3.80 \(\int \text{csch}^5(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\)

Optimal. Leaf size=135 \[ -\frac{3 (a-b)^2 \tanh ^{-1}\left (\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right )}{8 \sqrt{a} f}-\frac{\coth (e+f x) \text{csch}^3(e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{4 f}+\frac{3 (a-b) \coth (e+f x) \text{csch}(e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{8 f} \]

[Out]

(-3*(a - b)^2*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(8*Sqrt[a]*f) + (3*(a - b)*Sqr
t[a - b + b*Cosh[e + f*x]^2]*Coth[e + f*x]*Csch[e + f*x])/(8*f) - ((a - b + b*Cosh[e + f*x]^2)^(3/2)*Coth[e +
f*x]*Csch[e + f*x]^3)/(4*f)

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Rubi [A]  time = 0.140604, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3186, 378, 377, 206} \[ -\frac{3 (a-b)^2 \tanh ^{-1}\left (\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right )}{8 \sqrt{a} f}-\frac{\coth (e+f x) \text{csch}^3(e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{4 f}+\frac{3 (a-b) \coth (e+f x) \text{csch}(e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{8 f} \]

Antiderivative was successfully verified.

[In]

Int[Csch[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

(-3*(a - b)^2*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(8*Sqrt[a]*f) + (3*(a - b)*Sqr
t[a - b + b*Cosh[e + f*x]^2]*Coth[e + f*x]*Csch[e + f*x])/(8*f) - ((a - b + b*Cosh[e + f*x]^2)^(3/2)*Coth[e +
f*x]*Csch[e + f*x]^3)/(4*f)

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 378

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

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 \text{csch}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a-b+b x^2\right )^{3/2}}{\left (1-x^2\right )^3} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=-\frac{\left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text{csch}^3(e+f x)}{4 f}-\frac{(3 (a-b)) \operatorname{Subst}\left (\int \frac{\sqrt{a-b+b x^2}}{\left (1-x^2\right )^2} \, dx,x,\cosh (e+f x)\right )}{4 f}\\ &=\frac{3 (a-b) \sqrt{a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text{csch}(e+f x)}{8 f}-\frac{\left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text{csch}^3(e+f x)}{4 f}-\frac{\left (3 (a-b)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\cosh (e+f x)\right )}{8 f}\\ &=\frac{3 (a-b) \sqrt{a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text{csch}(e+f x)}{8 f}-\frac{\left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text{csch}^3(e+f x)}{4 f}-\frac{\left (3 (a-b)^2\right ) \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 )}{8 f}\\ &=-\frac{3 (a-b)^2 \tanh ^{-1}\left (\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a-b+b \cosh ^2(e+f x)}}\right )}{8 \sqrt{a} f}+\frac{3 (a-b) \sqrt{a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text{csch}(e+f x)}{8 f}-\frac{\left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text{csch}^3(e+f x)}{4 f}\\ \end{align*}

Mathematica [A]  time = 0.630421, size = 123, normalized size = 0.91 \[ \frac{\sqrt{2} \sqrt{a} \coth (e+f x) \text{csch}(e+f x) \sqrt{2 a+b \cosh (2 (e+f x))-b} \left (-2 a \text{csch}^2(e+f x)+3 a-5 b\right )-6 (a-b)^2 \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right )}{16 \sqrt{a} f} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

(-6*(a - b)^2*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] + Sqrt[2]*Sqrt[a]*S
qrt[2*a - b + b*Cosh[2*(e + f*x)]]*Coth[e + f*x]*Csch[e + f*x]*(3*a - 5*b - 2*a*Csch[e + f*x]^2))/(16*Sqrt[a]*
f)

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Maple [B]  time = 0.095, size = 379, normalized size = 2.8 \begin{align*} -{\frac{1}{16\, \left ( \sinh \left ( fx+e \right ) \right ) ^{4}\cosh \left ( fx+e \right ) f}\sqrt{ \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}} \left ( 3\,{a}^{2}\ln \left ({\frac{ \left ( a+b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}+2\,\sqrt{a}\sqrt{b \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+ \left ( a-b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}+a-b}{ \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}} \right ) \left ( \sinh \left ( fx+e \right ) \right ) ^{4}-6\,ab\ln \left ({\frac{ \left ( a+b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}+2\,\sqrt{a}\sqrt{b \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+ \left ( a-b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}+a-b}{ \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}} \right ) \left ( \sinh \left ( fx+e \right ) \right ) ^{4}+3\,{b}^{2}\ln \left ({\frac{ \left ( a+b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}+2\,\sqrt{a}\sqrt{b \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+ \left ( a-b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}+a-b}{ \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}} \right ) \left ( \sinh \left ( fx+e \right ) \right ) ^{4}-6\,{a}^{3/2}\sqrt{ \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}} \left ( \sinh \left ( fx+e \right ) \right ) ^{2}+10\,b\sqrt{ \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}\sqrt{a} \left ( \sinh \left ( fx+e \right ) \right ) ^{2}+4\,{a}^{3/2}\sqrt{ \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}} \right ){\frac{1}{\sqrt{a}}}{\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)^5*(a+b*sinh(f*x+e)^2)^(3/2),x)

