3.5.34 \(\int \coth ^2(e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx\) [434]

Optimal. Leaf size=56 \[ -\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}(e+f x) \text {sech}(e+f x)}{f}+\frac {\sqrt {a \cosh ^2(e+f x)} \tanh (e+f x)}{f} \]

[Out]

-csch(f*x+e)*sech(f*x+e)*(a*cosh(f*x+e)^2)^(1/2)/f+(a*cosh(f*x+e)^2)^(1/2)*tanh(f*x+e)/f

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3255, 3286, 2670, 14} \begin {gather*} \frac {\tanh (e+f x) \sqrt {a \cosh ^2(e+f x)}}{f}-\frac {\text {csch}(e+f x) \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)}}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2670

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

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 35, normalized size = 0.62 \begin {gather*} -\frac {\sqrt {a \cosh ^2(e+f x)} \left (-1+\text {csch}^2(e+f x)\right ) \tanh (e+f x)}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.93, size = 42, normalized size = 0.75

method	result	size
default	\(\frac {\cosh \left (f x +e \right ) a \left (\sinh ^{2}\left (f x +e \right )-1\right )}{\sinh \left (f x +e \right ) \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}\, f}\)	\(42\)
risch	\(\frac {\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{2 f x +2 e}}{2 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}-\frac {\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}}{2 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}-\frac {2 \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{2 f x +2 e}}{\left ({\mathrm e}^{2 f x +2 e}-1\right ) f \left ({\mathrm e}^{2 f x +2 e}+1\right )}\)	\(165\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cosh(f*x+e)*a*(sinh(f*x+e)^2-1)/sinh(f*x+e)/(a*cosh(f*x+e)^2)^(1/2)/f

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (57) = 114\).
time = 0.50, size = 133, normalized size = 2.38 \begin {gather*} \frac {\sqrt {a} e^{\left (-f x - e\right )}}{f {\left (e^{\left (-2 \, f x - 2 \, e\right )} - 1\right )}} - \frac {2 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} - \sqrt {a}}{2 \, f {\left (e^{\left (-f x - e\right )} - e^{\left (-3 \, f x - 3 \, e\right )}\right )}} + \frac {2 \, \sqrt {a} e^{\left (-f x - e\right )} - \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )}}{2 \, f {\left (e^{\left (-2 \, f x - 2 \, e\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (52) = 104\).
time = 0.47, size = 317, normalized size = 5.66 \begin {gather*} \frac {{\left (4 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{4} + 6 \, {\left (\cosh \left (f x + e\right )^{2} - 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (\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 ) + {\left (\cosh \left (f x + e\right )^{4} - 6 \, \cosh \left (f x + e\right )^{2} + 1\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 )}}{2 \, {\left (f \cosh \left (f x + e\right )^{3} + {\left (f e^{\left (2 \, f x + 2 \, e\right )} + f\right )} \sinh \left (f x + e\right )^{3} + 3 \, {\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 )^{2} - f \cosh \left (f x + e\right ) + {\left (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 (3 \, f \cosh \left (f x + e\right )^{2} + {\left (3 \, 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)^2*(a+a*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")

[Out]

1/2*(4*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^3 + e^(f*x + e)*sinh(f*x + e)^4 + 6*(cosh(f*x + e)^2 - 1)*e^(f*
x + e)*sinh(f*x + e)^2 + 4*(cosh(f*x + e)^3 - 3*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e) + (cosh(f*x + e)^4 -
6*cosh(f*x + e)^2 + 1)*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)^3 + (f*e^(2*f*x + 2*e) + f)*sinh(f*x + e)^3 + 3*(f*cosh(f*x + e)*e^(2*f*x + 2*e) + f*cosh(f*x + e))*sinh
(f*x + e)^2 - f*cosh(f*x + e) + (f*cosh(f*x + e)^3 - f*cosh(f*x + e))*e^(2*f*x + 2*e) + (3*f*cosh(f*x + e)^2 +
(3*f*cosh(f*x + e)^2 - f)*e^(2*f*x + 2*e) - f)*sinh(f*x + e))

________________________________________________________________________________________

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 ^{2}{\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.43, size = 48, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {a} {\left (\frac {4}{e^{\left (f x + e\right )} - e^{\left (-f x - e\right )}} - e^{\left (f x + e\right )} + e^{\left (-f x - e\right )}\right )}}{2 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.93, size = 67, normalized size = 1.20 \begin {gather*} \frac {\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}\,\left ({\mathrm {e}}^{4\,e+4\,f\,x}-6\,{\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}{f\,\left ({\mathrm {e}}^{4\,e+4\,f\,x}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2)*(exp(4*e + 4*f*x) - 6*exp(2*e + 2*f*x) + 1))/(f*(exp(4*e
+ 4*f*x) - 1))

________________________________________________________________________________________