3.5.54 \(\int \frac {\coth ^4(e+f x)}{(a+a \sinh ^2(e+f x))^{3/2}} \, dx\) [454]

Optimal. Leaf size=38 \[ -\frac {\coth (e+f x) \text {csch}^2(e+f x)}{3 a f \sqrt {a \cosh ^2(e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3255, 3286, 2686, 30} \begin {gather*} -\frac {\coth (e+f x) \text {csch}^2(e+f x)}{3 a f \sqrt {a \cosh ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[e + f*x]^4/(a + a*Sinh[e + f*x]^2)^(3/2),x]

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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

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Mathematica [A]
time = 0.04, size = 29, normalized size = 0.76 \begin {gather*} -\frac {\coth ^3(e+f x)}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[e + f*x]^4/(a + a*Sinh[e + f*x]^2)^(3/2),x]

[Out]

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

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Maple [A]
time = 1.25, size = 35, normalized size = 0.92

method	result	size
default	\(-\frac {\cosh \left (f x +e \right )}{3 a \sinh \left (f x +e \right )^{3} \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}\, f}\)	\(35\)
risch	\(-\frac {8 \left ({\mathrm e}^{2 f x +2 e}+1\right ) {\mathrm e}^{2 f x +2 e}}{3 \left ({\mathrm e}^{2 f x +2 e}-1\right )^{3} f \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, a}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(f*x+e)^4/(a+a*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*cosh(f*x+e)/a/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. 881 vs. \(2 (37) = 74\).
time = 0.53, size = 881, normalized size = 23.18 \begin {gather*} \frac {\frac {21 \, e^{\left (-f x - e\right )} - 16 \, e^{\left (-3 \, f x - 3 \, e\right )} + 34 \, e^{\left (-5 \, f x - 5 \, e\right )} + 8 \, e^{\left (-7 \, f x - 7 \, e\right )} - 15 \, e^{\left (-9 \, f x - 9 \, e\right )}}{a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 2 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} - 2 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} - a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} - a^{\frac {3}{2}}} + \frac {3 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{a^{\frac {3}{2}}} + \frac {9 \, \log \left (e^{\left (-f x - e\right )} + 1\right )}{a^{\frac {3}{2}}} - \frac {9 \, \log \left (e^{\left (-f x - e\right )} - 1\right )}{a^{\frac {3}{2}}}}{12 \, f} - \frac {\frac {15 \, e^{\left (-f x - e\right )} - 8 \, e^{\left (-3 \, f x - 3 \, e\right )} - 34 \, e^{\left (-5 \, f x - 5 \, e\right )} + 16 \, e^{\left (-7 \, f x - 7 \, e\right )} - 21 \, e^{\left (-9 \, f x - 9 \, e\right )}}{a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 2 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} - 2 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} - a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} - a^{\frac {3}{2}}} - \frac {3 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{a^{\frac {3}{2}}} + \frac {9 \, \log \left (e^{\left (-f x - e\right )} + 1\right )}{a^{\frac {3}{2}}} - \frac {9 \, \log \left (e^{\left (-f x - e\right )} - 1\right )}{a^{\frac {3}{2}}}}{12 \, f} - \frac {\frac {15 \, e^{\left (-f x - e\right )} - 20 \, e^{\left (-3 \, f x - 3 \, e\right )} - 22 \, e^{\left (-5 \, f x - 5 \, e\right )} - 20 \, e^{\left (-7 \, f x - 7 \, e\right )} + 15 \, e^{\left (-9 \, f x - 9 \, e\right )}}{a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 2 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} - 2 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} - a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} - a^{\frac {3}{2}}} + \frac {15 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{a^{\frac {3}{2}}}}{8 \, f} + \frac {45 \, e^{\left (-f x - e\right )} - 52 \, e^{\left (-3 \, f x - 3 \, e\right )} - 74 \, e^{\left (-5 \, f x - 5 \, e\right )} + 92 \, e^{\left (-7 \, f x - 7 \, e\right )} + 21 \, e^{\left (-9 \, f x - 9 \, e\right )}}{48 \, {\left (a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 2 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} - 2 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} - a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} - a^{\frac {3}{2}}\right )} f} + \frac {21 \, e^{\left (-f x - e\right )} + 92 \, e^{\left (-3 \, f x - 3 \, e\right )} - 74 \, e^{\left (-5 \, f x - 5 \, e\right )} - 52 \, e^{\left (-7 \, f x - 7 \, e\right )} + 45 \, e^{\left (-9 \, f x - 9 \, e\right )}}{48 \, {\left (a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 2 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} - 2 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} - a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} - a^{\frac {3}{2}}\right )} f} + \frac {11 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{8 \, a^{\frac {3}{2}} 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)^(3/2),x, algorithm="maxima")

