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

Optimal. Leaf size=203 \[ \frac{\left (8 a^2+3 b (8 a+b)\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 a^2 f}+\frac{\left (8 a^2+3 b (8 a+b)\right ) \sqrt{a+b \sinh ^2(e+f x)}}{8 a f}-\frac{\left (8 a^2+3 b (8 a+b)\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right )}{8 \sqrt{a} f}-\frac{(8 a+b) \text{csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{8 a^2 f}-\frac{\text{csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 a f} \]

[Out]

-((8*a^2 + 3*b*(8*a + b))*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(8*Sqrt[a]*f) + ((8*a^2 + 3*b*(8*a + b
))*Sqrt[a + b*Sinh[e + f*x]^2])/(8*a*f) + ((8*a^2 + 3*b*(8*a + b))*(a + b*Sinh[e + f*x]^2)^(3/2))/(24*a^2*f) -
((8*a + b)*Csch[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(5/2))/(8*a^2*f) - (Csch[e + f*x]^4*(a + b*Sinh[e + f*x]^2
)^(5/2))/(4*a*f)

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Rubi [A]  time = 0.23248, antiderivative size = 199, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3194, 89, 78, 50, 63, 208} \[ \frac{\left (\frac{3 b (8 a+b)}{a^2}+8\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 f}+\frac{\left (8 a^2+3 b (8 a+b)\right ) \sqrt{a+b \sinh ^2(e+f x)}}{8 a f}-\frac{\left (8 a^2+3 b (8 a+b)\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right )}{8 \sqrt{a} f}-\frac{(8 a+b) \text{csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{8 a^2 f}-\frac{\text{csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((8*a^2 + 3*b*(8*a + b))*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(8*Sqrt[a]*f) + ((8*a^2 + 3*b*(8*a + b
))*Sqrt[a + b*Sinh[e + f*x]^2])/(8*a*f) + ((8 + (3*b*(8*a + b))/a^2)*(a + b*Sinh[e + f*x]^2)^(3/2))/(24*f) - (
(8*a + b)*Csch[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(5/2))/(8*a^2*f) - (Csch[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^
(5/2))/(4*a*f)

Rule 3194

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

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 \coth ^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(1+x)^2 (a+b x)^{3/2}}{x^3} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=-\frac{\text{csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 a f}+\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{1}{2} (8 a+b)+2 a x\right ) (a+b x)^{3/2}}{x^2} \, dx,x,\sinh ^2(e+f x)\right )}{4 a f}\\ &=-\frac{(8 a+b) \text{csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{8 a^2 f}-\frac{\text{csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 a f}+\frac{\left (8 a^2+3 b (8 a+b)\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,\sinh ^2(e+f x)\right )}{16 a^2 f}\\ &=\frac{\left (8 a^2+3 b (8 a+b)\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 a^2 f}-\frac{(8 a+b) \text{csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{8 a^2 f}-\frac{\text{csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 a f}+\frac{\left (8 a^2+3 b (8 a+b)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\sinh ^2(e+f x)\right )}{16 a f}\\ &=\frac{\left (8 a^2+3 b (8 a+b)\right ) \sqrt{a+b \sinh ^2(e+f x)}}{8 a f}+\frac{\left (8 a^2+3 b (8 a+b)\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 a^2 f}-\frac{(8 a+b) \text{csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{8 a^2 f}-\frac{\text{csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 a f}+\frac{\left (8 a^2+3 b (8 a+b)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{16 f}\\ &=\frac{\left (8 a^2+3 b (8 a+b)\right ) \sqrt{a+b \sinh ^2(e+f x)}}{8 a f}+\frac{\left (8 a^2+3 b (8 a+b)\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 a^2 f}-\frac{(8 a+b) \text{csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{8 a^2 f}-\frac{\text{csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 a f}+\frac{\left (8 a^2+3 b (8 a+b)\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^2(e+f x)}\right )}{8 b f}\\ &=-\frac{\left (8 a^2+3 b (8 a+b)\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right )}{8 \sqrt{a} f}+\frac{\left (8 a^2+3 b (8 a+b)\right ) \sqrt{a+b \sinh ^2(e+f x)}}{8 a f}+\frac{\left (8 a^2+3 b (8 a+b)\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 a^2 f}-\frac{(8 a+b) \text{csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{8 a^2 f}-\frac{\text{csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 a f}\\ \end{align*}

