∫ coth6 (e+fx)___ (a+a sinh2(e+fx))3∕2 dx [455]

3.5.55 \(\int \frac {\coth ^6(e+f x)}{(a+a \sinh ^2(e+f x))^{3/2}} \, dx\) [455]

Optimal. Leaf size=77 \[ -\frac {\coth (e+f x) \text {csch}^2(e+f x)}{3 a f \sqrt {a \cosh ^2(e+f x)}}-\frac {\coth (e+f x) \text {csch}^4(e+f x)}{5 a f \sqrt {a \cosh ^2(e+f x)}} \]

[Out]

-1/3*coth(f*x+e)*csch(f*x+e)^2/a/f/(a*cosh(f*x+e)^2)^(1/2)-1/5*coth(f*x+e)*csch(f*x+e)^4/a/f/(a*cosh(f*x+e)^2)

^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3255, 3286, 2686, 14} \begin {gather*} -\frac {\coth (e+f x) \text {csch}^4(e+f x)}{5 a f \sqrt {a \cosh ^2(e+f x)}}-\frac {\coth (e+f x) \text {csch}^2(e+f x)}{3 a f \sqrt {a \cosh ^2(e+f x)}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[Coth[e + f*x]^6/(a + a*Sinh[e + f*x]^2)^(3/2),x]

[Out]

-1/3*(Coth[e + f*x]*Csch[e + f*x]^2)/(a*f*Sqrt[a*Cosh[e + f*x]^2]) - (Coth[e + f*x]*Csch[e + f*x]^4)/(5*a*f*Sq

rt[a*Cosh[e + f*x]^2])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]

&& !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[

Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -

1)/2] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 3255

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p]

, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3286

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di

st[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]

*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |

| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig

]])

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 41, normalized size = 0.53 \begin {gather*} -\frac {\coth ^3(e+f x) \left (5+3 \text {csch}^2(e+f x)\right )}{15 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[Coth[e + f*x]^6/(a + a*Sinh[e + f*x]^2)^(3/2),x]

[Out]

-1/15*(Coth[e + f*x]^3*(5 + 3*Csch[e + f*x]^2))/(f*(a*Cosh[e + f*x]^2)^(3/2))

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Maple [A]
time = 1.50, size = 67, normalized size = 0.87


method	result	size

default	\(-\frac {\cosh \left (f x +e \right ) \left (5 \left (\cosh ^{2}\left (f x +e \right )\right )-2\right )}{15 \left (\cosh \left (f x +e \right )+1\right )^{2} \left (\cosh \left (f x +e \right )-1\right )^{2} a \sinh \left (f x +e \right ) \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}\, f}\)	\(67\)
risch	\(-\frac {8 \left (5 \,{\mathrm e}^{4 f x +4 e}+2 \,{\mathrm e}^{2 f x +2 e}+5\right ) \left ({\mathrm e}^{2 f x +2 e}+1\right ) {\mathrm e}^{2 f x +2 e}}{15 \left ({\mathrm e}^{2 f x +2 e}-1\right )^{5} f \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, a}\)	\(92\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(coth(f*x+e)^6/(a+a*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/15*cosh(f*x+e)*(5*cosh(f*x+e)^2-2)/(cosh(f*x+e)+1)^2/(cosh(f*x+e)-1)^2/a/sinh(f*x+e)/(a*cosh(f*x+e)^2)^(1/2

)/f

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1641 vs. \(2 (75) = 150\).
time = 0.58, size = 1641, normalized size = 21.31 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^6/(a+a*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

-3/256*(2*(105*e^(-f*x - e) - 300*e^(-3*f*x - 3*e) + 81*e^(-5*f*x - 5*e) - 248*e^(-7*f*x - 7*e) + 51*e^(-9*f*x

- 9*e) + 100*e^(-11*f*x - 11*e) - 45*e^(-13*f*x - 13*e))/(3*a^(3/2)*e^(-2*f*x - 2*e) - a^(3/2)*e^(-4*f*x - 4*

e) - 5*a^(3/2)*e^(-6*f*x - 6*e) + 5*a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2)*e^(-10*f*x - 10*e) - 3*a^(3/2)*e^(-12*f

*x - 12*e) + a^(3/2)*e^(-14*f*x - 14*e) - a^(3/2)) + 60*arctan(e^(-f*x - e))/a^(3/2) + 75*log(e^(-f*x - e) + 1

)/a^(3/2) - 75*log(e^(-f*x - e) - 1)/a^(3/2))/f + 1/48*((105*e^(-f*x - e) - 350*e^(-3*f*x - 3*e) + 231*e^(-5*f

*x - 5*e) + 412*e^(-7*f*x - 7*e) + 231*e^(-9*f*x - 9*e) - 350*e^(-11*f*x - 11*e) + 105*e^(-13*f*x - 13*e))/(3*

a^(3/2)*e^(-2*f*x - 2*e) - a^(3/2)*e^(-4*f*x - 4*e) - 5*a^(3/2)*e^(-6*f*x - 6*e) + 5*a^(3/2)*e^(-8*f*x - 8*e)

