∫ coth6 (e+fx)______∘ a + a sinh2(e + fx) dx [446]

3.5.46 \(\int \frac {\coth ^6(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx\) [446]

Optimal. Leaf size=96 \[ -\frac {\coth (e+f x)}{f \sqrt {a \cosh ^2(e+f x)}}-\frac {2 \coth (e+f x) \text {csch}^2(e+f x)}{3 f \sqrt {a \cosh ^2(e+f x)}}-\frac {\coth (e+f x) \text {csch}^4(e+f x)}{5 f \sqrt {a \cosh ^2(e+f x)}} \]

[Out]

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

*csch(f*x+e)^4/f/(a*cosh(f*x+e)^2)^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 96, 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, 200} \begin {gather*} -\frac {\coth (e+f x)}{f \sqrt {a \cosh ^2(e+f x)}}-\frac {\coth (e+f x) \text {csch}^4(e+f x)}{5 f \sqrt {a \cosh ^2(e+f x)}}-\frac {2 \coth (e+f x) \text {csch}^2(e+f x)}{3 f \sqrt {a \cosh ^2(e+f x)}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]

&& IGtQ[n, 0] && IGtQ[p, 0]

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

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Mathematica [A]
time = 0.07, size = 49, normalized size = 0.51 \begin {gather*} -\frac {\coth (e+f x) \left (15+10 \text {csch}^2(e+f x)+3 \text {csch}^4(e+f x)\right )}{15 f \sqrt {a \cosh ^2(e+f x)}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/15*(Coth[e + f*x]*(15 + 10*Csch[e + f*x]^2 + 3*Csch[e + f*x]^4))/(f*Sqrt[a*Cosh[e + f*x]^2])

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Maple [A]
time = 1.58, size = 54, normalized size = 0.56