Mathematica [A]  time = 0.789817, size = 123, normalized size = 0.61 \[ \frac{\sqrt{a} \sqrt{a+b \sinh ^2(e+f x)} \left (8 \left (4 a+b \sinh ^2(e+f x)+6 b\right )-3 (8 a+5 b) \text{csch}^2(e+f x)-6 a \text{csch}^4(e+f x)\right )-3 \left (8 a^2+24 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a}}\right )}{24 \sqrt{a} f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(8*a^2 + 24*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]] + Sqrt[a]*Sqrt[a + b*Sinh[e + f*x]^2
]*(-3*(8*a + 5*b)*Csch[e + f*x]^2 - 6*a*Csch[e + f*x]^4 + 8*(4*a + 6*b + b*Sinh[e + f*x]^2)))/(24*Sqrt[a]*f)

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Maple [C]  time = 0.254, size = 113, normalized size = 0.6 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ({\frac{ \left ( \cosh \left ( fx+e \right ) \right ) ^{4} \left ({b}^{2} \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+2\,ab \left ( \cosh \left ( fx+e \right ) \right ) ^{2}-2\,{b}^{2} \left ( \cosh \left ( fx+e \right ) \right ) ^{2}+{a}^{2}-2\,ab+{b}^{2} \right ) }{\sinh \left ( fx+e \right ) \left ( \left ( \cosh \left ( fx+e \right ) \right ) ^{4}-2\, \left ( \cosh \left ( fx+e \right ) \right ) ^{2}+1 \right ) }{\frac{1}{\sqrt{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}}}},\sinh \left ( fx+e \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

`int/indef0`(1/sinh(f*x+e)/(cosh(f*x+e)^4-2*cosh(f*x+e)^2+1)*cosh(f*x+e)^4*(b^2*cosh(f*x+e)^4+2*a*b*cosh(f*x+e
)^2-2*b^2*cosh(f*x+e)^2+a^2-2*a*b+b^2)/(a+b*sinh(f*x+e)^2)^(1/2),sinh(f*x+e))/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}} \coth \left (f x + e\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(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)*coth(f*x + e)^5, x)

