3\,a \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) \left ({a}^{2}-2\,ab+{b}^{2} \right ) }{\frac{1}{\sqrt{ \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}}}}-{\frac{b \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{{a}^{2} \left ( a-b \right ) }{\frac{1}{\sqrt{ \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}}}}-{\frac{1}{2}\ln \left ({\frac{1}{ \left ( \sinh \left ( fx+e \right ) \right ) ^{2}} \left ( 2\,a+ \left ( a+b \right ) \left ( \sinh \left ( fx+e \right ) \right ) ^{2}+2\,\sqrt{a}\sqrt{ \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}} \right ) } \right ){a}^{-{\frac{5}{2}}}} \right ){\frac{1}{\sqrt{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(f*x+e)/(a+b*sinh(f*x+e)^2)^(5/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}\left (f x + e\right )}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(csch(f*x + e)/(b*sinh(f*x + e)^2 + a)^(5/2), x)

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Fricas [B]  time = 5.34341, size = 12269, normalized size = 90.21 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="fricas")

[Out]

[1/6*(3*((a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^8 + 8*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)*sinh(f*x + e)^7
+ (a^2*b^2 - 2*a*b^3 + b^4)*sinh(f*x + e)^8 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^6 + 4*(2*
a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4 + 7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 8*(7*(a^2*b
^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^3 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e))*sinh(f*x + e)^5 +
2*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x + e)^4 + 2*(35*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f
*x + e)^4 + 8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4 + 30*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f
*x + e)^2)*sinh(f*x + e)^4 + a^2*b^2 - 2*a*b^3 + b^4 + 8*(7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^5 + 10*(2*
a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^3 + (8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f
*x + e))*sinh(f*x + e)^3 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^2 + 4*(7*(a^2*b^2 - 2*a*b^3 +
b^4)*cosh(f*x + e)^6 + 15*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^4 + 2*a^3*b - 5*a^2*b^2 + 4*a*b
^3 - b^4 + 3*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*((a^2*b^2
- 2*a*b^3 + b^4)*cosh(f*x + e)^7 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^5 + (8*a^4 - 24*a^3*
b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x + e)^3 + (2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*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)*cosh(f*x + e)^2 + 2*(3*(a + b)*cosh(f*x + e)^2 + 3*a - b)*sinh(f*x + e)^2 - 2*sqrt(2)*(co
sh(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)*((5*a^2*b^2 - 3*a*b^3)*cosh(f*x + e)^6 + 6*(5*a^2*b^2 - 3*a*b^3)*co
sh(f*x + e)*sinh(f*x + e)^5 + (5*a^2*b^2 - 3*a*b^3)*sinh(f*x + e)^6 + 3*(8*a^3*b - 7*a^2*b^2 + a*b^3)*cosh(f*x
+ e)^4 + 3*(8*a^3*b - 7*a^2*b^2 + a*b^3 + 5*(5*a^2*b^2 - 3*a*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 5*a^2*b^
2 - 3*a*b^3 + 4*(5*(5*a^2*b^2 - 3*a*b^3)*cosh(f*x + e)^3 + 3*(8*a^3*b - 7*a^2*b^2 + a*b^3)*cosh(f*x + e))*sinh
(f*x + e)^3 + 3*(8*a^3*b - 7*a^2*b^2 + a*b^3)*cosh(f*x + e)^2 + 3*(5*(5*a^2*b^2 - 3*a*b^3)*cosh(f*x + e)^4 + 8
*a^3*b - 7*a^2*b^2 + a*b^3 + 6*(8*a^3*b - 7*a^2*b^2 + a*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 6*((5*a^2*b^2
- 3*a*b^3)*cosh(f*x + e)^5 + 2*(8*a^3*b - 7*a^2*b^2 + a*b^3)*cosh(f*x + e)^3 + (8*a^3*b - 7*a^2*b^2 + a*b^3)*c
osh(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^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)^8 + 8*(a^5*b^2 -
2*a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)*sinh(f*x + e)^7 + (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*sinh(f*x + e)^8 + 4*(
2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*x + e)^6 + 4*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*cosh(f*x
+ e)^2 + (2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f)*sinh(f*x + e)^6 + 2*(8*a^7 - 24*a^6*b + 27*a^5*b^2 -
14*a^4*b^3 + 3*a^3*b^4)*f*cosh(f*x + e)^4 + 8*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)^3 + 3*(2*a^6*
b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*
f*cosh(f*x + e)^4 + 30*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*x + e)^2 + (8*a^7 - 24*a^6*b + 27*
a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*f)*sinh(f*x + e)^4 + 4*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*
x + e)^2 + 8*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)^5 + 10*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*
b^4)*f*cosh(f*x + e)^3 + (8*a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*f*cosh(f*x + e))*sinh(f*x +
