3.5.99 \(\int \frac {\coth ^2(e+f x)}{(a+b \sinh ^2(e+f x))^{3/2}} \, dx\) [499]

Optimal. Leaf size=237 \[ \frac {\coth (e+f x)}{a f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 f}-\frac {2 E\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {2 \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{a^2 f} \]

[Out]

coth(f*x+e)/a/f/(a+b*sinh(f*x+e)^2)^(1/2)-2*coth(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a^2/f-2*(1/(1+sinh(f*x+e)^2)
)^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1/2),(1-b/a)^(1/2))*sech(f*x+e)*(a+b*
sinh(f*x+e)^2)^(1/2)/a^2/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x
+e)^2)^(1/2)*EllipticF(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1/2),(1-b/a)^(1/2))*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2
)/a^2/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+2*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/a^2/f

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Rubi [A]
time = 0.18, antiderivative size = 237, 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 = {3275, 480, 597, 545, 429, 506, 422} \begin {gather*} \frac {\text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {2 \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} E\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {2 \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 f}-\frac {2 \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 f}+\frac {\coth (e+f x)}{a f \sqrt {a+b \sinh ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Coth[e + f*x]/(a*f*Sqrt[a + b*Sinh[e + f*x]^2]) - (2*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a^2*f) - (2*E
llipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a^2*f*Sqrt[(Sech[e + f*x]
^2*(a + b*Sinh[e + f*x]^2))/a]) + (EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e +
f*x]^2])/(a^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (2*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e +
f*x])/(a^2*f)

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*
Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 480

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-(e*x)^
(m + 1))*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*e*n*(p + 1))), x] + Dist[1/(a*n*(p + 1)), Int[(e*x)^m*(a + b*x^
n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ
[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b,
c, d, e, m, n, p, q, x]

Rule 506

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[c + d*x^2])), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 545

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 3275

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_), x_Symbol] :> With[{ff = FreeF
actors[Sin[e + f*x], x]}, Dist[ff^(m + 1)*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])), Subst[Int[x^m*((a + b*ff^2*
x^2)^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2]
&&  !IntegerQ[p]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.60, size = 153, normalized size = 0.65 \begin {gather*} \frac {-2 (a-b+b \cosh (2 (e+f x))) \coth (e+f x)-2 i \sqrt {2} a \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+i \sqrt {2} a \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )}{a^2 f \sqrt {4 a-2 b+2 b \cosh (2 (e+f x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a - b + b*Cosh[2*(e + f*x)])*Coth[e + f*x] - (2*I)*Sqrt[2]*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*Elli
pticE[I*(e + f*x), b/a] + I*Sqrt[2]*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(a^
2*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])

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Maple [A]
time = 2.11, size = 218, normalized size = 0.92