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Fricas [B]  time = 6.13721, size = 14438, normalized size = 71.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(3*((8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^11 + 11*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)*sinh(f*x + e)^
10 + (8*a^2 + 24*a*b + 3*b^2)*sinh(f*x + e)^11 - 4*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^9 + (55*(8*a^2 + 24*
a*b + 3*b^2)*cosh(f*x + e)^2 - 32*a^2 - 96*a*b - 12*b^2)*sinh(f*x + e)^9 + 3*(55*(8*a^2 + 24*a*b + 3*b^2)*cosh
(f*x + e)^3 - 12*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e))*sinh(f*x + e)^8 + 6*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x
+ e)^7 + 6*(55*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^4 - 24*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^2 + 8*a^2
+ 24*a*b + 3*b^2)*sinh(f*x + e)^7 + 42*(11*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^5 - 8*(8*a^2 + 24*a*b + 3*b
^2)*cosh(f*x + e)^3 + (8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e))*sinh(f*x + e)^6 - 4*(8*a^2 + 24*a*b + 3*b^2)*cos
h(f*x + e)^5 + 2*(231*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^6 - 252*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^4
+ 63*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^2 - 16*a^2 - 48*a*b - 6*b^2)*sinh(f*x + e)^5 + 2*(165*(8*a^2 + 24*
a*b + 3*b^2)*cosh(f*x + e)^7 - 252*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^5 + 105*(8*a^2 + 24*a*b + 3*b^2)*cos
h(f*x + e)^3 - 10*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e))*sinh(f*x + e)^4 + (8*a^2 + 24*a*b + 3*b^2)*cosh(f*x
+ e)^3 + (165*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^8 - 336*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^6 + 210*(8
*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^4 - 40*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^2 + 8*a^2 + 24*a*b + 3*b^2)
*sinh(f*x + e)^3 + (55*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^9 - 144*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^7
+ 126*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^5 - 40*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^3 + 3*(8*a^2 + 24*
a*b + 3*b^2)*cosh(f*x + e))*sinh(f*x + e)^2 + (11*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^10 - 36*(8*a^2 + 24*a
*b + 3*b^2)*cosh(f*x + e)^8 + 42*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^6 - 20*(8*a^2 + 24*a*b + 3*b^2)*cosh(f
*x + e)^4 + 3*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^2)*sinh(f*x + e))*sqrt(a)*log((b*cosh(f*x + e)^4 + 4*b*co
sh(f*x + e)*sinh(f*x + e)^3 + b*sinh(f*x + e)^4 + 2*(4*a - b)*cosh(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2 + 4*a -
b)*sinh(f*x + e)^2 - 4*sqrt(2)*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))*(cosh(f*x + e) + sinh(f*x + e)) + 4*(b*cosh(f*x + e)^3 +
