3.446 \(\int \frac{\coth ^6(e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx\)
Optimal. Leaf size=96 \[ -\frac{\coth (e+f x)}{f \sqrt{a \cosh ^2(e+f x)}}-\frac{\coth (e+f x) \text{csch}^4(e+f x)}{5 f \sqrt{a \cosh ^2(e+f x)}}-\frac{2 \coth (e+f x) \text{csch}^2(e+f x)}{3 f \sqrt{a \cosh ^2(e+f x)}} \]
[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])
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Rubi [A] time = 0.122724, antiderivative size = 96, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used =
{3176, 3207, 2606, 194} \[ -\frac{\coth (e+f x)}{f \sqrt{a \cosh ^2(e+f x)}}-\frac{\coth (e+f x) \text{csch}^4(e+f x)}{5 f \sqrt{a \cosh ^2(e+f x)}}-\frac{2 \coth (e+f x) \text{csch}^2(e+f x)}{3 f \sqrt{a \cosh ^2(e+f x)}} \]
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 3176
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 3207
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
]])
Rule 2606
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 194
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]
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)) \operatorname{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)) \operatorname{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*}
Mathematica [A] time = 0.0819224, size = 49, normalized size = 0.51 \[ -\frac{\coth (e+f x) \left (3 \text{csch}^4(e+f x)+10 \text{csch}^2(e+f x)+15\right )}{15 f \sqrt{a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[e + f*x]^6/Sqrt[a + a*Sinh[e + f*x]^2],x]
[Out]
-(Coth[e + f*x]*(15 + 10*Csch[e + f*x]^2 + 3*Csch[e + f*x]^4))/(15*f*Sqrt[a*Cosh[e + f*x]^2])
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Maple [A] time = 0.113, size = 54, normalized size = 0.6 \begin{align*} -{\frac{\cosh \left ( fx+e \right ) \left ( 15\, \left ( \sinh \left ( fx+e \right ) \right ) ^{4}+10\, \left ( \sinh \left ( fx+e \right ) \right ) ^{2}+3 \right ) }{15\, \left ( \sinh \left ( fx+e \right ) \right ) ^{5}f}{\frac{1}{\sqrt{a \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}
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)
[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] time = 2.24241, size = 1662, normalized size = 17.31 \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)^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] time = 1.89456, size = 3776, normalized size = 39.33 \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)^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(-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)**6/(a+a*sinh(f*x+e)**2)**(1/2),x)
[Out]
Timed out
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Giac [A] time = 1.56128, size = 128, normalized size = 1.33 \begin{align*} -\frac{2 \,{\left (15 \, \sqrt{a} e^{\left (9 \, f x + 9 \, e\right )} - 20 \, \sqrt{a} e^{\left (7 \, f x + 7 \, e\right )} + 58 \, \sqrt{a} e^{\left (5 \, f x + 5 \, e\right )} - 20 \, \sqrt{a} e^{\left (3 \, f x + 3 \, e\right )} + 15 \, \sqrt{a} e^{\left (f x + e\right )}\right )}}{15 \, a f{\left (e^{\left (2 \, f x + 2 \, e\right )} - 1\right )}^{5}} \end{align*}
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]
-2/15*(15*sqrt(a)*e^(9*f*x + 9*e) - 20*sqrt(a)*e^(7*f*x + 7*e) + 58*sqrt(a)*e^(5*f*x + 5*e) - 20*sqrt(a)*e^(3*
f*x + 3*e) + 15*sqrt(a)*e^(f*x + e))/(a*f*(e^(2*f*x + 2*e) - 1)^5)