3.5.30 \(\int \coth ^3(e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx\) [430]

Optimal. Leaf size=87 \[ -\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{2 f}+\frac {3 \sqrt {a \cosh ^2(e+f x)}}{2 f}-\frac {\left (a \cosh ^2(e+f x)\right )^{3/2} \text {csch}^2(e+f x)}{2 a f} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3255, 3284, 16, 43, 52, 65, 212} \begin {gather*} \frac {3 \sqrt {a \cosh ^2(e+f x)}}{2 f}-\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{2 f}-\frac {\text {csch}^2(e+f x) \left (a \cosh ^2(e+f x)\right )^{3/2}}{2 a f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[e + f*x]^3*Sqrt[a + a*Sinh[e + f*x]^2],x]

[Out]

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

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 3284

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFact
ors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[x^((m - 1)/2)*((b*ff^(n/2)*x^(n/2))^p/(1 - ff*x)
^((m + 1)/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2
]

Rubi steps

\begin {align*} \int \coth ^3(e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx &=\int \sqrt {a \cosh ^2(e+f x)} \coth ^3(e+f x) \, dx\\ &=\frac {\text {Subst}\left (\int \frac {x \sqrt {a x}}{(1-x)^2} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac {\text {Subst}\left (\int \frac {(a x)^{3/2}}{(1-x)^2} \, dx,x,\cosh ^2(e+f x)\right )}{2 a f}\\ &=-\frac {\left (a \cosh ^2(e+f x)\right )^{3/2} \text {csch}^2(e+f x)}{2 a f}-\frac {3 \text {Subst}\left (\int \frac {\sqrt {a x}}{1-x} \, dx,x,\cosh ^2(e+f x)\right )}{4 f}\\ &=\frac {3 \sqrt {a \cosh ^2(e+f x)}}{2 f}-\frac {\left (a \cosh ^2(e+f x)\right )^{3/2} \text {csch}^2(e+f x)}{2 a f}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\cosh ^2(e+f x)\right )}{4 f}\\ &=\frac {3 \sqrt {a \cosh ^2(e+f x)}}{2 f}-\frac {\left (a \cosh ^2(e+f x)\right )^{3/2} \text {csch}^2(e+f x)}{2 a f}-\frac {3 \text {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \cosh ^2(e+f x)}\right )}{2 f}\\ &=-\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{2 f}+\frac {3 \sqrt {a \cosh ^2(e+f x)}}{2 f}-\frac {\left (a \cosh ^2(e+f x)\right )^{3/2} \text {csch}^2(e+f x)}{2 a f}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 77, normalized size = 0.89 \begin {gather*} \frac {\sqrt {a \cosh ^2(e+f x)} \left (8 \cosh (e+f x)-\text {csch}^2\left (\frac {1}{2} (e+f x)\right )+12 \log \left (\tanh \left (\frac {1}{2} (e+f x)\right )\right )-\text {sech}^2\left (\frac {1}{2} (e+f x)\right )\right ) \text {sech}(e+f x)}{8 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[e + f*x]^3*Sqrt[a + a*Sinh[e + f*x]^2],x]

[Out]

(Sqrt[a*Cosh[e + f*x]^2]*(8*Cosh[e + f*x] - Csch[(e + f*x)/2]^2 + 12*Log[Tanh[(e + f*x)/2]] - Sech[(e + f*x)/2
]^2)*Sech[e + f*x])/(8*f)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.10, size = 54, normalized size = 0.62

method	result	size
default	\(\frac {\mathit {`\,int/indef0`\,}\left (\frac {a \left (\cosh ^{4}\left (f x +e \right )\right )}{\sinh \left (f x +e \right ) \left (\cosh ^{2}\left (f x +e \right )-1\right ) \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\)	\(54\)
risch	\(\frac {\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{2 f x +2 e}}{2 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}+\frac {\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}}{2 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}-\frac {\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{2 f x +2 e}}{\left ({\mathrm e}^{2 f x +2 e}-1\right )^{2} f}+\frac {3 \ln \left ({\mathrm e}^{f x}-{\mathrm e}^{-e}\right ) \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{f x +e}}{2 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}-\frac {3 \ln \left ({\mathrm e}^{f x}+{\mathrm e}^{-e}\right ) \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{f x +e}}{2 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}\)	\(274\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(f*x+e)^3*(a+a*sinh(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