method	result	size

default	\(-\frac {\cosh \left (f x +e \right ) \left (15 \left (\sinh ^{4}\left (f x +e \right )\right )+10 \left (\sinh ^{2}\left (f x +e \right )\right )+3\right )}{15 \sinh \left (f x +e \right )^{5} \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}\, f}\)	\(54\)
risch	\(-\frac {2 \left ({\mathrm e}^{2 f x +2 e}+1\right ) \left (15 \,{\mathrm e}^{8 f x +8 e}-20 \,{\mathrm e}^{6 f x +6 e}+58 \,{\mathrm e}^{4 f x +4 e}-20 \,{\mathrm e}^{2 f x +2 e}+15\right )}{15 \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, f \left ({\mathrm e}^{2 f x +2 e}-1\right )^{5}}\)	\(102\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(coth(f*x+e)^6/(a+a*sinh(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/15*cosh(f*x+e)*(15*sinh(f*x+e)^4+10*sinh(f*x+e)^2+3)/sinh(f*x+e)^5/(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. 1315 vs. \(2 (94) = 188\).
time = 0.57, size = 1315, normalized size = 13.70 \begin {gather*} -\frac {\frac {120 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{\sqrt {a}} + \frac {45 \, \log \left (e^{\left (-f x - e\right )} + 1\right )}{\sqrt {a}} - \frac {45 \, \log \left (e^{\left (-f x - e\right )} - 1\right )}{\sqrt {a}} + \frac {2 \, {\left (105 \, \sqrt {a} e^{\left (-f x - e\right )} - 530 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 328 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} - 110 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )} + 15 \, \sqrt {a} e^{\left (-9 \, f x - 9 \, e\right )}\right )}}{5 \, a e^{\left (-2 \, f x - 2 \, e\right )} - 10 \, a e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, a e^{\left (-6 \, f x - 6 \, e\right )} - 5 \, a e^{\left (-8 \, f x - 8 \, e\right )} + a e^{\left (-10 \, f x - 10 \, e\right )} - a}}{256 \, f} - \frac {\frac {120 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{\sqrt {a}} - \frac {45 \, \log \left (e^{\left (-f x - e\right )} + 1\right )}{\sqrt {a}} + \frac {45 \, \log \left (e^{\left (-f x - e\right )} - 1\right )}{\sqrt {a}} + \frac {2 \, {\left (15 \, \sqrt {a} e^{\left (-f x - e\right )} - 110 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 328 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} - 530 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )} + 105 \, \sqrt {a} e^{\left (-9 \, f x - 9 \, e\right )}\right )}}{5 \, a e^{\left (-2 \, f x - 2 \, e\right )} - 10 \, a e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, a e^{\left (-6 \, f x - 6 \, e\right )} - 5 \, a e^{\left (-8 \, f x - 8 \, e\right )} + a e^{\left (-10 \, f x - 10 \, e\right )} - a}}{256 \, f} + \frac {\frac {60 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{\sqrt {a}} + \frac {75 \, \log \left (e^{\left (-f x - e\right )} + 1\right )}{\sqrt {a}} - \frac {75 \, \log \left (e^{\left (-f x - e\right )} - 1\right )}{\sqrt {a}} + \frac {2 \, {\left (105 \, \sqrt {a} e^{\left (-f x - e\right )} + 130 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} - 284 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} + 190 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )} - 45 \, \sqrt {a} e^{\left (-9 \, f x - 9 \, e\right )}\right )}}{5 \, a e^{\left (-2 \, f x - 2 \, e\right )} - 10 \, a e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, a e^{\left (-6 \, f x - 6 \, e\right )} - 5 \, a e^{\left (-8 \, f x - 8 \, e\right )} + a e^{\left (-10 \, f x - 10 \, e\right )} - a}}{320 \, f} + \frac {\frac {60 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{\sqrt {a}} - \frac {75 \, \log \left (e^{\left (-f x - e\right )} + 1\right )}{\sqrt {a}} + \frac {75 \, \log \left (e^{\left (-f x - e\right )} - 1\right )}{\sqrt {a}} - \frac {2 \, {\left (45 \, \sqrt {a} e^{\left (-f x - e\right )} - 190 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 284 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} - 130 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )} - 105 \, \sqrt {a} e^{\left (-9 \, f x - 9 \, e\right )}\right )}}{5 \, a e^{\left (-2 \, f x - 2 \, e\right )} - 10 \, a e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, a e^{\left (-6 \, f x - 6 \, e\right )} - 5 \, a e^{\left (-8 \, f x - 8 \, e\right )} + a e^{\left (-10 \, f x - 10 \, e\right )} - a}}{320 \, f} + \frac {\frac {15 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{\sqrt {a}} + \frac {15 \, \sqrt {a} e^{\left (-f x - e\right )} - 80 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 178 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} - 80 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )} + 15 \, \sqrt {a} e^{\left (-9 \, f x - 9 \, e\right )}}{5 \, a e^{\left (-2 \, f x - 2 \, e\right )} - 10 \, a e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, a e^{\left (-6 \, f x - 6 \, e\right )} - 5 \, a e^{\left (-8 \, f x - 8 \, e\right )} + a e^{\left (-10 \, f x - 10 \, e\right )} - a}}{24 \, f} - \frac {\arctan \left (e^{\left (-f x - e\right )}\right )}{16 \, \sqrt {a} f} + \frac {2685 \, \sqrt {a} e^{\left (-f x - e\right )} - 7370 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 8632 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} - 4790 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )} + 1035 \, \sqrt {a} e^{\left (-9 \, f x - 9 \, e\right )}}{1920 \, {\left (5 \, a e^{\left (-2 \, f x - 2 \, e\right )} - 10 \, a e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, a e^{\left (-6 \, f x - 6 \, e\right )} - 5 \, a e^{\left (-8 \, f x - 8 \, e\right )} + a e^{\left (-10 \, f x - 10 \, e\right )} - a\right )} f} + \frac {1035 \, \sqrt {a} e^{\left (-f x - e\right )} - 4790 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 8632 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} - 7370 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )} + 2685 \, \sqrt {a} e^{\left (-9 \, f x - 9 \, e\right )}}{1920 \, {\left (5 \, a e^{\left (-2 \, f x - 2 \, e\right )} - 10 \, a e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, a e^{\left (-6 \, f x - 6 \, e\right )} - 5 \, a e^{\left (-8 \, f x - 8 \, e\right )} + a e^{\left (-10 \, f x - 10 \, e\right )} - a\right )} f} \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)^(1/2),x, algorithm="maxima")