(4*a - b)*cosh(f*x + e))*sinh(f*x + e) + 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*b*cosh(f*x + e)^12 + 12*a*b*cosh(f*x + e)*sinh(f*x + e)^11 + a*b*sinh(f*x + e)^
12 + 2*(8*a^2 + 9*a*b)*cosh(f*x + e)^10 + 2*(33*a*b*cosh(f*x + e)^2 + 8*a^2 + 9*a*b)*sinh(f*x + e)^10 + 20*(11
*a*b*cosh(f*x + e)^3 + (8*a^2 + 9*a*b)*cosh(f*x + e))*sinh(f*x + e)^9 - (112*a^2 + 111*a*b)*cosh(f*x + e)^8 +
(495*a*b*cosh(f*x + e)^4 + 90*(8*a^2 + 9*a*b)*cosh(f*x + e)^2 - 112*a^2 - 111*a*b)*sinh(f*x + e)^8 + 8*(99*a*b
*cosh(f*x + e)^5 + 30*(8*a^2 + 9*a*b)*cosh(f*x + e)^3 - (112*a^2 + 111*a*b)*cosh(f*x + e))*sinh(f*x + e)^7 + 8
*(18*a^2 + 23*a*b)*cosh(f*x + e)^6 + 4*(231*a*b*cosh(f*x + e)^6 + 105*(8*a^2 + 9*a*b)*cosh(f*x + e)^4 - 7*(112
*a^2 + 111*a*b)*cosh(f*x + e)^2 + 36*a^2 + 46*a*b)*sinh(f*x + e)^6 + 8*(99*a*b*cosh(f*x + e)^7 + 63*(8*a^2 + 9
*a*b)*cosh(f*x + e)^5 - 7*(112*a^2 + 111*a*b)*cosh(f*x + e)^3 + 6*(18*a^2 + 23*a*b)*cosh(f*x + e))*sinh(f*x +
e)^5 - (112*a^2 + 111*a*b)*cosh(f*x + e)^4 + (495*a*b*cosh(f*x + e)^8 + 420*(8*a^2 + 9*a*b)*cosh(f*x + e)^6 -
70*(112*a^2 + 111*a*b)*cosh(f*x + e)^4 + 120*(18*a^2 + 23*a*b)*cosh(f*x + e)^2 - 112*a^2 - 111*a*b)*sinh(f*x +
e)^4 + 4*(55*a*b*cosh(f*x + e)^9 + 60*(8*a^2 + 9*a*b)*cosh(f*x + e)^7 - 14*(112*a^2 + 111*a*b)*cosh(f*x + e)^
5 + 40*(18*a^2 + 23*a*b)*cosh(f*x + e)^3 - (112*a^2 + 111*a*b)*cosh(f*x + e))*sinh(f*x + e)^3 + 2*(8*a^2 + 9*a
*b)*cosh(f*x + e)^2 + 2*(33*a*b*cosh(f*x + e)^10 + 45*(8*a^2 + 9*a*b)*cosh(f*x + e)^8 - 14*(112*a^2 + 111*a*b)
*cosh(f*x + e)^6 + 60*(18*a^2 + 23*a*b)*cosh(f*x + e)^4 - 3*(112*a^2 + 111*a*b)*cosh(f*x + e)^2 + 8*a^2 + 9*a*
b)*sinh(f*x + e)^2 + a*b + 4*(3*a*b*cosh(f*x + e)^11 + 5*(8*a^2 + 9*a*b)*cosh(f*x + e)^9 - 2*(112*a^2 + 111*a*
b)*cosh(f*x + e)^7 + 12*(18*a^2 + 23*a*b)*cosh(f*x + e)^5 - (112*a^2 + 111*a*b)*cosh(f*x + e)^3 + (8*a^2 + 9*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)^11 + 11*a*f*cosh(f*x + e)*sinh(f*x + e)^10
+ a*f*sinh(f*x + e)^11 - 4*a*f*cosh(f*x + e)^9 + (55*a*f*cosh(f*x + e)^2 - 4*a*f)*sinh(f*x + e)^9 + 6*a*f*cos
h(f*x + e)^7 + 3*(55*a*f*cosh(f*x + e)^3 - 12*a*f*cosh(f*x + e))*sinh(f*x + e)^8 + 6*(55*a*f*cosh(f*x + e)^4 -
24*a*f*cosh(f*x + e)^2 + a*f)*sinh(f*x + e)^7 - 4*a*f*cosh(f*x + e)^5 + 42*(11*a*f*cosh(f*x + e)^5 - 8*a*f*co
sh(f*x + e)^3 + a*f*cosh(f*x + e))*sinh(f*x + e)^6 + 2*(231*a*f*cosh(f*x + e)^6 - 252*a*f*cosh(f*x + e)^4 + 63
*a*f*cosh(f*x + e)^2 - 2*a*f)*sinh(f*x + e)^5 + a*f*cosh(f*x + e)^3 + 2*(165*a*f*cosh(f*x + e)^7 - 252*a*f*cos
h(f*x + e)^5 + 105*a*f*cosh(f*x + e)^3 - 10*a*f*cosh(f*x + e))*sinh(f*x + e)^4 + (165*a*f*cosh(f*x + e)^8 - 33
6*a*f*cosh(f*x + e)^6 + 210*a*f*cosh(f*x + e)^4 - 40*a*f*cosh(f*x + e)^2 + a*f)*sinh(f*x + e)^3 + (55*a*f*cosh
(f*x + e)^9 - 144*a*f*cosh(f*x + e)^7 + 126*a*f*cosh(f*x + e)^5 - 40*a*f*cosh(f*x + e)^3 + 3*a*f*cosh(f*x + e)