method	result	size
default	\(\frac {-2 \sqrt {-\frac {b}{a}}\, b \left (\cosh ^{4}\left (f x +e \right )\right )+\left (-\sqrt {-\frac {b}{a}}\, a +2 \sqrt {-\frac {b}{a}}\, b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )+\sinh \left (f x +e \right ) \sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \left (a \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-2 b \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+2 b \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right )}{a^{2} \sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\)	\(218\)
risch	\(\text {Expression too large to display}\)	\(76390\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*(-1/a*b)^(1/2)*b*cosh(f*x+e)^4+(-(-1/a*b)^(1/2)*a+2*(-1/a*b)^(1/2)*b)*cosh(f*x+e)^2+sinh(f*x+e)*(b/a*cosh(
f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*(a*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))-2*b*Ellipti
cF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))+2*b*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))))/a^2/sinh(f*
x+e)/(-1/a*b)^(1/2)/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2436 vs. \(2 (251) = 502\).
time = 0.14, size = 2436, normalized size = 10.28 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(((2*a*b^2 - b^3)*cosh(f*x + e)^6 + 6*(2*a*b^2 - b^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (2*a*b^2 - b^3)*sinh(f
*x + e)^6 + (8*a^2*b - 10*a*b^2 + 3*b^3)*cosh(f*x + e)^4 + (8*a^2*b - 10*a*b^2 + 3*b^3 + 15*(2*a*b^2 - b^3)*co
sh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(5*(2*a*b^2 - b^3)*cosh(f*x + e)^3 + (8*a^2*b - 10*a*b^2 + 3*b^3)*cosh(f*x
+ e))*sinh(f*x + e)^3 - 2*a*b^2 + b^3 - (8*a^2*b - 10*a*b^2 + 3*b^3)*cosh(f*x + e)^2 + (15*(2*a*b^2 - b^3)*cos
h(f*x + e)^4 - 8*a^2*b + 10*a*b^2 - 3*b^3 + 6*(8*a^2*b - 10*a*b^2 + 3*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 +
2*(3*(2*a*b^2 - b^3)*cosh(f*x + e)^5 + 2*(8*a^2*b - 10*a*b^2 + 3*b^3)*cosh(f*x + e)^3 - (8*a^2*b - 10*a*b^2 +
3*b^3)*cosh(f*x + e))*sinh(f*x + e) - 2*(b^3*cosh(f*x + e)^6 + 6*b^3*cosh(f*x + e)*sinh(f*x + e)^5 + b^3*sinh(
f*x + e)^6 + (4*a*b^2 - 3*b^3)*cosh(f*x + e)^4 + (15*b^3*cosh(f*x + e)^2 + 4*a*b^2 - 3*b^3)*sinh(f*x + e)^4 +
4*(5*b^3*cosh(f*x + e)^3 + (4*a*b^2 - 3*b^3)*cosh(f*x + e))*sinh(f*x + e)^3 - b^3 - (4*a*b^2 - 3*b^3)*cosh(f*x
+ e)^2 + (15*b^3*cosh(f*x + e)^4 - 4*a*b^2 + 3*b^3 + 6*(4*a*b^2 - 3*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2
*(3*b^3*cosh(f*x + e)^5 + 2*(4*a*b^2 - 3*b^3)*cosh(f*x + e)^3 - (4*a*b^2 - 3*b^3)*cosh(f*x + e))*sinh(f*x + e)
)*sqrt((a^2 - a*b)/b^2))*sqrt(b)*sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqr
t((a^2 - a*b)/b^2) - 2*a + b)/b)*(cosh(f*x + e) + sinh(f*x + e))), (8*a^2 - 8*a*b + b^2 + 4*(2*a*b - b^2)*sqrt
((a^2 - a*b)/b^2))/b^2) - ((2*a^2*b - a*b^2)*cosh(f*x + e)^6 + 6*(2*a^2*b - a*b^2)*cosh(f*x + e)*sinh(f*x + e)
^5 + (2*a^2*b - a*b^2)*sinh(f*x + e)^6 + (8*a^3 - 10*a^2*b + 3*a*b^2)*cosh(f*x + e)^4 + (8*a^3 - 10*a^2*b + 3*
