∫ coth3 (e+fx)___ (a+b sinh2(e+fx))3∕2 dx [494]

Optimal. Leaf size=110 \[ -\frac {(2 a-3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{5/2} f}+\frac {2 a-3 b}{2 a^2 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\text {csch}^2(e+f x)}{2 a f \sqrt {a+b \sinh ^2(e+f x)}} \]

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

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Rubi [A]
time = 0.09, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, 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-3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{5/2} f}+\frac {2 a-3 b}{2 a^2 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\text {csch}^2(e+f x)}{2 a f \sqrt {a+b \sinh ^2(e+f x)}} \end {gather*}

-1/2*((2*a - 3*b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(a^(5/2)*f) + (2*a - 3*b)/(2*a^2*f*Sqrt[a + b*

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

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[

c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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 )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1+x}{x^2 (a+b x)^{3/2}} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=-\frac {\text {csch}^2(e+f x)}{2 a f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(2 a-3 b) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\sinh ^2(e+f x)\right )}{4 a f}\\ &=\frac {2 a-3 b}{2 a^2 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\text {csch}^2(e+f x)}{2 a f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(2 a-3 b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{4 a^2 f}\\ &=\frac {2 a-3 b}{2 a^2 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\text {csch}^2(e+f x)}{2 a f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(2 a-3 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^2 b f}\\ &=-\frac {(2 a-3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{5/2} f}+\frac {2 a-3 b}{2 a^2 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\text {csch}^2(e+f x)}{2 a 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 = 0.11, size = 69, normalized size = 0.63 \begin {gather*} \frac {-a \text {csch}^2(e+f x)+(2 a-3 b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1+\frac {b \sinh ^2(e+f x)}{a}\right )}{2 a^2 f \sqrt {a+b \sinh ^2(e+f x)}} \end {gather*}

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.23, size = 43, normalized size = 0.39


method	result	size

default	\(\frac {\mathit {`\,int/indef0`\,}\left (\frac {\cosh ^{2}\left (f x +e \right )}{\sinh \left (f x +e \right )^{3} \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}, \sinh \left (f x +e \right )\right )}{f}\)	\(43\)
risch	\(\text {Expression too large to display}\)	\(289430\)

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

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

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

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1513 vs. \(2 (94) = 188\).
time = 0.58, size = 3228, normalized size = 29.35 \begin {gather*} \text {Too large to display} \end {gather*}

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

[-1/4*(((2*a*b - 3*b^2)*cosh(f*x + e)^8 + 8*(2*a*b - 3*b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + (2*a*b - 3*b^2)*si

nh(f*x + e)^8 + 4*(2*a^2 - 5*a*b + 3*b^2)*cosh(f*x + e)^6 + 4*(7*(2*a*b - 3*b^2)*cosh(f*x + e)^2 + 2*a^2 - 5*a

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

sinh(f*x + e)^5 - 2*(8*a^2 - 18*a*b + 9*b^2)*cosh(f*x + e)^4 + 2*(35*(2*a*b - 3*b^2)*cosh(f*x + e)^4 + 30*(2*a

^2 - 5*a*b + 3*b^2)*cosh(f*x + e)^2 - 8*a^2 + 18*a*b - 9*b^2)*sinh(f*x + e)^4 + 8*(7*(2*a*b - 3*b^2)*cosh(f*x

+ e)^5 + 10*(2*a^2 - 5*a*b + 3*b^2)*cosh(f*x + e)^3 - (8*a^2 - 18*a*b + 9*b^2)*cosh(f*x + e))*sinh(f*x + e)^3

+ 4*(2*a^2 - 5*a*b + 3*b^2)*cosh(f*x + e)^2 + 4*(7*(2*a*b - 3*b^2)*cosh(f*x + e)^6 + 15*(2*a^2 - 5*a*b + 3*b^2

)*cosh(f*x + e)^4 - 3*(8*a^2 - 18*a*b + 9*b^2)*cosh(f*x + e)^2 + 2*a^2 - 5*a*b + 3*b^2)*sinh(f*x + e)^2 + 2*a*

b - 3*b^2 + 8*((2*a*b - 3*b^2)*cosh(f*x + e)^7 + 3*(2*a^2 - 5*a*b + 3*b^2)*cosh(f*x + e)^5 - (8*a^2 - 18*a*b +