)*sinh(f*x + e)^2 + (11*a*f*cosh(f*x + e)^10 - 36*a*f*cosh(f*x + e)^8 + 42*a*f*cosh(f*x + e)^6 - 20*a*f*cosh(f
*x + e)^4 + 3*a*f*cosh(f*x + e)^2)*sinh(f*x + e)), 1/24*(3*((8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^11 + 11*(8*
a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)*sinh(f*x + e)^10 + (8*a^2 + 24*a*b + 3*b^2)*sinh(f*x + e)^11 - 4*(8*a^2 +
24*a*b + 3*b^2)*cosh(f*x + e)^9 + (55*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^2 - 32*a^2 - 96*a*b - 12*b^2)*sin
h(f*x + e)^9 + 3*(55*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^3 - 12*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e))*sin
h(f*x + e)^8 + 6*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^7 + 6*(55*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^4 - 2
4*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^2 + 8*a^2 + 24*a*b + 3*b^2)*sinh(f*x + e)^7 + 42*(11*(8*a^2 + 24*a*b
+ 3*b^2)*cosh(f*x + e)^5 - 8*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^3 + (8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)
)*sinh(f*x + e)^6 - 4*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^5 + 2*(231*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)
^6 - 252*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^4 + 63*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^2 - 16*a^2 - 48*
a*b - 6*b^2)*sinh(f*x + e)^5 + 2*(165*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^7 - 252*(8*a^2 + 24*a*b + 3*b^2)*
cosh(f*x + e)^5 + 105*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^3 - 10*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e))*si
nh(f*x + e)^4 + (8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^3 + (165*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^8 - 336
*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^6 + 210*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^4 - 40*(8*a^2 + 24*a*b
+ 3*b^2)*cosh(f*x + e)^2 + 8*a^2 + 24*a*b + 3*b^2)*sinh(f*x + e)^3 + (55*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e
)^9 - 144*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^7 + 126*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^5 - 40*(8*a^2
+ 24*a*b + 3*b^2)*cosh(f*x + e)^3 + 3*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e))*sinh(f*x + e)^2 + (11*(8*a^2 + 2
4*a*b + 3*b^2)*cosh(f*x + e)^10 - 36*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^8 + 42*(8*a^2 + 24*a*b + 3*b^2)*co
sh(f*x + e)^6 - 20*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^4 + 3*(8*a^2 + 24*a*b + 3*b^2)*cosh(f*x + e)^2)*sinh
(f*x + e))*sqrt(-a)*arctan(1/2*sqrt(2)*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))/(a*cosh(f*x + e) + a*sinh(f*x + e))) + sqrt(2)*(
a*b*cosh(f*x + e)^12 + 12*a*b*cosh(f*x + e)*sinh(f*x + e)^11 + a*b*sinh(f*x + e)^12 + 2*(8*a^2 + 9*a*b)*cosh(f
*x + e)^10 + 2*(33*a*b*cosh(f*x + e)^2 + 8*a^2 + 9*a*b)*sinh(f*x + e)^10 + 20*(11*a*b*cosh(f*x + e)^3 + (8*a^2
