∫ coth3 (e+fx)___ (a+b sinh2(e+fx))5∕2 dx [505]

Optimal. Leaf size=143 \[ -\frac {(2 a-5 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{7/2} f}+\frac {2 a-5 b}{6 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^2(e+f x)}{2 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {2 a-5 b}{2 a^3 f \sqrt {a+b \sinh ^2(e+f x)}} \]

-1/2*(2*a-5*b)*arctanh((a+b*sinh(f*x+e)^2)^(1/2)/a^(1/2))/a^(7/2)/f+1/6*(2*a-5*b)/a^2/f/(a+b*sinh(f*x+e)^2)^(3

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

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Rubi [A]
time = 0.10, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3273, 79, 53, 65, 214} \begin {gather*} -\frac {(2 a-5 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{7/2} f}+\frac {2 a-5 b}{2 a^3 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {2 a-5 b}{6 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^2(e+f x)}{2 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}

-1/2*((2*a - 5*b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(a^(7/2)*f) + (2*a - 5*b)/(6*a^2*f*(a + b*Sinh

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

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1

)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,

x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[

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,

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] || L

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

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]

\begin {align*} \int \frac {\coth ^3(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1+x}{x^2 (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=-\frac {\text {csch}^2(e+f x)}{2 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(2 a-5 b) \text {Subst}\left (\int \frac {1}{x (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{4 a f}\\ &=\frac {2 a-5 b}{6 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^2(e+f x)}{2 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(2 a-5 b) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\sinh ^2(e+f x)\right )}{4 a^2 f}\\ &=\frac {2 a-5 b}{6 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^2(e+f x)}{2 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {2 a-5 b}{2 a^3 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(2 a-5 b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{4 a^3 f}\\ &=\frac {2 a-5 b}{6 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^2(e+f x)}{2 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {2 a-5 b}{2 a^3 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(2 a-5 b) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^2(e+f x)}\right )}{2 a^3 b f}\\ &=-\frac {(2 a-5 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{7/2} f}+\frac {2 a-5 b}{6 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^2(e+f x)}{2 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {2 a-5 b}{2 a^3 f \sqrt {a+b \sinh ^2(e+f x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.21, size = 69, normalized size = 0.48 \begin {gather*} -\frac {3 a \text {csch}^2(e+f x)+(-2 a+5 b) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};1+\frac {b \sinh ^2(e+f x)}{a}\right )}{6 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}

-1/6*(3*a*Csch[e + f*x]^2 + (-2*a + 5*b)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Sinh[e + f*x]^2)/a])/(a^2*f*(

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.14, size = 73, normalized size = 0.51


method	result	size

default	\(\frac {\mathit {`\,int/indef0`\,}\left (\frac {\cosh ^{2}\left (f x +e \right )}{\left (b^{2} \left (\sinh ^{4}\left (f x +e \right )\right )+2 a b \left (\sinh ^{2}\left (f x +e \right )\right )+a^{2}\right ) \sinh \left (f x +e \right )^{3} \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\)	\(73\)
risch	\(\text {Expression too large to display}\)	\(309511\)

`int/indef0`(cosh(f*x+e)^2/(b^2*sinh(f*x+e)^4+2*a*b*sinh(f*x+e)^2+a^2)/sinh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3696 vs. \(2 (123) = 246\).
time = 0.66, size = 7594, normalized size = 53.10 \begin {gather*} \text {too large to display} \end {gather*}

[-1/12*(3*((2*a*b^2 - 5*b^3)*cosh(f*x + e)^12 + 12*(2*a*b^2 - 5*b^3)*cosh(f*x + e)*sinh(f*x + e)^11 + (2*a*b^2

- 5*b^3)*sinh(f*x + e)^12 + 2*(8*a^2*b - 26*a*b^2 + 15*b^3)*cosh(f*x + e)^10 + 2*(8*a^2*b - 26*a*b^2 + 15*b^3