a*b^2 + 15*(2*a^2*b - a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(5*(2*a^2*b - a*b^2)*cosh(f*x + e)^3 + (8*a^
3 - 10*a^2*b + 3*a*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 - 2*a^2*b + a*b^2 - (8*a^3 - 10*a^2*b + 3*a*b^2)*cosh(f
*x + e)^2 + (15*(2*a^2*b - a*b^2)*cosh(f*x + e)^4 - 8*a^3 + 10*a^2*b - 3*a*b^2 + 6*(8*a^3 - 10*a^2*b + 3*a*b^2
)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(3*(2*a^2*b - a*b^2)*cosh(f*x + e)^5 + 2*(8*a^3 - 10*a^2*b + 3*a*b^2)*c
osh(f*x + e)^3 - (8*a^3 - 10*a^2*b + 3*a*b^2)*cosh(f*x + e))*sinh(f*x + e) + 2*((a*b^2 - 2*b^3)*cosh(f*x + e)^
6 + 6*(a*b^2 - 2*b^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (a*b^2 - 2*b^3)*sinh(f*x + e)^6 + (4*a^2*b - 11*a*b^2 +
6*b^3)*cosh(f*x + e)^4 + (4*a^2*b - 11*a*b^2 + 6*b^3 + 15*(a*b^2 - 2*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4
*(5*(a*b^2 - 2*b^3)*cosh(f*x + e)^3 + (4*a^2*b - 11*a*b^2 + 6*b^3)*cosh(f*x + e))*sinh(f*x + e)^3 - a*b^2 + 2*
b^3 - (4*a^2*b - 11*a*b^2 + 6*b^3)*cosh(f*x + e)^2 + (15*(a*b^2 - 2*b^3)*cosh(f*x + e)^4 - 4*a^2*b + 11*a*b^2
- 6*b^3 + 6*(4*a^2*b - 11*a*b^2 + 6*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(3*(a*b^2 - 2*b^3)*cosh(f*x + e)
^5 + 2*(4*a^2*b - 11*a*b^2 + 6*b^3)*cosh(f*x + e)^3 - (4*a^2*b - 11*a*b^2 + 6*b^3)*cosh(f*x + e))*sinh(f*x + e
))*sqrt((a^2 - a*b)/b^2))*sqrt(b)*sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sq
rt((a^2 - a*b)/b^2) - 2*a + b)/b)*(cosh(f*x + e) + sinh(f*x + e))), (8*a^2 - 8*a*b + b^2 + 4*(2*a*b - b^2)*sqr
t((a^2 - a*b)/b^2))/b^2) - sqrt(2)*(b^3*cosh(f*x + e)^5 + 5*b^3*cosh(f*x + e)*sinh(f*x + e)^4 + b^3*sinh(f*x +
e)^5 + (3*a*b^2 - 2*b^3)*cosh(f*x + e)^3 + (10*b^3*cosh(f*x + e)^2 + 3*a*b^2 - 2*b^3)*sinh(f*x + e)^3 + (10*b
^3*cosh(f*x + e)^3 + 3*(3*a*b^2 - 2*b^3)*cosh(f*x + e))*sinh(f*x + e)^2 - (a*b^2 - b^3)*cosh(f*x + e) + (5*b^3
*cosh(f*x + e)^4 - a*b^2 + b^3 + 3*(3*a*b^2 - 2*b^3)*cosh(f*x + e)^2)*sinh(f*x + e))*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^2*b^3*f
*cosh(f*x + e)^6 + 6*a^2*b^3*f*cosh(f*x + e)*sinh(f*x + e)^5 + a^2*b^3*f*sinh(f*x + e)^6 - a^2*b^3*f + (4*a^3*
b^2 - 3*a^2*b^3)*f*cosh(f*x + e)^4 + (15*a^2*b^3*f*cosh(f*x + e)^2 + (4*a^3*b^2 - 3*a^2*b^3)*f)*sinh(f*x + e)^
4 - (4*a^3*b^2 - 3*a^2*b^3)*f*cosh(f*x + e)^2 + 4*(5*a^2*b^3*f*cosh(f*x + e)^3 + (4*a^3*b^2 - 3*a^2*b^3)*f*cos
h(f*x + e))*sinh(f*x + e)^3 + (15*a^2*b^3*f*cosh(f*x + e)^4 + 6*(4*a^3*b^2 - 3*a^2*b^3)*f*cosh(f*x + e)^2 - (4
*a^3*b^2 - 3*a^2*b^3)*f)*sinh(f*x + e)^2 + 2*(3*a^2*b^3*f*cosh(f*x + e)^5 + 2*(4*a^3*b^2 - 3*a^2*b^3)*f*cosh(f
*x + e)^3 - (4*a^3*b^2 - 3*a^2*b^3)*f*cosh(f*x + 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 ^{2}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(e + f*x)**2/(a + b*sinh(e + f*x)**2)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Error: Bad Argume
nt Type

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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