[Out]

1/12*((21*e^(-f*x - e) - 16*e^(-3*f*x - 3*e) + 34*e^(-5*f*x - 5*e) + 8*e^(-7*f*x - 7*e) - 15*e^(-9*f*x - 9*e))
/(a^(3/2)*e^(-2*f*x - 2*e) + 2*a^(3/2)*e^(-4*f*x - 4*e) - 2*a^(3/2)*e^(-6*f*x - 6*e) - a^(3/2)*e^(-8*f*x - 8*e
) + a^(3/2)*e^(-10*f*x - 10*e) - a^(3/2)) + 3*arctan(e^(-f*x - e))/a^(3/2) + 9*log(e^(-f*x - e) + 1)/a^(3/2) -
9*log(e^(-f*x - e) - 1)/a^(3/2))/f - 1/12*((15*e^(-f*x - e) - 8*e^(-3*f*x - 3*e) - 34*e^(-5*f*x - 5*e) + 16*e
^(-7*f*x - 7*e) - 21*e^(-9*f*x - 9*e))/(a^(3/2)*e^(-2*f*x - 2*e) + 2*a^(3/2)*e^(-4*f*x - 4*e) - 2*a^(3/2)*e^(-
6*f*x - 6*e) - a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2)*e^(-10*f*x - 10*e) - a^(3/2)) - 3*arctan(e^(-f*x - e))/a^(3/
2) + 9*log(e^(-f*x - e) + 1)/a^(3/2) - 9*log(e^(-f*x - e) - 1)/a^(3/2))/f - 1/8*((15*e^(-f*x - e) - 20*e^(-3*f
*x - 3*e) - 22*e^(-5*f*x - 5*e) - 20*e^(-7*f*x - 7*e) + 15*e^(-9*f*x - 9*e))/(a^(3/2)*e^(-2*f*x - 2*e) + 2*a^(
3/2)*e^(-4*f*x - 4*e) - 2*a^(3/2)*e^(-6*f*x - 6*e) - a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2)*e^(-10*f*x - 10*e) - a
^(3/2)) + 15*arctan(e^(-f*x - e))/a^(3/2))/f + 1/48*(45*e^(-f*x - e) - 52*e^(-3*f*x - 3*e) - 74*e^(-5*f*x - 5*
e) + 92*e^(-7*f*x - 7*e) + 21*e^(-9*f*x - 9*e))/((a^(3/2)*e^(-2*f*x - 2*e) + 2*a^(3/2)*e^(-4*f*x - 4*e) - 2*a^
(3/2)*e^(-6*f*x - 6*e) - a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2)*e^(-10*f*x - 10*e) - a^(3/2))*f) + 1/48*(21*e^(-f*
x - e) + 92*e^(-3*f*x - 3*e) - 74*e^(-5*f*x - 5*e) - 52*e^(-7*f*x - 7*e) + 45*e^(-9*f*x - 9*e))/((a^(3/2)*e^(-
2*f*x - 2*e) + 2*a^(3/2)*e^(-4*f*x - 4*e) - 2*a^(3/2)*e^(-6*f*x - 6*e) - a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2)*e^
(-10*f*x - 10*e) - a^(3/2))*f) + 11/8*arctan(e^(-f*x - e))/(a^(3/2)*f)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (34) = 68\).
time = 0.44, size = 612, normalized size = 16.11 \begin {gather*} -\frac {8 \, {\left (\cosh \left (f x + e\right )^{3} e^{\left (f x + e\right )} + 3 \, \cosh \left (f x + e\right )^{2} e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + 3 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3}\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^{2} f \cosh \left (f x + e\right )^{6} - 3 \, a^{2} f \cosh \left (f x + e\right )^{4} + {\left (a^{2} f e^{\left (2 \, f x + 2 \, e\right )} + a^{2} f\right )} \sinh \left (f x + e\right )^{6} + 6 \, {\left (a^{2} f \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + a^{2} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )^{5} + 3 \, a^{2} f \cosh \left (f x + e\right )^{2} + 3 \, {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f + {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{4} + 4 \, {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{3} - 3 \, a^{2} f \cosh \left (f x + e\right ) + {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{3} - 3 \, a^{2} f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{3} - a^{2} f + 3 \, {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{4} - 6 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f + {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{4} - 6 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{2} + {\left (a^{2} f \cosh \left (f x + e\right )^{6} - 3 \, a^{2} f \cosh \left (f x + e\right )^{4} + 3 \, a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )} + 6 \, {\left (a^{2} f \cosh \left (f x + e\right )^{5} - 2 \, a^{2} f \cosh \left (f x + e\right )^{3} + a^{2} f \cosh \left (f x + e\right ) + {\left (a^{2} f \cosh \left (f x + e\right )^{5} - 2 \, a^{2} f \cosh \left (f x + e\right )^{3} + a^{2} 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)^(3/2),x, algorithm="fricas")