[Out]

-1/16*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(3*a^2*ln(((a+b)*cosh(f*x+e)^2+2*a^(1/2)*(b*cosh(f*x+e)^4+(a-b
)*cosh(f*x+e)^2)^(1/2)+a-b)/sinh(f*x+e)^2)*sinh(f*x+e)^4-6*a*b*ln(((a+b)*cosh(f*x+e)^2+2*a^(1/2)*(b*cosh(f*x+e
)^4+(a-b)*cosh(f*x+e)^2)^(1/2)+a-b)/sinh(f*x+e)^2)*sinh(f*x+e)^4+3*b^2*ln(((a+b)*cosh(f*x+e)^2+2*a^(1/2)*(b*co
sh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)+a-b)/sinh(f*x+e)^2)*sinh(f*x+e)^4-6*a^(3/2)*((a+b*sinh(f*x+e)^2)*cosh(f
*x+e)^2)^(1/2)*sinh(f*x+e)^2+10*b*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*a^(1/2)*sinh(f*x+e)^2+4*a^(3/2)*((
a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2))/a^(1/2)/sinh(f*x+e)^4/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{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \operatorname{csch}\left (f x + e\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^(3/2)*csch(f*x + e)^5, x)

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Fricas [B]  time = 4.11192, size = 8128, normalized size = 60.21 \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)^5*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