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

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

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

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

+ e))*sinh(f*x + e)^2 + (2*a^2 - 3*a*b)*cosh(f*x + e) + (5*(2*a^2 - 3*a*b)*cosh(f*x + e)^4 - 6*(4*a^2 - 3*a*b)

*cosh(f*x + e)^2 + 2*a^2 - 3*a*b)*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^3*b*f*cosh(f*x + e)^8 + 8*a^3*b*f*cosh(f*x

+ e)*sinh(f*x + e)^7 + a^3*b*f*sinh(f*x + e)^8 + 4*(a^4 - a^3*b)*f*cosh(f*x + e)^6 + 4*(7*a^3*b*f*cosh(f*x + e

)^2 + (a^4 - a^3*b)*f)*sinh(f*x + e)^6 - 2*(4*a^4 - 3*a^3*b)*f*cosh(f*x + e)^4 + 8*(7*a^3*b*f*cosh(f*x + e)^3

+ 3*(a^4 - a^3*b)*f*cosh(f*x + e))*sinh(f*x + e)^5 + a^3*b*f + 2*(35*a^3*b*f*cosh(f*x + e)^4 + 30*(a^4 - a^3*b

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

cosh(f*x + e)^5 + 10*(a^4 - a^3*b)*f*cosh(f*x + e)^3 - (4*a^4 - 3*a^3*b)*f*cosh(f*x + e))*sinh(f*x + e)^3 + 4*

(7*a^3*b*f*cosh(f*x + e)^6 + 15*(a^4 - a^3*b)*f*cosh(f*x + e)^4 - 3*(4*a^4 - 3*a^3*b)*f*cosh(f*x + e)^2 + (a^4

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

*b)*f*cosh(f*x + e)^3 + (a^4 - a^3*b)*f*cosh(f*x + e))*sinh(f*x + e)), 1/2*(((2*a*b - 3*b^2)*cosh(f*x + e)^8 +

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

*cosh(f*x + e)^6 + 4*(7*(2*a*b - 3*b^2)*cosh(f*x + e)^2 + 2*a^2 - 5*a*b + 3*b^2)*sinh(f*x + e)^6 + 8*(7*(2*a*b

- 3*b^2)*cosh(f*x + e)^3 + 3*(2*a^2 - 5*a*b + 3*b^2)*cosh(f*x + e))*sinh(f*x + e)^5 - 2*(8*a^2 - 18*a*b + 9*b

^2)*cosh(f*x + e)^4 + 2*(35*(2*a*b - 3*b^2)*cosh(f*x + e)^4 + 30*(2*a^2 - 5*a*b + 3*b^2)*cosh(f*x + e)^2 - 8*a

^2 + 18*a*b - 9*b^2)*sinh(f*x + e)^4 + 8*(7*(2*a*b - 3*b^2)*cosh(f*x + e)^5 + 10*(2*a^2 - 5*a*b + 3*b^2)*cosh(

f*x + e)^3 - (8*a^2 - 18*a*b + 9*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(2*a^2 - 5*a*b + 3*b^2)*cosh(f*x + e)

^2 + 4*(7*(2*a*b - 3*b^2)*cosh(f*x + e)^6 + 15*(2*a^2 - 5*a*b + 3*b^2)*cosh(f*x + e)^4 - 3*(8*a^2 - 18*a*b + 9

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

e)^7 + 3*(2*a^2 - 5*a*b + 3*b^2)*cosh(f*x + e)^5 - (8*a^2 - 18*a*b + 9*b^2)*cosh(f*x + e)^3 + (2*a^2 - 5*a*b

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

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

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

nh(f*x + e)^2 + (2*a^2 - 3*a*b)*cosh(f*x + e) + (5*(2*a^2 - 3*a*b)*cosh(f*x + e)^4 - 6*(4*a^2 - 3*a*b)*cosh(f*

x + e)^2 + 2*a^2 - 3*a*b)*sinh(f*x + e))*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{3}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \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*}

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

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 )}^{3/2}} \,d x \end {gather*}

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3.5.94 \(\int \frac {\coth ^3(e+f x)}{(a+b \sinh ^2(e+f x))^{3/2}} \, dx\) [494]