3.435 \(\int \coth ^4(e+f x) \sqrt{a+a \sinh ^2(e+f x)} \, dx\)

Optimal. Leaf size=91 \[ \frac{\tanh (e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}-\frac{\text{csch}^3(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{3 f}-\frac{2 \text{csch}(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{f} \]

[Out]

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

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Rubi [A]  time = 0.121694, antiderivative size = 91, 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, 2590, 270} \[ \frac{\tanh (e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}-\frac{\text{csch}^3(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{3 f}-\frac{2 \text{csch}(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2590

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0729251, size = 47, normalized size = 0.52 \[ -\frac{\tanh (e+f x) \left (\text{csch}^4(e+f x)+6 \text{csch}^2(e+f x)-3\right ) \sqrt{a \cosh ^2(e+f x)}}{3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[a*Cosh[e + f*x]^2]*(-3 + 6*Csch[e + f*x]^2 + Csch[e + f*x]^4)*Tanh[e + f*x])/(3*f)

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Maple [A]  time = 0.092, size = 55, normalized size = 0.6 \begin{align*}{\frac{\cosh \left ( fx+e \right ) a \left ( 3\, \left ( \sinh \left ( fx+e \right ) \right ) ^{4}-6\, \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)*a*(3*sinh(f*x+e)^4-6*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 = 1.78277, size = 657, normalized size = 7.22 \begin{align*} -\frac{3 \, \sqrt{a} \log \left (e^{\left (-f x - e\right )} + 1\right ) - 3 \, \sqrt{a} \log \left (e^{\left (-f x - e\right )} - 1\right ) - \frac{2 \,{\left (9 \, \sqrt{a} e^{\left (-f x - e\right )} - 8 \, \sqrt{a} e^{\left (-3 \, f x - 3 \, e\right )} + 3 \, \sqrt{a} e^{\left (-5 \, f x - 5 \, e\right )}\right )}}{3 \, e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, e^{\left (-4 \, f x - 4 \, e\right )} + e^{\left (-6 \, f x - 6 \, e\right )} - 1}}{12 \, f} + \frac{3 \, \sqrt{a} \log \left (e^{\left (-f x - e\right )} + 1\right ) - 3 \, \sqrt{a} \log \left (e^{\left (-f x - e\right )} - 1\right ) + \frac{2 \,{\left (3 \, \sqrt{a} e^{\left (-f x - e\right )} - 8 \, \sqrt{a} e^{\left (-3 \, f x - 3 \, e\right )} + 9 \, \sqrt{a} e^{\left (-5 \, f x - 5 \, e\right )}\right )}}{3 \, e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, e^{\left (-4 \, f x - 4 \, e\right )} + e^{\left (-6 \, f x - 6 \, e\right )} - 1}}{12 \, f} + \frac{\sqrt{a} e^{\left (-3 \, f x - 3 \, e\right )}}{f{\left (3 \, e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, e^{\left (-4 \, f x - 4 \, e\right )} + e^{\left (-6 \, f x - 6 \, e\right )} - 1\right )}} - \frac{33 \, \sqrt{a} e^{\left (-2 \, f x - 2 \, e\right )} - 40 \, \sqrt{a} e^{\left (-4 \, f x - 4 \, e\right )} + 15 \, \sqrt{a} e^{\left (-6 \, f x - 6 \, e\right )} - 6 \, \sqrt{a}}{12 \, f{\left (e^{\left (-f x - e\right )} - 3 \, e^{\left (-3 \, f x - 3 \, e\right )} + 3 \, e^{\left (-5 \, f x - 5 \, e\right )} - e^{\left (-7 \, f x - 7 \, e\right )}\right )}} + \frac{15 \, \sqrt{a} e^{\left (-f x - e\right )} - 40 \, \sqrt{a} e^{\left (-3 \, f x - 3 \, e\right )} + 33 \, \sqrt{a} e^{\left (-5 \, f x - 5 \, e\right )} - 6 \, \sqrt{a} e^{\left (-7 \, f x - 7 \, e\right )}}{12 \, f{\left (3 \, e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, e^{\left (-4 \, f x - 4 \, e\right )} + e^{\left (-6 \, f x - 6 \, e\right )} - 1\right )}} \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*(3*sqrt(a)*log(e^(-f*x - e) + 1) - 3*sqrt(a)*log(e^(-f*x - e) - 1) - 2*(9*sqrt(a)*e^(-f*x - e) - 8*sqrt(
a)*e^(-3*f*x - 3*e) + 3*sqrt(a)*e^(-5*f*x - 5*e))/(3*e^(-2*f*x - 2*e) - 3*e^(-4*f*x - 4*e) + e^(-6*f*x - 6*e)
- 1))/f + 1/12*(3*sqrt(a)*log(e^(-f*x - e) + 1) - 3*sqrt(a)*log(e^(-f*x - e) - 1) + 2*(3*sqrt(a)*e^(-f*x - e)
- 8*sqrt(a)*e^(-3*f*x - 3*e) + 9*sqrt(a)*e^(-5*f*x - 5*e))/(3*e^(-2*f*x - 2*e) - 3*e^(-4*f*x - 4*e) + e^(-6*f*
x - 6*e) - 1))/f + sqrt(a)*e^(-3*f*x - 3*e)/(f*(3*e^(-2*f*x - 2*e) - 3*e^(-4*f*x - 4*e) + e^(-6*f*x - 6*e) - 1
)) - 1/12*(33*sqrt(a)*e^(-2*f*x - 2*e) - 40*sqrt(a)*e^(-4*f*x - 4*e) + 15*sqrt(a)*e^(-6*f*x - 6*e) - 6*sqrt(a)
)/(f*(e^(-f*x - e) - 3*e^(-3*f*x - 3*e) + 3*e^(-5*f*x - 5*e) - e^(-7*f*x - 7*e))) + 1/12*(15*sqrt(a)*e^(-f*x -
e) - 40*sqrt(a)*e^(-3*f*x - 3*e) + 33*sqrt(a)*e^(-5*f*x - 5*e) - 6*sqrt(a)*e^(-7*f*x - 7*e))/(f*(3*e^(-2*f*x
- 2*e) - 3*e^(-4*f*x - 4*e) + e^(-6*f*x - 6*e) - 1))