+ 33*(2*a*b^2 - 5*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^10 + 20*(11*(2*a*b^2 - 5*b^3)*cosh(f*x + e)^3 + (8*a^2*

b - 26*a*b^2 + 15*b^3)*cosh(f*x + e))*sinh(f*x + e)^9 + (32*a^3 - 144*a^2*b + 190*a*b^2 - 75*b^3)*cosh(f*x + e

)^8 + (495*(2*a*b^2 - 5*b^3)*cosh(f*x + e)^4 + 32*a^3 - 144*a^2*b + 190*a*b^2 - 75*b^3 + 90*(8*a^2*b - 26*a*b^

2 + 15*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^8 + 8*(99*(2*a*b^2 - 5*b^3)*cosh(f*x + e)^5 + 30*(8*a^2*b - 26*a*b^

2 + 15*b^3)*cosh(f*x + e)^3 + (32*a^3 - 144*a^2*b + 190*a*b^2 - 75*b^3)*cosh(f*x + e))*sinh(f*x + e)^7 - 4*(16

*a^3 - 64*a^2*b + 70*a*b^2 - 25*b^3)*cosh(f*x + e)^6 + 4*(231*(2*a*b^2 - 5*b^3)*cosh(f*x + e)^6 + 105*(8*a^2*b

- 26*a*b^2 + 15*b^3)*cosh(f*x + e)^4 - 16*a^3 + 64*a^2*b - 70*a*b^2 + 25*b^3 + 7*(32*a^3 - 144*a^2*b + 190*a*

b^2 - 75*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 8*(99*(2*a*b^2 - 5*b^3)*cosh(f*x + e)^7 + 63*(8*a^2*b - 26*a*

b^2 + 15*b^3)*cosh(f*x + e)^5 + 7*(32*a^3 - 144*a^2*b + 190*a*b^2 - 75*b^3)*cosh(f*x + e)^3 - 3*(16*a^3 - 64*a

^2*b + 70*a*b^2 - 25*b^3)*cosh(f*x + e))*sinh(f*x + e)^5 + (32*a^3 - 144*a^2*b + 190*a*b^2 - 75*b^3)*cosh(f*x

+ e)^4 + (495*(2*a*b^2 - 5*b^3)*cosh(f*x + e)^8 + 420*(8*a^2*b - 26*a*b^2 + 15*b^3)*cosh(f*x + e)^6 + 70*(32*a

^3 - 144*a^2*b + 190*a*b^2 - 75*b^3)*cosh(f*x + e)^4 + 32*a^3 - 144*a^2*b + 190*a*b^2 - 75*b^3 - 60*(16*a^3 -

64*a^2*b + 70*a*b^2 - 25*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(55*(2*a*b^2 - 5*b^3)*cosh(f*x + e)^9 + 60*

(8*a^2*b - 26*a*b^2 + 15*b^3)*cosh(f*x + e)^7 + 14*(32*a^3 - 144*a^2*b + 190*a*b^2 - 75*b^3)*cosh(f*x + e)^5 -

20*(16*a^3 - 64*a^2*b + 70*a*b^2 - 25*b^3)*cosh(f*x + e)^3 + (32*a^3 - 144*a^2*b + 190*a*b^2 - 75*b^3)*cosh(f

*x + e))*sinh(f*x + e)^3 + 2*a*b^2 - 5*b^3 + 2*(8*a^2*b - 26*a*b^2 + 15*b^3)*cosh(f*x + e)^2 + 2*(33*(2*a*b^2

- 5*b^3)*cosh(f*x + e)^10 + 45*(8*a^2*b - 26*a*b^2 + 15*b^3)*cosh(f*x + e)^8 + 14*(32*a^3 - 144*a^2*b + 190*a*

b^2 - 75*b^3)*cosh(f*x + e)^6 - 30*(16*a^3 - 64*a^2*b + 70*a*b^2 - 25*b^3)*cosh(f*x + e)^4 + 8*a^2*b - 26*a*b^

2 + 15*b^3 + 3*(32*a^3 - 144*a^2*b + 190*a*b^2 - 75*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*(3*(2*a*b^2 - 5*

b^3)*cosh(f*x + e)^11 + 5*(8*a^2*b - 26*a*b^2 + 15*b^3)*cosh(f*x + e)^9 + 2*(32*a^3 - 144*a^2*b + 190*a*b^2 -

