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

Optimal. Leaf size=270 \[ \frac {(7 a+b) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {\coth ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {(3 a+b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}+\frac {(3 a+5 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(7 a+b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \]

[Out]

-1/3*(3*a+b)*coth(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a/f-1/3*coth(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(1/2)/f-1/3*(7*a+
b)*(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/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+1/3*(3*a+5*b)*(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))*s
ech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+1/3*(7*a+b)*(a+b*sinh(f*x
+e)^2)^(1/2)*tanh(f*x+e)/a/f

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Rubi [A]  time = 0.30, antiderivative size = 270, 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 = {3196, 473, 580, 531, 418, 492, 411} \[ \frac {(7 a+b) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {\coth ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {(3 a+b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}+\frac {(3 a+5 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(7 a+b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3*a + b)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*f) - (Coth[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2])
/(3*f) - ((7*a + b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*
f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a + 5*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*
Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((7*a +
b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*a*f)

Rule 411

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

Rule 418

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

Rule 473

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*(c + d*x^n)^q)/(e*(m + 1)), x] - Dist[n/(e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^(p -
1)*(c + d*x^n)^(q - 1)*Simp[b*c*p + a*d*q + b*d*(p + q)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*
c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] && LtQ[m, -1] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x
]

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[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 531

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 580

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)/(a*g*(m + 1)), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 3196

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 \coth ^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx &=\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^{3/2} \sqrt {a+b x^2}}{x^4} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac {\coth ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {\left (2 \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2} \left (\frac {1}{2} (3 a+b)+2 b x^2\right )}{x^2 \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 f}\\ &=-\frac {(3 a+b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {\coth ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {\left (2 \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{2} a (3 a+5 b)+\frac {1}{2} b (7 a+b) x^2}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a f}\\ &=-\frac {(3 a+b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {\coth ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {\left (b (7 a+b) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a f}+\frac {\left ((3 a+5 b) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 f}\\ &=-\frac {(3 a+b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {\coth ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {(3 a+5 b) 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)}}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(7 a+b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a f}-\frac {\left ((7 a+b) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\left (1+x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 a f}\\ &=-\frac {(3 a+b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {\coth ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {(7 a+b) 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)}}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(3 a+5 b) 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)}}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(7 a+b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a f}\\ \end {align*}

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Mathematica [C]  time = 3.10, size = 210, normalized size = 0.78 \[ \frac {-\frac {\coth (e+f x) \text {csch}^2(e+f x) \left (4 \left (4 a^2-2 a b-b^2\right ) \cosh (2 (e+f x))-8 a^2+b (4 a+b) \cosh (4 (e+f x))+4 a b+3 b^2\right )}{2 \sqrt {2}}+8 i a (a-b) \sqrt {\frac {2 a+b \cosh (2 (e+f x))-b}{a}} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )-2 i a (7 a+b) \sqrt {\frac {2 a+b \cosh (2 (e+f x))-b}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )}{6 a f \sqrt {2 a+b \cosh (2 (e+f x))-b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1/2*((-8*a^2 + 4*a*b + 3*b^2 + 4*(4*a^2 - 2*a*b - b^2)*Cosh[2*(e + f*x)] + b*(4*a + b)*Cosh[4*(e + f*x)])*Co
th[e + f*x]*Csch[e + f*x]^2)/Sqrt[2] - (2*I)*a*(7*a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(
e + f*x), b/a] + (8*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(6*a*f*S
qrt[2*a - b + b*Cosh[2*(e + f*x)]])