[1/16*(3*((a^2 - 2*a*b + b^2)*cosh(f*x + e)^8 + 8*(a^2 - 2*a*b + b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + (a^2 - 2
*a*b + b^2)*sinh(f*x + e)^8 - 4*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^6 + 4*(7*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2
- a^2 + 2*a*b - b^2)*sinh(f*x + e)^6 + 8*(7*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^3 - 3*(a^2 - 2*a*b + b^2)*cosh(
f*x + e))*sinh(f*x + e)^5 + 6*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^4 + 2*(35*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^4
- 30*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 + 3*a^2 - 6*a*b + 3*b^2)*sinh(f*x + e)^4 + 8*(7*(a^2 - 2*a*b + b^2)*c
osh(f*x + e)^5 - 10*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^3 + 3*(a^2 - 2*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^3
- 4*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 + 4*(7*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^6 - 15*(a^2 - 2*a*b + b^2)*c
osh(f*x + e)^4 + 9*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 - a^2 + 2*a*b - b^2)*sinh(f*x + e)^2 + a^2 - 2*a*b + b^
2 + 8*((a^2 - 2*a*b + b^2)*cosh(f*x + e)^7 - 3*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^5 + 3*(a^2 - 2*a*b + b^2)*cos
h(f*x + e)^3 - (a^2 - 2*a*b + b^2)*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)*(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)*((3*a^2 - 5*a
*b)*cosh(f*x + e)^6 + 6*(3*a^2 - 5*a*b)*cosh(f*x + e)*sinh(f*x + e)^5 + (3*a^2 - 5*a*b)*sinh(f*x + e)^6 - (11*
a^2 - 5*a*b)*cosh(f*x + e)^4 + (15*(3*a^2 - 5*a*b)*cosh(f*x + e)^2 - 11*a^2 + 5*a*b)*sinh(f*x + e)^4 + 4*(5*(3
*a^2 - 5*a*b)*cosh(f*x + e)^3 - (11*a^2 - 5*a*b)*cosh(f*x + e))*sinh(f*x + e)^3 - (11*a^2 - 5*a*b)*cosh(f*x +
e)^2 + (15*(3*a^2 - 5*a*b)*cosh(f*x + e)^4 - 6*(11*a^2 - 5*a*b)*cosh(f*x + e)^2 - 11*a^2 + 5*a*b)*sinh(f*x + e
)^2 + 3*a^2 - 5*a*b + 2*(3*(3*a^2 - 5*a*b)*cosh(f*x + e)^5 - 2*(11*a^2 - 5*a*b)*cosh(f*x + e)^3 - (11*a^2 - 5*
a*b)*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*f*cosh(f*x + e)^8 + 8*a*f*cosh(f*x + e)*sinh(f*x + e)^7 +
a*f*sinh(f*x + e)^8 - 4*a*f*cosh(f*x + e)^6 + 4*(7*a*f*cosh(f*x + e)^2 - a*f)*sinh(f*x + e)^6 + 6*a*f*cosh(f*
x + e)^4 + 8*(7*a*f*cosh(f*x + e)^3 - 3*a*f*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35*a*f*cosh(f*x + e)^4 - 30*a*
f*cosh(f*x + e)^2 + 3*a*f)*sinh(f*x + e)^4 - 4*a*f*cosh(f*x + e)^2 + 8*(7*a*f*cosh(f*x + e)^5 - 10*a*f*cosh(f*
x + e)^3 + 3*a*f*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(7*a*f*cosh(f*x + e)^6 - 15*a*f*cosh(f*x + e)^4 + 9*a*f*co
sh(f*x + e)^2 - a*f)*sinh(f*x + e)^2 + a*f + 8*(a*f*cosh(f*x + e)^7 - 3*a*f*cosh(f*x + e)^5 + 3*a*f*cosh(f*x +
e)^3 - a*f*cosh(f*x + e))*sinh(f*x + e)), 1/8*(3*((a^2 - 2*a*b + b^2)*cosh(f*x + e)^8 + 8*(a^2 - 2*a*b + b^2)
*cosh(f*x + e)*sinh(f*x + e)^7 + (a^2 - 2*a*b + b^2)*sinh(f*x + e)^8 - 4*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^6 +
4*(7*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 - a^2 + 2*a*b - b^2)*sinh(f*x + e)^6 + 8*(7*(a^2 - 2*a*b + b^2)*cosh
(f*x + e)^3 - 3*(a^2 - 2*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 6*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^4 + 2
*(35*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^4 - 30*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 + 3*a^2 - 6*a*b + 3*b^2)*sin
h(f*x + e)^4 + 8*(7*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^5 - 10*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^3 + 3*(a^2 - 2*
a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^3 - 4*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 + 4*(7*(a^2 - 2*a*b + b^2)*c
osh(f*x + e)^6 - 15*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^4 + 9*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 - a^2 + 2*a*b
- b^2)*sinh(f*x + e)^2 + a^2 - 2*a*b + b^2 + 8*((a^2 - 2*a*b + b^2)*cosh(f*x + e)^7 - 3*(a^2 - 2*a*b + b^2)*co
sh(f*x + e)^5 + 3*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^3 - (a^2 - 2*a*b + b^2)*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*c
osh(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*cosh(f*x + e)^3 + (2*a - b)*cosh(f*x + e))*sinh(f*x
+ e) + b)) + sqrt(2)*((3*a^2 - 5*a*b)*cosh(f*x + e)^6 + 6*(3*a^2 - 5*a*b)*cosh(f*x + e)*sinh(f*x + e)^5 + (3*
a^2 - 5*a*b)*sinh(f*x + e)^6 - (11*a^2 - 5*a*b)*cosh(f*x + e)^4 + (15*(3*a^2 - 5*a*b)*cosh(f*x + e)^2 - 11*a^2
+ 5*a*b)*sinh(f*x + e)^4 + 4*(5*(3*a^2 - 5*a*b)*cosh(f*x + e)^3 - (11*a^2 - 5*a*b)*cosh(f*x + e))*sinh(f*x +
e)^3 - (11*a^2 - 5*a*b)*cosh(f*x + e)^2 + (15*(3*a^2 - 5*a*b)*cosh(f*x + e)^4 - 6*(11*a^2 - 5*a*b)*cosh(f*x +
e)^2 - 11*a^2 + 5*a*b)*sinh(f*x + e)^2 + 3*a^2 - 5*a*b + 2*(3*(3*a^2 - 5*a*b)*cosh(f*x + e)^5 - 2*(11*a^2 - 5*
a*b)*cosh(f*x + e)^3 - (11*a^2 - 5*a*b)*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*f*cosh(f*x + e)^8 + 8*
a*f*cosh(f*x + e)*sinh(f*x + e)^7 + a*f*sinh(f*x + e)^8 - 4*a*f*cosh(f*x + e)^6 + 4*(7*a*f*cosh(f*x + e)^2 - a
*f)*sinh(f*x + e)^6 + 6*a*f*cosh(f*x + e)^4 + 8*(7*a*f*cosh(f*x + e)^3 - 3*a*f*cosh(f*x + e))*sinh(f*x + e)^5
+ 2*(35*a*f*cosh(f*x + e)^4 - 30*a*f*cosh(f*x + e)^2 + 3*a*f)*sinh(f*x + e)^4 - 4*a*f*cosh(f*x + e)^2 + 8*(7*a
*f*cosh(f*x + e)^5 - 10*a*f*cosh(f*x + e)^3 + 3*a*f*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(7*a*f*cosh(f*x + e)^6
- 15*a*f*cosh(f*x + e)^4 + 9*a*f*cosh(f*x + e)^2 - a*f)*sinh(f*x + e)^2 + a*f + 8*(a*f*cosh(f*x + e)^7 - 3*a*f
*cosh(f*x + e)^5 + 3*a*f*cosh(f*x + e)^3 - a*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)**5*(a+b*sinh(f*x+e)**2)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \operatorname{csch}\left (f x + e\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^(3/2)*csch(f*x + e)^5, x)