75*b^3)*cosh(f*x + e)^7 - 6*(16*a^3 - 64*a^2*b + 70*a*b^2 - 25*b^3)*cosh(f*x + e)^5 + (32*a^3 - 144*a^2*b + 19

0*a*b^2 - 75*b^3)*cosh(f*x + e)^3 + (8*a^2*b - 26*a*b^2 + 15*b^3)*cosh(f*x + e))*sinh(f*x + e))*sqrt(a)*log((b

*cosh(f*x + e)^4 + 4*b*cosh(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)) - 4*sqrt(2)*(3*(2*a^2*b - 5*a*b^2)*cosh(f*x + e)^9 + 27*(2*a^2*b - 5*

a*b^2)*cosh(f*x + e)*sinh(f*x + e)^8 + 3*(2*a^2*b - 5*a*b^2)*sinh(f*x + e)^9 + 4*(8*a^3 - 26*a^2*b + 15*a*b^2)

*cosh(f*x + e)^7 + 4*(8*a^3 - 26*a^2*b + 15*a*b^2 + 27*(2*a^2*b - 5*a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^7 +

28*(9*(2*a^2*b - 5*a*b^2)*cosh(f*x + e)^3 + (8*a^3 - 26*a^2*b + 15*a*b^2)*cosh(f*x + e))*sinh(f*x + e)^6 - 2*(

56*a^3 - 98*a^2*b + 45*a*b^2)*cosh(f*x + e)^5 + 2*(189*(2*a^2*b - 5*a*b^2)*cosh(f*x + e)^4 - 56*a^3 + 98*a^2*b

- 45*a*b^2 + 42*(8*a^3 - 26*a^2*b + 15*a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^5 + 2*(189*(2*a^2*b - 5*a*b^2)*c

osh(f*x + e)^5 + 70*(8*a^3 - 26*a^2*b + 15*a*b^2)*cosh(f*x + e)^3 - 5*(56*a^3 - 98*a^2*b + 45*a*b^2)*cosh(f*x

+ e))*sinh(f*x + e)^4 + 4*(8*a^3 - 26*a^2*b + 15*a*b^2)*cosh(f*x + e)^3 + 4*(63*(2*a^2*b - 5*a*b^2)*cosh(f*x +

e)^6 + 35*(8*a^3 - 26*a^2*b + 15*a*b^2)*cosh(f*x + e)^4 + 8*a^3 - 26*a^2*b + 15*a*b^2 - 5*(56*a^3 - 98*a^2*b

+ 45*a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^3 + 4*(27*(2*a^2*b - 5*a*b^2)*cosh(f*x + e)^7 + 21*(8*a^3 - 26*a^2*

b + 15*a*b^2)*cosh(f*x + e)^5 - 5*(56*a^3 - 98*a^2*b + 45*a*b^2)*cosh(f*x + e)^3 + 3*(8*a^3 - 26*a^2*b + 15*a*

b^2)*cosh(f*x + e))*sinh(f*x + e)^2 + 3*(2*a^2*b - 5*a*b^2)*cosh(f*x + e) + (27*(2*a^2*b - 5*a*b^2)*cosh(f*x +

e)^8 + 28*(8*a^3 - 26*a^2*b + 15*a*b^2)*cosh(f*x + e)^6 - 10*(56*a^3 - 98*a^2*b + 45*a*b^2)*cosh(f*x + e)^4 +

6*a^2*b - 15*a*b^2 + 12*(8*a^3 - 26*a^2*b + 15*a*b^2)*cosh(f*x + e)^2)*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^2

*f*cosh(f*x + e)^12 + 12*a^4*b^2*f*cosh(f*x + e)*sinh(f*x + e)^11 + a^4*b^2*f*sinh(f*x + e)^12 + 2*(4*a^5*b -

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {coth}\left (e+f\,x\right )}^3}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \end {gather*}

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3.6.5 \(\int \frac {\coth ^3(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\) [505]