[Out]

-1/256*(120*arctan(e^(-f*x - e))/sqrt(a) + 45*log(e^(-f*x - e) + 1)/sqrt(a) - 45*log(e^(-f*x - e) - 1)/sqrt(a)

+ 2*(105*sqrt(a)*e^(-f*x - e) - 530*sqrt(a)*e^(-3*f*x - 3*e) + 328*sqrt(a)*e^(-5*f*x - 5*e) - 110*sqrt(a)*e^(

-7*f*x - 7*e) + 15*sqrt(a)*e^(-9*f*x - 9*e))/(5*a*e^(-2*f*x - 2*e) - 10*a*e^(-4*f*x - 4*e) + 10*a*e^(-6*f*x -

6*e) - 5*a*e^(-8*f*x - 8*e) + a*e^(-10*f*x - 10*e) - a))/f - 1/256*(120*arctan(e^(-f*x - e))/sqrt(a) - 45*log(

e^(-f*x - e) + 1)/sqrt(a) + 45*log(e^(-f*x - e) - 1)/sqrt(a) + 2*(15*sqrt(a)*e^(-f*x - e) - 110*sqrt(a)*e^(-3*

f*x - 3*e) + 328*sqrt(a)*e^(-5*f*x - 5*e) - 530*sqrt(a)*e^(-7*f*x - 7*e) + 105*sqrt(a)*e^(-9*f*x - 9*e))/(5*a*

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

- a))/f + 1/320*(60*arctan(e^(-f*x - e))/sqrt(a) + 75*log(e^(-f*x - e) + 1)/sqrt(a) - 75*log(e^(-f*x - e) - 1

)/sqrt(a) + 2*(105*sqrt(a)*e^(-f*x - e) + 130*sqrt(a)*e^(-3*f*x - 3*e) - 284*sqrt(a)*e^(-5*f*x - 5*e) + 190*sq

rt(a)*e^(-7*f*x - 7*e) - 45*sqrt(a)*e^(-9*f*x - 9*e))/(5*a*e^(-2*f*x - 2*e) - 10*a*e^(-4*f*x - 4*e) + 10*a*e^(

-6*f*x - 6*e) - 5*a*e^(-8*f*x - 8*e) + a*e^(-10*f*x - 10*e) - a))/f + 1/320*(60*arctan(e^(-f*x - e))/sqrt(a) -

75*log(e^(-f*x - e) + 1)/sqrt(a) + 75*log(e^(-f*x - e) - 1)/sqrt(a) - 2*(45*sqrt(a)*e^(-f*x - e) - 190*sqrt(a

)*e^(-3*f*x - 3*e) + 284*sqrt(a)*e^(-5*f*x - 5*e) - 130*sqrt(a)*e^(-7*f*x - 7*e) - 105*sqrt(a)*e^(-9*f*x - 9*e

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

- 10*e) - a))/f + 1/24*(15*arctan(e^(-f*x - e))/sqrt(a) + (15*sqrt(a)*e^(-f*x - e) - 80*sqrt(a)*e^(-3*f*x - 3

*e) + 178*sqrt(a)*e^(-5*f*x - 5*e) - 80*sqrt(a)*e^(-7*f*x - 7*e) + 15*sqrt(a)*e^(-9*f*x - 9*e))/(5*a*e^(-2*f*x

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

- 1/16*arctan(e^(-f*x - e))/(sqrt(a)*f) + 1/1920*(2685*sqrt(a)*e^(-f*x - e) - 7370*sqrt(a)*e^(-3*f*x - 3*e) +

8632*sqrt(a)*e^(-5*f*x - 5*e) - 4790*sqrt(a)*e^(-7*f*x - 7*e) + 1035*sqrt(a)*e^(-9*f*x - 9*e))/((5*a*e^(-2*f*x

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

+ 1/1920*(1035*sqrt(a)*e^(-f*x - e) - 4790*sqrt(a)*e^(-3*f*x - 3*e) + 8632*sqrt(a)*e^(-5*f*x - 5*e) - 7370*sqr

t(a)*e^(-7*f*x - 7*e) + 2685*sqrt(a)*e^(-9*f*x - 9*e))/((5*a*e^(-2*f*x - 2*e) - 10*a*e^(-4*f*x - 4*e) + 10*a*e

