3.445 \(\int \frac{\coth ^4(e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx\)
Optimal. Leaf size=61 \[ -\frac{\coth (e+f x)}{f \sqrt{a \cosh ^2(e+f x)}}-\frac{\coth (e+f x) \text{csch}^2(e+f x)}{3 f \sqrt{a \cosh ^2(e+f x)}} \]
[Out]
-(Coth[e + f*x]/(f*Sqrt[a*Cosh[e + f*x]^2])) - (Coth[e + f*x]*Csch[e + f*x]^2)/(3*f*Sqrt[a*Cosh[e + f*x]^2])
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Rubi [A] time = 0.114711, antiderivative size = 61, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used =
{3176, 3207, 2606} \[ -\frac{\coth (e+f x)}{f \sqrt{a \cosh ^2(e+f x)}}-\frac{\coth (e+f x) \text{csch}^2(e+f x)}{3 f \sqrt{a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
Int[Coth[e + f*x]^4/Sqrt[a + a*Sinh[e + f*x]^2],x]
[Out]
-(Coth[e + f*x]/(f*Sqrt[a*Cosh[e + f*x]^2])) - (Coth[e + f*x]*Csch[e + f*x]^2)/(3*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])
Rubi steps
\begin{align*} \int \frac{\coth ^4(e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx &=\int \frac{\coth ^4(e+f x)}{\sqrt{a \cosh ^2(e+f x)}} \, dx\\ &=\frac{\cosh (e+f x) \int \coth ^3(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 ) \, 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{\coth (e+f x) \text{csch}^2(e+f x)}{3 f \sqrt{a \cosh ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.0614957, size = 37, normalized size = 0.61 \[ -\frac{\coth (e+f x) \left (\text{csch}^2(e+f x)+3\right )}{3 f \sqrt{a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[e + f*x]^4/Sqrt[a + a*Sinh[e + f*x]^2],x]
[Out]
-(Coth[e + f*x]*(3 + Csch[e + f*x]^2))/(3*f*Sqrt[a*Cosh[e + f*x]^2])
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Maple [A] time = 0.098, size = 44, normalized size = 0.7 \begin{align*} -{\frac{\cosh \left ( fx+e \right ) \left ( 3\, \left ( \sinh \left ( fx+e \right ) \right ) ^{2}+1 \right ) }{3\, \left ( \sinh \left ( fx+e \right ) \right ) ^{3}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)^4/(a+a*sinh(f*x+e)^2)^(1/2),x)
[Out]
-1/3*cosh(f*x+e)*(3*sinh(f*x+e)^2+1)/sinh(f*x+e)^3/(a*cosh(f*x+e)^2)^(1/2)/f
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Maxima [B] time = 2.00025, size = 751, normalized size = 12.31 \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)^4/(a+a*sinh(f*x+e)^2)^(1/2),x, algorithm="maxima")
[Out]
1/12*(6*arctan(e^(-f*x - e))/sqrt(a) + 3*log(e^(-f*x - e) + 1)/sqrt(a) - 3*log(e^(-f*x - e) - 1)/sqrt(a) + 4*(
3*sqrt(a)*e^(-f*x - e) - sqrt(a)*e^(-3*f*x - 3*e))/(3*a*e^(-2*f*x - 2*e) - 3*a*e^(-4*f*x - 4*e) + a*e^(-6*f*x
- 6*e) - a))/f + 1/12*(6*arctan(e^(-f*x - e))/sqrt(a) - 3*log(e^(-f*x - e) + 1)/sqrt(a) + 3*log(e^(-f*x - e) -
1)/sqrt(a) - 4*(sqrt(a)*e^(-3*f*x - 3*e) - 3*sqrt(a)*e^(-5*f*x - 5*e))/(3*a*e^(-2*f*x - 2*e) - 3*a*e^(-4*f*x