`int/indef0`(a*cosh(f*x+e)^4/sinh(f*x+e)/(cosh(f*x+e)^2-1)/(a*cosh(f*x+e)^2)^(1/2),sinh(f*x+e))/f

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Maxima [A]
time = 0.51, size = 134, normalized size = 1.54 \begin {gather*} -\frac {3 \, \sqrt {a} \log \left (e^{\left (-f x - e\right )} + 1\right )}{2 \, f} + \frac {3 \, \sqrt {a} \log \left (e^{\left (-f x - e\right )} - 1\right )}{2 \, f} - \frac {3 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + 3 \, \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} - \sqrt {a} e^{\left (-6 \, f x - 6 \, e\right )} - \sqrt {a}}{2 \, f {\left (e^{\left (-f x - e\right )} - 2 \, e^{\left (-3 \, f x - 3 \, e\right )} + e^{\left (-5 \, f x - 5 \, e\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^3*(a+a*sinh(f*x+e)^2)^(1/2),x, algorithm="maxima")

[Out]

-3/2*sqrt(a)*log(e^(-f*x - e) + 1)/f + 3/2*sqrt(a)*log(e^(-f*x - e) - 1)/f - 1/2*(3*sqrt(a)*e^(-2*f*x - 2*e) +
3*sqrt(a)*e^(-4*f*x - 4*e) - sqrt(a)*e^(-6*f*x - 6*e) - sqrt(a))/(f*(e^(-f*x - e) - 2*e^(-3*f*x - 3*e) + e^(-
5*f*x - 5*e)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 764 vs. \(2 (71) = 142\).
time = 0.52, size = 764, normalized size = 8.78 \begin {gather*} \frac {{\left (6 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{5} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{6} + 3 \, {\left (5 \, \cosh \left (f x + e\right )^{2} - 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{4} + 4 \, {\left (5 \, \cosh \left (f x + e\right )^{3} - 3 \, \cosh \left (f x + e\right )\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3} + 3 \, {\left (5 \, \cosh \left (f x + e\right )^{4} - 6 \, \cosh \left (f x + e\right )^{2} - 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + 6 \, {\left (\cosh \left (f x + e\right )^{5} - 2 \, \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 ) + {\left (\cosh \left (f x + e\right )^{6} - 3 \, \cosh \left (f x + e\right )^{4} - 3 \, \cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )} + 3 \, {\left (5 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{4} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{5} + 2 \, {\left (5 \, \cosh \left (f x + e\right )^{2} - 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3} + 2 \, {\left (5 \, \cosh \left (f x + e\right )^{3} - 3 \, \cosh \left (f x + e\right )\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + {\left (5 \, \cosh \left (f x + e\right )^{4} - 6 \, \cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + {\left (\cosh \left (f x + e\right )^{5} - 2 \, \cosh \left (f x + e\right )^{3} + \cosh \left (f x + e\right )\right )} e^{\left (f x + e\right )}\right )} \log \left (\frac {\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) - 1}{\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1}\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 )}}{2 \, {\left (f \cosh \left (f x + e\right )^{5} + {\left (f e^{\left (2 \, f x + 2 \, e\right )} + f\right )} \sinh \left (f x + e\right )^{5} + 5 \, {\left (f \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )^{4} - 2 \, f \cosh \left (f x + e\right )^{3} + 2 \, {\left (5 \, f \cosh \left (f x + e\right )^{2} + {\left (5 \, f \cosh \left (f x + e\right )^{2} - f\right )} e^{\left (2 \, f x + 2 \, e\right )} - f\right )} \sinh \left (f x + e\right )^{3} + 2 \, {\left (5 \, f \cosh \left (f x + e\right )^{3} - 3 \, f \cosh \left (f x + e\right ) + {\left (5 \, f \cosh \left (f x + e\right )^{3} - 3 \, f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{2} + f \cosh \left (f x + e\right ) + {\left (f \cosh \left (f x + e\right )^{5} - 2 \, f \cosh \left (f x + e\right )^{3} + f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )} + {\left (5 \, f \cosh \left (f x + e\right )^{4} - 6 \, f \cosh \left (f x + e\right )^{2} + {\left (5 \, f \cosh \left (f x + e\right )^{4} - 6 \, f \cosh \left (f x + e\right )^{2} + f\right )} e^{\left (2 \, f x + 2 \, e\right )} + f\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)^3*(a+a*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")

