3.455 \(\int \frac{\coth ^6(e+f x)}{(a+a \sinh ^2(e+f x))^{3/2}} \, dx\)
Optimal. Leaf size=77 \[ -\frac{\coth (e+f x) \text{csch}^4(e+f x)}{5 a f \sqrt{a \cosh ^2(e+f x)}}-\frac{\coth (e+f x) \text{csch}^2(e+f x)}{3 a f \sqrt{a \cosh ^2(e+f x)}} \]
[Out]
-(Coth[e + f*x]*Csch[e + f*x]^2)/(3*a*f*Sqrt[a*Cosh[e + f*x]^2]) - (Coth[e + f*x]*Csch[e + f*x]^4)/(5*a*f*Sqrt
[a*Cosh[e + f*x]^2])
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Rubi [A] time = 0.143745, antiderivative size = 77, 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, 14} \[ -\frac{\coth (e+f x) \text{csch}^4(e+f x)}{5 a f \sqrt{a \cosh ^2(e+f x)}}-\frac{\coth (e+f x) \text{csch}^2(e+f x)}{3 a f \sqrt{a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
Int[Coth[e + f*x]^6/(a + a*Sinh[e + f*x]^2)^(3/2),x]
[Out]
-(Coth[e + f*x]*Csch[e + f*x]^2)/(3*a*f*Sqrt[a*Cosh[e + f*x]^2]) - (Coth[e + f*x]*Csch[e + f*x]^4)/(5*a*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 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]]
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)) \operatorname{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)) \operatorname{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*}
Mathematica [A] time = 0.108842, size = 41, normalized size = 0.53 \[ -\frac{\coth ^3(e+f x) \left (3 \text{csch}^2(e+f x)+5\right )}{15 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[e + f*x]^6/(a + a*Sinh[e + f*x]^2)^(3/2),x]
[Out]
-(Coth[e + f*x]^3*(5 + 3*Csch[e + f*x]^2))/(15*f*(a*Cosh[e + f*x]^2)^(3/2))
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Maple [A] time = 0.132, size = 67, normalized size = 0.9 \begin{align*} -{\frac{\cosh \left ( fx+e \right ) \left ( 5\, \left ( \cosh \left ( fx+e \right ) \right ) ^{2}-2 \right ) }{15\, \left ( \cosh \left ( fx+e \right ) -1 \right ) ^{2} \left ( \cosh \left ( fx+e \right ) +1 \right ) ^{2}a\sinh \left ( fx+e \right ) 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)^(3/2),x)
[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] time = 2.32633, size = 2067, normalized size = 26.84 \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)^(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] time = 1.94361, size = 3571, normalized size = 46.38 \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)^(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)] 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)**(3/2),x)
[Out]
Timed out
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Giac [A] time = 1.80338, size = 92, normalized size = 1.19 \begin{align*} -\frac{8 \,{\left (5 \, \sqrt{a} e^{\left (7 \, f x + 7 \, e\right )} + 2 \, \sqrt{a} e^{\left (5 \, f x + 5 \, e\right )} + 5 \, \sqrt{a} e^{\left (3 \, f x + 3 \, e\right )}\right )}}{15 \, a^{2} 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)^(3/2),x, algorithm="giac")
[Out]
-8/15*(5*sqrt(a)*e^(7*f*x + 7*e) + 2*sqrt(a)*e^(5*f*x + 5*e) + 5*sqrt(a)*e^(3*f*x + 3*e))/(a^2*f*(e^(2*f*x + 2
*e) - 1)^5)