+ 9*a*b)*cosh(f*x + e))*sinh(f*x + e)^9 - (112*a^2 + 111*a*b)*cosh(f*x + e)^8 + (495*a*b*cosh(f*x + e)^4 + 90
*(8*a^2 + 9*a*b)*cosh(f*x + e)^2 - 112*a^2 - 111*a*b)*sinh(f*x + e)^8 + 8*(99*a*b*cosh(f*x + e)^5 + 30*(8*a^2
+ 9*a*b)*cosh(f*x + e)^3 - (112*a^2 + 111*a*b)*cosh(f*x + e))*sinh(f*x + e)^7 + 8*(18*a^2 + 23*a*b)*cosh(f*x +
e)^6 + 4*(231*a*b*cosh(f*x + e)^6 + 105*(8*a^2 + 9*a*b)*cosh(f*x + e)^4 - 7*(112*a^2 + 111*a*b)*cosh(f*x + e)
^2 + 36*a^2 + 46*a*b)*sinh(f*x + e)^6 + 8*(99*a*b*cosh(f*x + e)^7 + 63*(8*a^2 + 9*a*b)*cosh(f*x + e)^5 - 7*(11
2*a^2 + 111*a*b)*cosh(f*x + e)^3 + 6*(18*a^2 + 23*a*b)*cosh(f*x + e))*sinh(f*x + e)^5 - (112*a^2 + 111*a*b)*co
sh(f*x + e)^4 + (495*a*b*cosh(f*x + e)^8 + 420*(8*a^2 + 9*a*b)*cosh(f*x + e)^6 - 70*(112*a^2 + 111*a*b)*cosh(f
*x + e)^4 + 120*(18*a^2 + 23*a*b)*cosh(f*x + e)^2 - 112*a^2 - 111*a*b)*sinh(f*x + e)^4 + 4*(55*a*b*cosh(f*x +
e)^9 + 60*(8*a^2 + 9*a*b)*cosh(f*x + e)^7 - 14*(112*a^2 + 111*a*b)*cosh(f*x + e)^5 + 40*(18*a^2 + 23*a*b)*cosh
(f*x + e)^3 - (112*a^2 + 111*a*b)*cosh(f*x + e))*sinh(f*x + e)^3 + 2*(8*a^2 + 9*a*b)*cosh(f*x + e)^2 + 2*(33*a
*b*cosh(f*x + e)^10 + 45*(8*a^2 + 9*a*b)*cosh(f*x + e)^8 - 14*(112*a^2 + 111*a*b)*cosh(f*x + e)^6 + 60*(18*a^2
+ 23*a*b)*cosh(f*x + e)^4 - 3*(112*a^2 + 111*a*b)*cosh(f*x + e)^2 + 8*a^2 + 9*a*b)*sinh(f*x + e)^2 + a*b + 4*
(3*a*b*cosh(f*x + e)^11 + 5*(8*a^2 + 9*a*b)*cosh(f*x + e)^9 - 2*(112*a^2 + 111*a*b)*cosh(f*x + e)^7 + 12*(18*a
^2 + 23*a*b)*cosh(f*x + e)^5 - (112*a^2 + 111*a*b)*cosh(f*x + e)^3 + (8*a^2 + 9*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)^11 + 11*a*f*cosh(f*x + e)*sinh(f*x + e)^10 + a*f*sinh(f*x + e)^11 - 4*a
*f*cosh(f*x + e)^9 + (55*a*f*cosh(f*x + e)^2 - 4*a*f)*sinh(f*x + e)^9 + 6*a*f*cosh(f*x + e)^7 + 3*(55*a*f*cosh
(f*x + e)^3 - 12*a*f*cosh(f*x + e))*sinh(f*x + e)^8 + 6*(55*a*f*cosh(f*x + e)^4 - 24*a*f*cosh(f*x + e)^2 + a*f
)*sinh(f*x + e)^7 - 4*a*f*cosh(f*x + e)^5 + 42*(11*a*f*cosh(f*x + e)^5 - 8*a*f*cosh(f*x + e)^3 + a*f*cosh(f*x
+ e))*sinh(f*x + e)^6 + 2*(231*a*f*cosh(f*x + e)^6 - 252*a*f*cosh(f*x + e)^4 + 63*a*f*cosh(f*x + e)^2 - 2*a*f)
*sinh(f*x + e)^5 + a*f*cosh(f*x + e)^3 + 2*(165*a*f*cosh(f*x + e)^7 - 252*a*f*cosh(f*x + e)^5 + 105*a*f*cosh(f
*x + e)^3 - 10*a*f*cosh(f*x + e))*sinh(f*x + e)^4 + (165*a*f*cosh(f*x + e)^8 - 336*a*f*cosh(f*x + e)^6 + 210*a
*f*cosh(f*x + e)^4 - 40*a*f*cosh(f*x + e)^2 + a*f)*sinh(f*x + e)^3 + (55*a*f*cosh(f*x + e)^9 - 144*a*f*cosh(f*
x + e)^7 + 126*a*f*cosh(f*x + e)^5 - 40*a*f*cosh(f*x + e)^3 + 3*a*f*cosh(f*x + e))*sinh(f*x + e)^2 + (11*a*f*c
osh(f*x + e)^10 - 36*a*f*cosh(f*x + e)^8 + 42*a*f*cosh(f*x + e)^6 - 20*a*f*cosh(f*x + e)^4 + 3*a*f*cosh(f*x +
e)^2)*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(coth(f*x+e)**5*(a+b*sinh(f*x+e)**2)**(3/2),x)

[Out]

Timed out

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

[Out]

Exception raised: TypeError