e)^3 + 4*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)^6 + 15*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)
*f*cosh(f*x + e)^4 + 3*(8*a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*f*cosh(f*x + e)^2 + (2*a^6*b -
5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f)*sinh(f*x + e)^2 + (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f + 8*((a^5*b^2 - 2*a^4
*b^3 + a^3*b^4)*f*cosh(f*x + e)^7 + 3*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*x + e)^5 + (8*a^7 -
24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*f*cosh(f*x + e)^3 + (2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^
4)*f*cosh(f*x + e))*sinh(f*x + e)), 1/3*(3*((a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^8 + 8*(a^2*b^2 - 2*a*b^3 +
b^4)*cosh(f*x + e)*sinh(f*x + e)^7 + (a^2*b^2 - 2*a*b^3 + b^4)*sinh(f*x + e)^8 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a
*b^3 - b^4)*cosh(f*x + e)^6 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4 + 7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x +
e)^2)*sinh(f*x + e)^6 + 8*(7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^3 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^
4)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x + e)^4 + 2*(
35*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^4 + 8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4 + 30*(2*a^3*b
- 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + a^2*b^2 - 2*a*b^3 + b^4 + 8*(7*(a^2*b^2 - 2*a*
b^3 + b^4)*cosh(f*x + e)^5 + 10*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^3 + (8*a^4 - 24*a^3*b + 27
*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x
+ e)^2 + 4*(7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^6 + 15*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x +
e)^4 + 2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4 + 3*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x +
e)^2)*sinh(f*x + e)^2 + 8*((a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^7 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)
*cosh(f*x + e)^5 + (8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x + e)^3 + (2*a^3*b - 5*a^2*b^2 +
4*a*b^3 - b^4)*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*c
osh(f*x + e)^3 + (2*a - b)*cosh(f*x + e))*sinh(f*x + e) + b)) - sqrt(2)*((5*a^2*b^2 - 3*a*b^3)*cosh(f*x + e)^6
+ 6*(5*a^2*b^2 - 3*a*b^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (5*a^2*b^2 - 3*a*b^3)*sinh(f*x + e)^6 + 3*(8*a^3*b
- 7*a^2*b^2 + a*b^3)*cosh(f*x + e)^4 + 3*(8*a^3*b - 7*a^2*b^2 + a*b^3 + 5*(5*a^2*b^2 - 3*a*b^3)*cosh(f*x + e)^
2)*sinh(f*x + e)^4 + 5*a^2*b^2 - 3*a*b^3 + 4*(5*(5*a^2*b^2 - 3*a*b^3)*cosh(f*x + e)^3 + 3*(8*a^3*b - 7*a^2*b^2
+ a*b^3)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(8*a^3*b - 7*a^2*b^2 + a*b^3)*cosh(f*x + e)^2 + 3*(5*(5*a^2*b^2 -
3*a*b^3)*cosh(f*x + e)^4 + 8*a^3*b - 7*a^2*b^2 + a*b^3 + 6*(8*a^3*b - 7*a^2*b^2 + a*b^3)*cosh(f*x + e)^2)*sin
h(f*x + e)^2 + 6*((5*a^2*b^2 - 3*a*b^3)*cosh(f*x + e)^5 + 2*(8*a^3*b - 7*a^2*b^2 + a*b^3)*cosh(f*x + e)^3 + (8
*a^3*b - 7*a^2*b^2 + a*b^3)*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^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*co
sh(f*x + e)^8 + 8*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)*sinh(f*x + e)^7 + (a^5*b^2 - 2*a^4*b^3 + a^3
*b^4)*f*sinh(f*x + e)^8 + 4*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*x + e)^6 + 4*(7*(a^5*b^2 - 2*
a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)^2 + (2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f)*sinh(f*x + e)^6 + 2*(8*a
^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*f*cosh(f*x + e)^4 + 8*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*
f*cosh(f*x + e)^3 + 3*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35*(a^
5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)^4 + 30*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*x + e
)^2 + (8*a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*f)*sinh(f*x + e)^4 + 4*(2*a^6*b - 5*a^5*b^2 + 4
*a^4*b^3 - a^3*b^4)*f*cosh(f*x + e)^2 + 8*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)^5 + 10*(2*a^6*b -
5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*x + e)^3 + (8*a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)
*f*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)^6 + 15*(2*a^6*b - 5*a
^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*x + e)^4 + 3*(8*a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*f
*cosh(f*x + e)^2 + (2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f)*sinh(f*x + e)^2 + (a^5*b^2 - 2*a^4*b^3 + a^3
*b^4)*f + 8*((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)^7 + 3*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)
*f*cosh(f*x + e)^5 + (8*a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*f*cosh(f*x + e)^3 + (2*a^6*b - 5
*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*x + e))*sinh(f*x + e))]