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1399 vs. \(2 (86) = 172\).
time = 0.45, size = 1399, normalized size = 14.57 \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)^(1/2),x, algorithm="fricas")

[Out]

-2/15*(135*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^8 + 15*e^(f*x + e)*sinh(f*x + e)^9 + 20*(27*cosh(f*x + e)^2

- 1)*e^(f*x + e)*sinh(f*x + e)^7 + 140*(9*cosh(f*x + e)^3 - cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^6 + 2*(9

45*cosh(f*x + e)^4 - 210*cosh(f*x + e)^2 + 29)*e^(f*x + e)*sinh(f*x + e)^5 + 10*(189*cosh(f*x + e)^5 - 70*cosh

(f*x + e)^3 + 29*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^4 + 20*(63*cosh(f*x + e)^6 - 35*cosh(f*x + e)^4 + 29

*cosh(f*x + e)^2 - 1)*e^(f*x + e)*sinh(f*x + e)^3 + 20*(27*cosh(f*x + e)^7 - 21*cosh(f*x + e)^5 + 29*cosh(f*x

+ e)^3 - 3*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^2 + 5*(27*cosh(f*x + e)^8 - 28*cosh(f*x + e)^6 + 58*cosh(f

*x + e)^4 - 12*cosh(f*x + e)^2 + 3)*e^(f*x + e)*sinh(f*x + e) + (15*cosh(f*x + e)^9 - 20*cosh(f*x + e)^7 + 58*

cosh(f*x + e)^5 - 20*cosh(f*x + e)^3 + 15*cosh(f*x + e))*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*f*cosh(f*x + e)^10 + (a*f*e^(2*f*x + 2*e) + a*f)*sinh(f*x + e)^10 - 5*a*f*cosh(f*x +

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

- a*f + (9*a*f*cosh(f*x + e)^2 - a*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^8 + 10*a*f*cosh(f*x + e)^6 + 40*(3*a*f*co

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

^7 + 10*(21*a*f*cosh(f*x + e)^4 - 14*a*f*cosh(f*x + e)^2 + a*f + (21*a*f*cosh(f*x + e)^4 - 14*a*f*cosh(f*x + e

)^2 + a*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^6 - 10*a*f*cosh(f*x + e)^4 + 4*(63*a*f*cosh(f*x + e)^5 - 70*a*f*cosh

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

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

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

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

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

x + e))*e^(2*f*x + 2*e))*sinh(f*x + e)^3 + 5*(9*a*f*cosh(f*x + e)^8 - 28*a*f*cosh(f*x + e)^6 + 30*a*f*cosh(f*x

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

)^4 - 12*a*f*cosh(f*x + e)^2 + a*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^2 - a*f + (a*f*cosh(f*x + e)^10 - 5*a*f*cos

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

+ 10*(a*f*cosh(f*x + e)^9 - 4*a*f*cosh(f*x + e)^7 + 6*a*f*cosh(f*x + e)^5 - 4*a*f*cosh(f*x + e)^3 + a*f*cosh(f

*x + e) + (a*f*cosh(f*x + e)^9 - 4*a*f*cosh(f*x + e)^7 + 6*a*f*cosh(f*x + e)^5 - 4*a*f*cosh(f*x + e)^3 + a*f*c

osh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{6}{\left (e + f x \right )}}{\sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )}}\, dx \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)**(1/2),x)

[Out]

Integral(coth(e + f*x)**6/sqrt(a*(sinh(e + f*x)**2 + 1)), x)

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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)^(1/2),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 = 0.90, size = 381, normalized size = 3.97 \begin {gather*} -\frac {4\,{\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}}{a\,f\,\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {32\,{\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\,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 {352\,{\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\,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\,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\,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)^(1/2),x)

[Out]

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

+ f*x) + exp(3*e + 3*f*x))) - (32*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(3*a*f

*(exp(2*e + 2*f*x) - 1)^2*(exp(e + f*x) + exp(3*e + 3*f*x))) - (352*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 -

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

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

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