[Out]

-8/3*(cosh(f*x + e)^3*e^(f*x + e) + 3*cosh(f*x + e)^2*e^(f*x + e)*sinh(f*x + e) + 3*cosh(f*x + e)*e^(f*x + e)*
sinh(f*x + e)^2 + e^(f*x + e)*sinh(f*x + e)^3)*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)^6 - 3*a^2*f*cosh(f*x + e)^4 + (a^2*f*e^(2*f*x + 2*e) + a^2*f)*sinh(f*x + e)^6 + 6*(a^2*f*
cosh(f*x + e)*e^(2*f*x + 2*e) + a^2*f*cosh(f*x + e))*sinh(f*x + e)^5 + 3*a^2*f*cosh(f*x + e)^2 + 3*(5*a^2*f*co
sh(f*x + e)^2 - a^2*f + (5*a^2*f*cosh(f*x + e)^2 - a^2*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^4 + 4*(5*a^2*f*cosh(f
*x + e)^3 - 3*a^2*f*cosh(f*x + e) + (5*a^2*f*cosh(f*x + e)^3 - 3*a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*
x + e)^3 - a^2*f + 3*(5*a^2*f*cosh(f*x + e)^4 - 6*a^2*f*cosh(f*x + e)^2 + a^2*f + (5*a^2*f*cosh(f*x + e)^4 - 6
*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)^6 - 3*a^2*f*cosh(f*x +
e)^4 + 3*a^2*f*cosh(f*x + e)^2 - a^2*f)*e^(2*f*x + 2*e) + 6*(a^2*f*cosh(f*x + e)^5 - 2*a^2*f*cosh(f*x + e)^3
+ a^2*f*cosh(f*x + e) + (a^2*f*cosh(f*x + e)^5 - 2*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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{4}{\left (e + f x \right )}}{\left (a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, 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)**(3/2),x)

[Out]

Integral(coth(e + f*x)**4/(a*(sinh(e + f*x)**2 + 1))**(3/2), 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)^(3/2),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 = 71, normalized size = 1.87 \begin {gather*} -\frac {16\,{\mathrm {e}}^{4\,e+4\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{3\,a^2\,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)^(3/2),x)

[Out]

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

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