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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sinh \left (f x + e\right )^{2} + a} \coth \left (f x + e\right )^{4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sinh(f*x + e)^2 + a)*coth(f*x + e)^4, x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.95Unable to divide, perhaps due to rounding error%%%{1024,[8,16,8]%%%}+%%%{%%%{-4096,[1]%%%},[8
,16,7]%%%}+%%%{%%%{6144,[2]%%%},[8,16,6]%%%}+%%%{%%%{-4096,[3]%%%},[8,16,5]%%%}+%%%{%%%{1024,[4]%%%},[8,16,4]%
%%}+%%%{%%{[8192,0]:[1,0,%%%{-1,[1]%%%}]%%},[7,16,8]%%%}+%%%{%%{[%%%{-32768,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%}
,[7,16,7]%%%}+%%%{%%{[%%%{49152,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,16,6]%%%}+%%%{%%{[%%%{-32768,[3]%%%},0]:
[1,0,%%%{-1,[1]%%%}]%%},[7,16,5]%%%}+%%%{%%{[%%%{8192,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,16,4]%%%}+%%%{-163
84,[6,16,9]%%%}+%%%{%%%{94208,[1]%%%},[6,16,8]%%%}+%%%{%%%{-212992,[2]%%%},[6,16,7]%%%}+%%%{%%%{237568,[3]%%%}
,[6,16,6]%%%}+%%%{%%%{-131072,[4]%%%},[6,16,5]%%%}+%%%{%%%{28672,[5]%%%},[6,16,4]%%%}+%%%{%%{[-98304,0]:[1,0,%
%%{-1,[1]%%%}]%%},[5,16,9]%%%}+%%%{%%{[%%%{450560,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,16,8]%%%}+%%%{%%{[%%%{
-819200,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,16,7]%%%}+%%%{%%{[%%%{737280,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[5,16,6]%%%}+%%%{%%{[%%%{-327680,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,16,5]%%%}+%%%{%%{[%%%{57344,[5]%%%},0]:
[1,0,%%%{-1,[1]%%%}]%%},[5,16,4]%%%}+%%%{98304,[4,16,10]%%%}+%%%{%%%{-638976,[1]%%%},[4,16,9]%%%}+%%%{%%%{1644
544,[2]%%%},[4,16,8]%%%}+%%%{%%%{-2154496,[3]%%%},[4,16,7]%%%}+%%%{%%%{1511424,[4]%%%},[4,16,6]%%%}+%%%{%%%{-5
32480,[5]%%%},[4,16,5]%%%}+%%%{%%%{71680,[6]%%%},[4,16,4]%%%}+%%%{%%{[393216,0]:[1,0,%%%{-1,[1]%%%}]%%},[3,16,
10]%%%}+%%%{%%{[%%%{-1900544,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,16,9]%%%}+%%%{%%{[%%%{3727360,[2]%%%},0]:[1
,0,%%%{-1,[1]%%%}]%%},[3,16,8]%%%}+%%%{%%{[%%%{-3768320,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,16,7]%%%}+%%%{%%
{[%%%{2048000,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,16,6]%%%}+%%%{%%{[%%%{-557056,[5]%%%},0]:[1,0,%%%{-1,[1]%%
%}]%%},[3,16,5]%%%}+%%%{%%{[%%%{57344,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,16,4]%%%}+%%%{-262144,[2,16,11]%%%
}+%%%{%%%{1638400,[1]%%%},[2,16,10]%%%}+%%%{%%%{-4177920,[2]%%%},[2,16,9]%%%}+%%%{%%%{5599232,[3]%%%},[2,16,8]
%%%}+%%%{%%%{-4210688,[4]%%%},[2,16,7]%%%}+%%%{%%%{1744896,[5]%%%},[2,16,6]%%%}+%%%{%%%{-360448,[6]%%%},[2,16,
5]%%%}+%%%{%%%{28672,[7]%%%},[2,16,4]%%%}+%%%{%%{[-524288,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,16,11]%%%}+%%%{%%{[%%%
{2490368,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,16,10]%%%}+%%%{%%{[%%%{-4816896,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]
%%},[1,16,9]%%%}+%%%{%%{[%%%{4857856,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,16,8]%%%}+%%%{%%{[%%%{-2719744,[4]%
%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,16,7]%%%}+%%%{%%{[%%%{835584,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,16,6]%%%}
+%%%{%%{[%%%{-131072,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,16,5]%%%}+%%%{%%{[%%%{8192,[7]%%%},0]:[1,0,%%%{-1,[
1]%%%}]%%},[1,16,4]%%%}+%%%{262144,[0,16,12]%%%}+%%%{%%%{-1310720,[1]%%%},[0,16,11]%%%}+%%%{%%%{2719744,[2]%%%
},[0,16,10]%%%}+%%%{%%%{-3031040,[3]%%%},[0,16,9]%%%}+%%%{%%%{1967104,[4]%%%},[0,16,8]%%%}+%%%{%%%{-757760,[5]
%%%},[0,16,7]%%%}+%%%{%%%{169984,[6]%%%},[0,16,6]%%%}+%%%{%%%{-20480,[7]%%%},[0,16,5]%%%}+%%%{%%%{1024,[8]%%%}
,[0,16,4]%%%} / %%%{%%%{1,[2]%%%},[8,0,0]%%%}+%%%{%%{poly1[%%%{8,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,0,0]%%%
}+%%%{%%%{-16,[2]%%%},[6,0,1]%%%}+%%%{%%%{28,[3]%%%},[6,0,0]%%%}+%%%{%%{[%%%{-96,[2]%%%},0]:[1,0,%%%{-1,[1]%%%
}]%%},[5,0,1]%%%}+%%%{%%{poly1[%%%{56,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0,0]%%%}+%%%{%%%{96,[2]%%%},[4,0,2
]%%%}+%%%{%%%{-240,[3]%%%},[4,0,1]%%%}+%%%{%%%{70,[4]%%%},[4,0,0]%%%}+%%%{%%{poly1[%%%{384,[2]%%%},0]:[1,0,%%%
{-1,[1]%%%}]%%},[3,0,2]%%%}+%%%{%%{[%%%{-320,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,1]%%%}+%%%{%%{poly1[%%%{5
6,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,0]%%%}+%%%{%%%{-256,[2]%%%},[2,0,3]%%%}+%%%{%%%{576,[3]%%%},[2,0,2]%
%%}+%%%{%%%{-240,[4]%%%},[2,0,1]%%%}+%%%{%%%{28,[5]%%%},[2,0,0]%%%}+%%%{%%{[%%%{-512,[2]%%%},0]:[1,0,%%%{-1,[1
]%%%}]%%},[1,0,3]%%%}+%%%{%%{poly1[%%%{384,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,2]%%%}+%%%{%%{[%%%{-96,[4]%
%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,1]%%%}+%%%{%%{poly1[%%%{8,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,0]%%%}+%
%%{%%%{256,[2]%%%},[0,0,4]%%%}+%%%{%%%{-256,[3]%%%},[0,0,3]%%%}+%%%{%%%{96,[4]%%%},[0,0,2]%%%}+%%%{%%%{-16,[5]
%%%},[0,0,1]%%%}+%%%{%%%{1,[6]%%%},[0,0,0]%%%} Error: Bad Argument Value