+ a^(3/2)*e^(-10*f*x - 10*e) - 3*a^(3/2)*e^(-12*f*x - 12*e) + a^(3/2)*e^(-14*f*x - 14*e) - a^(3/2)) + 105*arct

an(e^(-f*x - e))/a^(3/2))/f + 3/256*(2*(45*e^(-f*x - e) - 100*e^(-3*f*x - 3*e) - 51*e^(-5*f*x - 5*e) + 248*e^(

-7*f*x - 7*e) - 81*e^(-9*f*x - 9*e) + 300*e^(-11*f*x - 11*e) - 105*e^(-13*f*x - 13*e))/(3*a^(3/2)*e^(-2*f*x -

2*e) - a^(3/2)*e^(-4*f*x - 4*e) - 5*a^(3/2)*e^(-6*f*x - 6*e) + 5*a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2)*e^(-10*f*x

- 10*e) - 3*a^(3/2)*e^(-12*f*x - 12*e) + a^(3/2)*e^(-14*f*x - 14*e) - a^(3/2)) - 60*arctan(e^(-f*x - e))/a^(3

/2) + 75*log(e^(-f*x - e) + 1)/a^(3/2) - 75*log(e^(-f*x - e) - 1)/a^(3/2))/f - 3/320*(4*(45*e^(-f*x - e) - 135

*e^(-3*f*x - 3*e) + 54*e^(-5*f*x - 5*e) + 198*e^(-7*f*x - 7*e) - 211*e^(-9*f*x - 9*e) - 15*e^(-11*f*x - 11*e))

/(3*a^(3/2)*e^(-2*f*x - 2*e) - a^(3/2)*e^(-4*f*x - 4*e) - 5*a^(3/2)*e^(-6*f*x - 6*e) + 5*a^(3/2)*e^(-8*f*x - 8

*e) + a^(3/2)*e^(-10*f*x - 10*e) - 3*a^(3/2)*e^(-12*f*x - 12*e) + a^(3/2)*e^(-14*f*x - 14*e) - a^(3/2)) + 90*a

rctan(e^(-f*x - e))/a^(3/2) + 45*log(e^(-f*x - e) + 1)/a^(3/2) - 45*log(e^(-f*x - e) - 1)/a^(3/2))/f + 3/320*(

4*(15*e^(-3*f*x - 3*e) + 211*e^(-5*f*x - 5*e) - 198*e^(-7*f*x - 7*e) - 54*e^(-9*f*x - 9*e) + 135*e^(-11*f*x -

11*e) - 45*e^(-13*f*x - 13*e))/(3*a^(3/2)*e^(-2*f*x - 2*e) - a^(3/2)*e^(-4*f*x - 4*e) - 5*a^(3/2)*e^(-6*f*x -

6*e) + 5*a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2)*e^(-10*f*x - 10*e) - 3*a^(3/2)*e^(-12*f*x - 12*e) + a^(3/2)*e^(-14

*f*x - 14*e) - a^(3/2)) - 90*arctan(e^(-f*x - e))/a^(3/2) + 45*log(e^(-f*x - e) + 1)/a^(3/2) - 45*log(e^(-f*x

- e) - 1)/a^(3/2))/f + 1/1920*(1155*e^(-f*x - e) + 1460*e^(-3*f*x - 3*e) - 4173*e^(-5*f*x - 5*e) + 2024*e^(-7*

f*x - 7*e) + 1857*e^(-9*f*x - 9*e) - 2140*e^(-11*f*x - 11*e) + 585*e^(-13*f*x - 13*e))/((3*a^(3/2)*e^(-2*f*x -

2*e) - a^(3/2)*e^(-4*f*x - 4*e) - 5*a^(3/2)*e^(-6*f*x - 6*e) + 5*a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2)*e^(-10*f*

x - 10*e) - 3*a^(3/2)*e^(-12*f*x - 12*e) + a^(3/2)*e^(-14*f*x - 14*e) - a^(3/2))*f) + 1/1920*(585*e^(-f*x - e)

- 2140*e^(-3*f*x - 3*e) + 1857*e^(-5*f*x - 5*e) + 2024*e^(-7*f*x - 7*e) - 4173*e^(-9*f*x - 9*e) + 1460*e^(-11

*f*x - 11*e) + 1155*e^(-13*f*x - 13*e))/((3*a^(3/2)*e^(-2*f*x - 2*e) - a^(3/2)*e^(-4*f*x - 4*e) - 5*a^(3/2)*e^

(-6*f*x - 6*e) + 5*a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2)*e^(-10*f*x - 10*e) - 3*a^(3/2)*e^(-12*f*x - 12*e) + a^(3

/2)*e^(-14*f*x - 14*e) - a^(3/2))*f) + 29/32*arctan(e^(-f*x - e))/(a^(3/2)*f)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1410 vs. \(2 (69) = 138\).
time = 0.43, size = 1410, normalized size = 18.31 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^6/(a+a*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