- 4*e) + a*e^(-6*f*x - 6*e) - a))/f - 1/4*(3*arctan(e^(-f*x - e))/sqrt(a) + (3*sqrt(a)*e^(-f*x - e) - 10*sqrt(
a)*e^(-3*f*x - 3*e) + 3*sqrt(a)*e^(-5*f*x - 5*e))/(3*a*e^(-2*f*x - 2*e) - 3*a*e^(-4*f*x - 4*e) + a*e^(-6*f*x -
6*e) - a))/f - 1/4*arctan(e^(-f*x - e))/(sqrt(a)*f) + 1/24*(27*sqrt(a)*e^(-f*x - e) - 38*sqrt(a)*e^(-3*f*x -
3*e) + 15*sqrt(a)*e^(-5*f*x - 5*e))/((3*a*e^(-2*f*x - 2*e) - 3*a*e^(-4*f*x - 4*e) + a*e^(-6*f*x - 6*e) - a)*f)
+ 1/24*(15*sqrt(a)*e^(-f*x - e) - 38*sqrt(a)*e^(-3*f*x - 3*e) + 27*sqrt(a)*e^(-5*f*x - 5*e))/((3*a*e^(-2*f*x
- 2*e) - 3*a*e^(-4*f*x - 4*e) + a*e^(-6*f*x - 6*e) - a)*f)
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Fricas [B] time = 1.75004, size = 1692, normalized size = 27.74 \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)^4/(a+a*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")
[Out]
-2/3*(15*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^4 + 3*e^(f*x + e)*sinh(f*x + e)^5 + 2*(15*cosh(f*x + e)^2 - 1
)*e^(f*x + e)*sinh(f*x + e)^3 + 6*(5*cosh(f*x + e)^3 - cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^2 + 3*(5*cosh(
f*x + e)^4 - 2*cosh(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e) + (3*cosh(f*x + e)^5 - 2*cosh(f*x + e)^3 + 3*cos
h(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)^6 +
(a*f*e^(2*f*x + 2*e) + a*f)*sinh(f*x + e)^6 - 3*a*f*cosh(f*x + e)^4 + 6*(a*f*cosh(f*x + e)*e^(2*f*x + 2*e) +
a*f*cosh(f*x + e))*sinh(f*x + e)^5 + 3*(5*a*f*cosh(f*x + e)^2 - a*f + (5*a*f*cosh(f*x + e)^2 - a*f)*e^(2*f*x +
2*e))*sinh(f*x + e)^4 + 3*a*f*cosh(f*x + e)^2 + 4*(5*a*f*cosh(f*x + e)^3 - 3*a*f*cosh(f*x + e) + (5*a*f*cosh(
f*x + e)^3 - 3*a*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e)^3 + 3*(5*a*f*cosh(f*x + e)^4 - 6*a*f*cosh(f*x
+ e)^2 + a*f + (5*a*f*cosh(f*x + e)^4 - 6*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)^6 - 3*a*f*cosh(f*x + e)^4 + 3*a*f*cosh(f*x + e)^2 - a*f)*e^(2*f*x + 2*e) + 6*(a*f*cosh(f*x
+ e)^5 - 2*a*f*cosh(f*x + e)^3 + a*f*cosh(f*x + e) + (a*f*cosh(f*x + e)^5 - 2*a*f*cosh(f*x + e)^3 + a*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)**4/(a+a*sinh(f*x+e)**2)**(1/2),x)
[Out]
Timed out
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Giac [A] time = 1.49729, size = 88, normalized size = 1.44 \begin{align*} -\frac{2 \,{\left (3 \, \sqrt{a} e^{\left (5 \, f x + 5 \, e\right )} - 2 \, \sqrt{a} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, \sqrt{a} e^{\left (f x + e\right )}\right )}}{3 \, a f{\left (e^{\left (2 \, f x + 2 \, e\right )} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(f*x+e)^4/(a+a*sinh(f*x+e)^2)^(1/2),x, algorithm="giac")
[Out]
-2/3*(3*sqrt(a)*e^(5*f*x + 5*e) - 2*sqrt(a)*e^(3*f*x + 3*e) + 3*sqrt(a)*e^(f*x + e))/(a*f*(e^(2*f*x + 2*e) - 1
)^3)