[Out]

1/2*(6*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^5 + e^(f*x + e)*sinh(f*x + e)^6 + 3*(5*cosh(f*x + e)^2 - 1)*e^(
f*x + e)*sinh(f*x + e)^4 + 4*(5*cosh(f*x + e)^3 - 3*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^3 + 3*(5*cosh(f*x
+ e)^4 - 6*cosh(f*x + e)^2 - 1)*e^(f*x + e)*sinh(f*x + e)^2 + 6*(cosh(f*x + e)^5 - 2*cosh(f*x + e)^3 - cosh(f
*x + e))*e^(f*x + e)*sinh(f*x + e) + (cosh(f*x + e)^6 - 3*cosh(f*x + e)^4 - 3*cosh(f*x + e)^2 + 1)*e^(f*x + e)
+ 3*(5*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^4 + e^(f*x + e)*sinh(f*x + e)^5 + 2*(5*cosh(f*x + e)^2 - 1)*e^
(f*x + e)*sinh(f*x + e)^3 + 2*(5*cosh(f*x + e)^3 - 3*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^2 + (5*cosh(f*x
+ e)^4 - 6*cosh(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e) + (cosh(f*x + e)^5 - 2*cosh(f*x + e)^3 + cosh(f*x +
e))*e^(f*x + e))*log((cosh(f*x + e) + sinh(f*x + e) - 1)/(cosh(f*x + e) + sinh(f*x + e) + 1)))*sqrt(a*e^(4*f*x
+ 4*e) + 2*a*e^(2*f*x + 2*e) + a)*e^(-f*x - e)/(f*cosh(f*x + e)^5 + (f*e^(2*f*x + 2*e) + f)*sinh(f*x + e)^5 +
5*(f*cosh(f*x + e)*e^(2*f*x + 2*e) + f*cosh(f*x + e))*sinh(f*x + e)^4 - 2*f*cosh(f*x + e)^3 + 2*(5*f*cosh(f*x
+ e)^2 + (5*f*cosh(f*x + e)^2 - f)*e^(2*f*x + 2*e) - f)*sinh(f*x + e)^3 + 2*(5*f*cosh(f*x + e)^3 - 3*f*cosh(f
*x + e) + (5*f*cosh(f*x + e)^3 - 3*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e)^2 + f*cosh(f*x + e) + (f*co
sh(f*x + e)^5 - 2*f*cosh(f*x + e)^3 + f*cosh(f*x + e))*e^(2*f*x + 2*e) + (5*f*cosh(f*x + e)^4 - 6*f*cosh(f*x +
e)^2 + (5*f*cosh(f*x + e)^4 - 6*f*cosh(f*x + e)^2 + f)*e^(2*f*x + 2*e) + f)*sinh(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )} \coth ^{3}{\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)**3*(a+a*sinh(f*x+e)**2)**(1/2),x)

[Out]

Integral(sqrt(a*(sinh(e + f*x)**2 + 1))*coth(e + f*x)**3, x)

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Giac [A]
time = 0.43, size = 108, normalized size = 1.24 \begin {gather*} -\frac {\sqrt {a} {\left (\frac {4 \, {\left (e^{\left (f x + e\right )} + e^{\left (-f x - e\right )}\right )}}{{\left (e^{\left (f x + e\right )} + e^{\left (-f x - e\right )}\right )}^{2} - 4} - 2 \, e^{\left (f x + e\right )} - 2 \, e^{\left (-f x - e\right )} + 3 \, \log \left (e^{\left (f x + e\right )} + e^{\left (-f x - e\right )} + 2\right ) - 3 \, \log \left (e^{\left (f x + e\right )} + e^{\left (-f x - e\right )} - 2\right )\right )}}{4 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^3*(a+a*sinh(f*x+e)^2)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {coth}\left (e+f\,x\right )}^3\,\sqrt {a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(e + f*x)^3*(a + a*sinh(e + f*x)^2)^(1/2),x)

[Out]

int(coth(e + f*x)^3*(a + a*sinh(e + f*x)^2)^(1/2), x)

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