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Sympy [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(f*x+e)/(a+b*sinh(f*x+e)**2)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.51809, size = 857, normalized size = 6.3 \begin{align*} \frac{5 \, a b - 3 \, b^{2}}{3 \,{\left (a^{4} \sqrt{b} f - 2 \, a^{3} b^{\frac{3}{2}} f + a^{2} b^{\frac{5}{2}} f\right )}} - \frac{{\left ({\left (\frac{{\left (5 \, a^{13} b^{4} f^{3} - 13 \, a^{12} b^{5} f^{3} + 11 \, a^{11} b^{6} f^{3} - 3 \, a^{10} b^{7} f^{3}\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a^{16} b^{2} f^{4} - 4 \, a^{15} b^{3} f^{4} + 6 \, a^{14} b^{4} f^{4} - 4 \, a^{13} b^{5} f^{4} + a^{12} b^{6} f^{4}} + \frac{3 \,{\left (8 \, a^{14} b^{3} f^{3} - 23 \, a^{13} b^{4} f^{3} + 23 \, a^{12} b^{5} f^{3} - 9 \, a^{11} b^{6} f^{3} + a^{10} b^{7} f^{3}\right )}}{a^{16} b^{2} f^{4} - 4 \, a^{15} b^{3} f^{4} + 6 \, a^{14} b^{4} f^{4} - 4 \, a^{13} b^{5} f^{4} + a^{12} b^{6} f^{4}}\right )} e^{\left (2 \, f x + 2 \, e\right )} + \frac{3 \,{\left (8 \, a^{14} b^{3} f^{3} - 23 \, a^{13} b^{4} f^{3} + 23 \, a^{12} b^{5} f^{3} - 9 \, a^{11} b^{6} f^{3} + a^{10} b^{7} f^{3}\right )}}{a^{16} b^{2} f^{4} - 4 \, a^{15} b^{3} f^{4} + 6 \, a^{14} b^{4} f^{4} - 4 \, a^{13} b^{5} f^{4} + a^{12} b^{6} f^{4}}\right )} e^{\left (2 \, f x + 2 \, e\right )} + \frac{5 \, a^{13} b^{4} f^{3} - 13 \, a^{12} b^{5} f^{3} + 11 \, a^{11} b^{6} f^{3} - 3 \, a^{10} b^{7} f^{3}}{a^{16} b^{2} f^{4} - 4 \, a^{15} b^{3} f^{4} + 6 \, a^{14} b^{4} f^{4} - 4 \, a^{13} b^{5} f^{4} + a^{12} b^{6} f^{4}}}{3 \,{\left (b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b\right )}^{\frac{3}{2}}} + \frac{2 \, \arctan \left (-\frac{\sqrt{b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt{b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b} - \sqrt{b}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a} a^{2} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="giac")

[Out]

1/3*(5*a*b - 3*b^2)/(a^4*sqrt(b)*f - 2*a^3*b^(3/2)*f + a^2*b^(5/2)*f) - 1/3*((((5*a^13*b^4*f^3 - 13*a^12*b^5*f
^3 + 11*a^11*b^6*f^3 - 3*a^10*b^7*f^3)*e^(2*f*x + 2*e)/(a^16*b^2*f^4 - 4*a^15*b^3*f^4 + 6*a^14*b^4*f^4 - 4*a^1
3*b^5*f^4 + a^12*b^6*f^4) + 3*(8*a^14*b^3*f^3 - 23*a^13*b^4*f^3 + 23*a^12*b^5*f^3 - 9*a^11*b^6*f^3 + a^10*b^7*
f^3)/(a^16*b^2*f^4 - 4*a^15*b^3*f^4 + 6*a^14*b^4*f^4 - 4*a^13*b^5*f^4 + a^12*b^6*f^4))*e^(2*f*x + 2*e) + 3*(8*
a^14*b^3*f^3 - 23*a^13*b^4*f^3 + 23*a^12*b^5*f^3 - 9*a^11*b^6*f^3 + a^10*b^7*f^3)/(a^16*b^2*f^4 - 4*a^15*b^3*f
^4 + 6*a^14*b^4*f^4 - 4*a^13*b^5*f^4 + a^12*b^6*f^4))*e^(2*f*x + 2*e) + (5*a^13*b^4*f^3 - 13*a^12*b^5*f^3 + 11
*a^11*b^6*f^3 - 3*a^10*b^7*f^3)/(a^16*b^2*f^4 - 4*a^15*b^3*f^4 + 6*a^14*b^4*f^4 - 4*a^13*b^5*f^4 + a^12*b^6*f^
4))/(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b)^(3/2) + 2*arctan(-1/2*(sqrt(b)*e^(2*f*
x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b) - sqrt(b))/sqrt(-a))/(sqrt(
-a)*a^2*f)