3.5.45 \(\int \frac {\coth ^4(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx\) [445]

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

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Rubi [A]
time = 0.08, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3255, 3286, 2686} \begin {gather*} -\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 {gather*}

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 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 ^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)) \text {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*}

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Mathematica [A]
time = 0.05, size = 37, normalized size = 0.61 \begin {gather*} -\frac {\coth (e+f x) \left (3+\text {csch}^2(e+f x)\right )}{3 f \sqrt {a \cosh ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[e + f*x]^4/Sqrt[a + a*Sinh[e + f*x]^2],x]

[Out]

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

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Maple [A]
time = 1.21, size = 44, normalized size = 0.72

method	result	size
default	\(-\frac {\cosh \left (f x +e \right ) \left (3 \left (\sinh ^{2}\left (f x +e \right )\right )+1\right )}{3 \sinh \left (f x +e \right )^{3} \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}\, f}\)	\(44\)
risch	\(-\frac {2 \left ({\mathrm e}^{2 f x +2 e}+1\right ) \left (3 \,{\mathrm e}^{4 f x +4 e}-2 \,{\mathrm e}^{2 f x +2 e}+3\right )}{3 \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, f \left ({\mathrm e}^{2 f x +2 e}-1\right )^{3}}\)	\(80\)

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,method=_RETURNVERBOSE)

[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] Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (60) = 120\).
time = 0.53, size = 592, normalized size = 9.70 \begin {gather*} \frac {\frac {6 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{\sqrt {a}} + \frac {3 \, \log \left (e^{\left (-f x - e\right )} + 1\right )}{\sqrt {a}} - \frac {3 \, \log \left (e^{\left (-f x - e\right )} - 1\right )}{\sqrt {a}} + \frac {4 \, {\left (3 \, \sqrt {a} e^{\left (-f x - e\right )} - \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )}\right )}}{3 \, a e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, a e^{\left (-4 \, f x - 4 \, e\right )} + a e^{\left (-6 \, f x - 6 \, e\right )} - a}}{12 \, f} + \frac {\frac {6 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{\sqrt {a}} - \frac {3 \, \log \left (e^{\left (-f x - e\right )} + 1\right )}{\sqrt {a}} + \frac {3 \, \log \left (e^{\left (-f x - e\right )} - 1\right )}{\sqrt {a}} - \frac {4 \, {\left (\sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} - 3 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )}\right )}}{3 \, a e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, a e^{\left (-4 \, f x - 4 \, e\right )} + a e^{\left (-6 \, f x - 6 \, e\right )} - a}}{12 \, f} - \frac {\frac {3 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{\sqrt {a}} + \frac {3 \, \sqrt {a} e^{\left (-f x - e\right )} - 10 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 3 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )}}{3 \, a e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, a e^{\left (-4 \, f x - 4 \, e\right )} + a e^{\left (-6 \, f x - 6 \, e\right )} - a}}{4 \, f} - \frac {\arctan \left (e^{\left (-f x - e\right )}\right )}{4 \, \sqrt {a} f} + \frac {27 \, \sqrt {a} e^{\left (-f x - e\right )} - 38 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 15 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )}}{24 \, {\left (3 \, a e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, a e^{\left (-4 \, f x - 4 \, e\right )} + a e^{\left (-6 \, f x - 6 \, e\right )} - a\right )} f} + \frac {15 \, \sqrt {a} e^{\left (-f x - e\right )} - 38 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 27 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )}}{24 \, {\left (3 \, a e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, a e^{\left (-4 \, f x - 4 \, e\right )} + a e^{\left (-6 \, f x - 6 \, e\right )} - a\right )} f} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 647 vs. \(2 (55) = 110\).
time = 0.44, size = 647, normalized size = 10.61 \begin {gather*} -\frac {2 \, {\left (15 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{4} + 3 \, e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{5} + 2 \, {\left (15 \, \cosh \left (f x + e\right )^{2} - 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3} + 6 \, {\left (5 \, \cosh \left (f x + e\right )^{3} - \cosh \left (f x + e\right )\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + 3 \, {\left (5 \, \cosh \left (f x + e\right )^{4} - 2 \, \cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + {\left (3 \, \cosh \left (f x + e\right )^{5} - 2 \, \cosh \left (f x + e\right )^{3} + 3 \, \cosh \left (f x + e\right )\right )} e^{\left (f x + e\right )}\right )} \sqrt {a e^{\left (4 \, f x + 4 \, e\right )} + 2 \, a e^{\left (2 \, f x + 2 \, e\right )} + a} e^{\left (-f x - e\right )}}{3 \, {\left (a f \cosh \left (f x + e\right )^{6} + {\left (a f e^{\left (2 \, f x + 2 \, e\right )} + a f\right )} \sinh \left (f x + e\right )^{6} - 3 \, a f \cosh \left (f x + e\right )^{4} + 6 \, {\left (a f \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + a f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )^{5} + 3 \, {\left (5 \, a f \cosh \left (f x + e\right )^{2} - a f + {\left (5 \, a f \cosh \left (f x + e\right )^{2} - a f\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{4} + 3 \, a f \cosh \left (f x + e\right )^{2} + 4 \, {\left (5 \, a f \cosh \left (f x + e\right )^{3} - 3 \, a f \cosh \left (f x + e\right ) + {\left (5 \, a f \cosh \left (f x + e\right )^{3} - 3 \, a f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{3} + 3 \, {\left (5 \, a f \cosh \left (f x + e\right )^{4} - 6 \, a f \cosh \left (f x + e\right )^{2} + a f + {\left (5 \, a f \cosh \left (f x + e\right )^{4} - 6 \, a f \cosh \left (f x + e\right )^{2} + a f\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{2} - a f + {\left (a f \cosh \left (f x + e\right )^{6} - 3 \, a f \cosh \left (f x + e\right )^{4} + 3 \, a f \cosh \left (f x + e\right )^{2} - a f\right )} e^{\left (2 \, f x + 2 \, e\right )} + 6 \, {\left (a f \cosh \left (f x + e\right )^{5} - 2 \, a f \cosh \left (f x + e\right )^{3} + a f \cosh \left (f x + e\right ) + {\left (a f \cosh \left (f x + e\right )^{5} - 2 \, a f \cosh \left (f x + e\right )^{3} + a f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )\right )}} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{4}{\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)**4/(a+a*sinh(f*x+e)**2)**(1/2),x)

[Out]

Integral(coth(e + f*x)**4/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)^4/(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 = 0.89, size = 95, normalized size = 1.56 \begin {gather*} -\frac {4\,{\mathrm {e}}^{2\,e+2\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}\,\left (3\,{\mathrm {e}}^{4\,e+4\,f\,x}-2\,{\mathrm {e}}^{2\,e+2\,f\,x}+3\right )}{3\,a\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^3\,\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(e + f*x)^4/(a + a*sinh(e + f*x)^2)^(1/2),x)

[Out]

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

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