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Fricas [B]  time = 1.97043, size = 2361, normalized size = 25.95 \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]

1/6*(24*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^7 + 3*e^(f*x + e)*sinh(f*x + e)^8 + 12*(7*cosh(f*x + e)^2 - 3)
*e^(f*x + e)*sinh(f*x + e)^6 + 24*(7*cosh(f*x + e)^3 - 9*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^5 + 10*(21*c
osh(f*x + e)^4 - 54*cosh(f*x + e)^2 + 5)*e^(f*x + e)*sinh(f*x + e)^4 + 8*(21*cosh(f*x + e)^5 - 90*cosh(f*x + e
)^3 + 25*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^3 + 12*(7*cosh(f*x + e)^6 - 45*cosh(f*x + e)^4 + 25*cosh(f*x
+ e)^2 - 3)*e^(f*x + e)*sinh(f*x + e)^2 + 8*(3*cosh(f*x + e)^7 - 27*cosh(f*x + e)^5 + 25*cosh(f*x + e)^3 - 9*
cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e) + (3*cosh(f*x + e)^8 - 36*cosh(f*x + e)^6 + 50*cosh(f*x + e)^4 - 36*c
osh(f*x + e)^2 + 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)/(f*cosh(f*x +
e)^7 + (f*e^(2*f*x + 2*e) + f)*sinh(f*x + e)^7 + 7*(f*cosh(f*x + e)*e^(2*f*x + 2*e) + f*cosh(f*x + e))*sinh(f*
x + e)^6 - 3*f*cosh(f*x + e)^5 + 3*(7*f*cosh(f*x + e)^2 + (7*f*cosh(f*x + e)^2 - f)*e^(2*f*x + 2*e) - f)*sinh(
f*x + e)^5 + 5*(7*f*cosh(f*x + e)^3 - 3*f*cosh(f*x + e) + (7*f*cosh(f*x + e)^3 - 3*f*cosh(f*x + e))*e^(2*f*x +
2*e))*sinh(f*x + e)^4 + 3*f*cosh(f*x + e)^3 + (35*f*cosh(f*x + e)^4 - 30*f*cosh(f*x + e)^2 + (35*f*cosh(f*x +
e)^4 - 30*f*cosh(f*x + e)^2 + 3*f)*e^(2*f*x + 2*e) + 3*f)*sinh(f*x + e)^3 + 3*(7*f*cosh(f*x + e)^5 - 10*f*cos
h(f*x + e)^3 + 3*f*cosh(f*x + e) + (7*f*cosh(f*x + e)^5 - 10*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*cosh(f*x + e)^7 - 3*f*cosh(f*x + e)^5 + 3*f*cosh(f*x + e)^3 - f*
cosh(f*x + e))*e^(2*f*x + 2*e) + (7*f*cosh(f*x + e)^6 - 15*f*cosh(f*x + e)^4 + 9*f*cosh(f*x + e)^2 + (7*f*cosh
(f*x + e)^6 - 15*f*cosh(f*x + e)^4 + 9*f*cosh(f*x + e)^2 - f)*e^(2*f*x + 2*e) - f)*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 = 1.27173, size = 108, normalized size = 1.19 \begin{align*} -\frac{\sqrt{a}{\left (\frac{8 \,{\left (3 \, e^{\left (5 \, f x + 5 \, e\right )} - 4 \, e^{\left (3 \, f x + 3 \, e\right )} + 3 \, e^{\left (f x + e\right )}\right )}}{{\left (e^{\left (2 \, f x + 2 \, e\right )} - 1\right )}^{3}} - 3 \, e^{\left (f x + e\right )} + 3 \, e^{\left (-f x - e\right )}\right )}}{6 \, f} \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]

-1/6*sqrt(a)*(8*(3*e^(5*f*x + 5*e) - 4*e^(3*f*x + 3*e) + 3*e^(f*x + e))/(e^(2*f*x + 2*e) - 1)^3 - 3*e^(f*x + e
) + 3*e^(-f*x - e))/f