∫ coth3(e + fx)∘ ____________ a + b sinh2(e + fx) dx [461]

Optimal. Leaf size=106 \[ -\frac {(2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{2 \sqrt {a} f}+\frac {(2 a+b) \sqrt {a+b \sinh ^2(e+f x)}}{2 a f}-\frac {\text {csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{2 a f} \]

-1/2*csch(f*x+e)^2*(a+b*sinh(f*x+e)^2)^(3/2)/a/f-1/2*(2*a+b)*arctanh((a+b*sinh(f*x+e)^2)^(1/2)/a^(1/2))/f/a^(1

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3273, 79, 52, 65, 214} \begin {gather*} \frac {(2 a+b) \sqrt {a+b \sinh ^2(e+f x)}}{2 a f}-\frac {(2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{2 \sqrt {a} f}-\frac {\text {csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{2 a f} \end {gather*}

-1/2*((2*a + b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(Sqrt[a]*f) + ((2*a + b)*Sqrt[a + b*Sinh[e + f*x

]^2])/(2*a*f) - (Csch[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2))/(2*a*f)

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]

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,

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f

))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1

) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,

f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || L

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

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = Free

Factors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[x^((m - 1)/2)*((a + b*ff*x)^p/(1 - ff*x)^((m

+ 1)/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.44, size = 69, normalized size = 0.65 \begin {gather*} -\frac {\frac {(2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\left (-2+\text {csch}^2(e+f x)\right ) \sqrt {a+b \sinh ^2(e+f x)}}{2 f} \end {gather*}

-1/2*(((2*a + b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/Sqrt[a] + (-2 + Csch[e + f*x]^2)*Sqrt[a + b*Sin

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.23, size = 58, normalized size = 0.55


method	result	size

default	\(\frac {\mathit {`\,int/indef0`\,}\left (\frac {b \sinh \left (f x +e \right )+\frac {a +b}{\sinh \left (f x +e \right )}+\frac {a}{\sinh \left (f x +e \right )^{3}}}{\sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\)	\(58\)

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

`int/indef0`((b*sinh(f*x+e)+(a+b)/sinh(f*x+e)+a/sinh(f*x+e)^3)/(a+b*sinh(f*x+e)^2)^(1/2),sinh(f*x+e))/f

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 621 vs. \(2 (90) = 180\).
time = 0.69, size = 1445, normalized size = 13.63 \begin {gather*} \text {Too large to display} \end {gather*}

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

[1/4*(((2*a + b)*cosh(f*x + e)^5 + 5*(2*a + b)*cosh(f*x + e)*sinh(f*x + e)^4 + (2*a + b)*sinh(f*x + e)^5 - 2*(

2*a + b)*cosh(f*x + e)^3 + 2*(5*(2*a + b)*cosh(f*x + e)^2 - 2*a - b)*sinh(f*x + e)^3 + 2*(5*(2*a + b)*cosh(f*x

+ e)^3 - 3*(2*a + b)*cosh(f*x + e))*sinh(f*x + e)^2 + (2*a + b)*cosh(f*x + e) + (5*(2*a + b)*cosh(f*x + e)^4

- 6*(2*a + b)*cosh(f*x + e)^2 + 2*a + b)*sinh(f*x + e))*sqrt(a)*log((b*cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*sin

h(f*x + e)^3 + b*sinh(f*x + e)^4 + 2*(4*a - b)*cosh(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2 + 4*a - b)*sinh(f*x +

e)^2 - 4*sqrt(2)*sqrt(a)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x

+ e)*sinh(f*x + e) + sinh(f*x + e)^2))*(cosh(f*x + e) + sinh(f*x + e)) + 4*(b*cosh(f*x + e)^3 + (4*a - b)*cosh

(f*x + e))*sinh(f*x + e) + b)/(cosh(f*x + e)^4 + 4*cosh(f*x + e)*sinh(f*x + e)^3 + sinh(f*x + e)^4 + 2*(3*cosh

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

+ 2*sqrt(2)*(a*cosh(f*x + e)^4 + 4*a*cosh(f*x + e)*sinh(f*x + e)^3 + a*sinh(f*x + e)^4 - 4*a*cosh(f*x + e)^2

+ 2*(3*a*cosh(f*x + e)^2 - 2*a)*sinh(f*x + e)^2 + 4*(a*cosh(f*x + e)^3 - 2*a*cosh(f*x + e))*sinh(f*x + e) + a)

*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sin

h(f*x + e)^2)))/(a*f*cosh(f*x + e)^5 + 5*a*f*cosh(f*x + e)*sinh(f*x + e)^4 + a*f*sinh(f*x + e)^5 - 2*a*f*cosh(

f*x + e)^3 + 2*(5*a*f*cosh(f*x + e)^2 - a*f)*sinh(f*x + e)^3 + a*f*cosh(f*x + e) + 2*(5*a*f*cosh(f*x + e)^3 -

3*a*f*cosh(f*x + e))*sinh(f*x + e)^2 + (5*a*f*cosh(f*x + e)^4 - 6*a*f*cosh(f*x + e)^2 + a*f)*sinh(f*x + e)), 1

/2*(((2*a + b)*cosh(f*x + e)^5 + 5*(2*a + b)*cosh(f*x + e)*sinh(f*x + e)^4 + (2*a + b)*sinh(f*x + e)^5 - 2*(2*

a + b)*cosh(f*x + e)^3 + 2*(5*(2*a + b)*cosh(f*x + e)^2 - 2*a - b)*sinh(f*x + e)^3 + 2*(5*(2*a + b)*cosh(f*x +

e)^3 - 3*(2*a + b)*cosh(f*x + e))*sinh(f*x + e)^2 + (2*a + b)*cosh(f*x + e) + (5*(2*a + b)*cosh(f*x + e)^4 -

6*(2*a + b)*cosh(f*x + e)^2 + 2*a + b)*sinh(f*x + e))*sqrt(-a)*arctan(1/2*sqrt(2)*sqrt(-a)*sqrt((b*cosh(f*x +

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

sh(f*x + e) + a*sinh(f*x + e))) + sqrt(2)*(a*cosh(f*x + e)^4 + 4*a*cosh(f*x + e)*sinh(f*x + e)^3 + a*sinh(f*x

+ e)^4 - 4*a*cosh(f*x + e)^2 + 2*(3*a*cosh(f*x + e)^2 - 2*a)*sinh(f*x + e)^2 + 4*(a*cosh(f*x + e)^3 - 2*a*cosh

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

(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/(a*f*cosh(f*x + e)^5 + 5*a*f*cosh(f*x + e)*sinh(f*x + e)^4 + a*f*

sinh(f*x + e)^5 - 2*a*f*cosh(f*x + e)^3 + 2*(5*a*f*cosh(f*x + e)^2 - a*f)*sinh(f*x + e)^3 + a*f*cosh(f*x + e)

+ 2*(5*a*f*cosh(f*x + e)^3 - 3*a*f*cosh(f*x + e))*sinh(f*x + e)^2 + (5*a*f*cosh(f*x + e)^4 - 6*a*f*cosh(f*x +

________________________________________________________________________________________

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

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP

UT:Unable to divide, perhaps due to rounding error%%%{128,[6,12,6]%%%}+%%%{%%%{-384,[1]%%%},[6,12,5]%%%}+%%%{%

________________________________________________________________________________________

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

________________________________________________________________________________________

3.5.61 \(\int \coth ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx\) [461]