-8/15*(35*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^6 + 5*e^(f*x + e)*sinh(f*x + e)^7 + (105*cosh(f*x + e)^2 + 2

)*e^(f*x + e)*sinh(f*x + e)^5 + 5*(35*cosh(f*x + e)^3 + 2*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^4 + 5*(35*c

osh(f*x + e)^4 + 4*cosh(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e)^3 + 5*(21*cosh(f*x + e)^5 + 4*cosh(f*x + e)^

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

2)*e^(f*x + e)*sinh(f*x + e) + (5*cosh(f*x + e)^7 + 2*cosh(f*x + e)^5 + 5*cosh(f*x + e)^3)*e^(f*x + e))*sqrt(a

*e^(4*f*x + 4*e) + 2*a*e^(2*f*x + 2*e) + a)*e^(-f*x - e)/(a^2*f*cosh(f*x + e)^10 - 5*a^2*f*cosh(f*x + e)^8 + (

a^2*f*e^(2*f*x + 2*e) + a^2*f)*sinh(f*x + e)^10 + 10*(a^2*f*cosh(f*x + e)*e^(2*f*x + 2*e) + a^2*f*cosh(f*x + e

))*sinh(f*x + e)^9 + 10*a^2*f*cosh(f*x + e)^6 + 5*(9*a^2*f*cosh(f*x + e)^2 - a^2*f + (9*a^2*f*cosh(f*x + e)^2

- a^2*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^8 + 40*(3*a^2*f*cosh(f*x + e)^3 - a^2*f*cosh(f*x + e) + (3*a^2*f*cosh(

f*x + e)^3 - a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e)^7 - 10*a^2*f*cosh(f*x + e)^4 + 10*(21*a^2*f*c

osh(f*x + e)^4 - 14*a^2*f*cosh(f*x + e)^2 + a^2*f + (21*a^2*f*cosh(f*x + e)^4 - 14*a^2*f*cosh(f*x + e)^2 + a^2

*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^6 + 4*(63*a^2*f*cosh(f*x + e)^5 - 70*a^2*f*cosh(f*x + e)^3 + 15*a^2*f*cosh(

f*x + e) + (63*a^2*f*cosh(f*x + e)^5 - 70*a^2*f*cosh(f*x + e)^3 + 15*a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sin

h(f*x + e)^5 + 5*a^2*f*cosh(f*x + e)^2 + 10*(21*a^2*f*cosh(f*x + e)^6 - 35*a^2*f*cosh(f*x + e)^4 + 15*a^2*f*co

sh(f*x + e)^2 - a^2*f + (21*a^2*f*cosh(f*x + e)^6 - 35*a^2*f*cosh(f*x + e)^4 + 15*a^2*f*cosh(f*x + e)^2 - a^2*

f)*e^(2*f*x + 2*e))*sinh(f*x + e)^4 + 40*(3*a^2*f*cosh(f*x + e)^7 - 7*a^2*f*cosh(f*x + e)^5 + 5*a^2*f*cosh(f*x

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

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

x + e)^6 + 30*a^2*f*cosh(f*x + e)^4 - 12*a^2*f*cosh(f*x + e)^2 + a^2*f + (9*a^2*f*cosh(f*x + e)^8 - 28*a^2*f*c

osh(f*x + e)^6 + 30*a^2*f*cosh(f*x + e)^4 - 12*a^2*f*cosh(f*x + e)^2 + a^2*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^2

+ (a^2*f*cosh(f*x + e)^10 - 5*a^2*f*cosh(f*x + e)^8 + 10*a^2*f*cosh(f*x + e)^6 - 10*a^2*f*cosh(f*x + e)^4 + 5

*a^2*f*cosh(f*x + e)^2 - a^2*f)*e^(2*f*x + 2*e) + 10*(a^2*f*cosh(f*x + e)^9 - 4*a^2*f*cosh(f*x + e)^7 + 6*a^2*

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

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

e))

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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*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)**6/(a+a*sinh(f*x+e)**2)**(3/2),x)

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^6/(a+a*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

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

UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [B]
time = 0.16, size = 305, normalized size = 3.96 \begin {gather*} -\frac {16\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^2\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {272\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{15\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^3\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {128\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^4\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {64\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^5\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(coth(e + f*x)^6/(a + a*sinh(e + f*x)^2)^(3/2),x)

[Out]

- (16*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(3*a^2*f*(exp(2*e + 2*f*x) - 1)^2*

(exp(e + f*x) + exp(3*e + 3*f*x))) - (272*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2)

)/(15*a^2*f*(exp(2*e + 2*f*x) - 1)^3*(exp(e + f*x) + exp(3*e + 3*f*x))) - (128*exp(3*e + 3*f*x)*(a + a*(exp(e

+ f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(5*a^2*f*(exp(2*e + 2*f*x) - 1)^4*(exp(e + f*x) + exp(3*e + 3*f*x))) -

(64*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(5*a^2*f*(exp(2*e + 2*f*x) - 1)^5*(e

xp(e + f*x) + exp(3*e + 3*f*x)))

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