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maple [A]  time = 0.31, size = 522, normalized size = 1.93 \[ \frac {-4 \sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{6}\left (f x +e \right )\right )-\sqrt {-\frac {b}{a}}\, b^{2} \left (\sinh ^{6}\left (f x +e \right )\right )+3 a^{2} \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) \left (\sinh ^{3}\left (f x +e \right )\right )-2 b \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \left (\sinh ^{3}\left (f x +e \right )\right )-\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2} \left (\sinh ^{3}\left (f x +e \right )\right )+7 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b \left (\sinh ^{3}\left (f x +e \right )\right )+\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2} \left (\sinh ^{3}\left (f x +e \right )\right )-4 \sqrt {-\frac {b}{a}}\, a^{2} \left (\sinh ^{4}\left (f x +e \right )\right )-6 \sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{4}\left (f x +e \right )\right )-\sqrt {-\frac {b}{a}}\, b^{2} \left (\sinh ^{4}\left (f x +e \right )\right )-5 \sqrt {-\frac {b}{a}}\, a^{2} \left (\sinh ^{2}\left (f x +e \right )\right )-2 \sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{2}\left (f x +e \right )\right )-\sqrt {-\frac {b}{a}}\, a^{2}}{3 a \sinh \left (f x +e \right )^{3} \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(1/2),x)

[Out]

1/3*(-4*(-1/a*b)^(1/2)*a*b*sinh(f*x+e)^6-(-1/a*b)^(1/2)*b^2*sinh(f*x+e)^6+3*a^2*((a+b*sinh(f*x+e)^2)/a)^(1/2)*
(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*sinh(f*x+e)^3-2*b*((a+b*sinh(f*x+e)^2)
/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*sinh(f*x+e)^3-((a+b*sinh(f
*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^2*sinh(f*x+e)^3+7*
((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*b*sinh
(f*x+e)^3+((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2)
)*b^2*sinh(f*x+e)^3-4*(-1/a*b)^(1/2)*a^2*sinh(f*x+e)^4-6*(-1/a*b)^(1/2)*a*b*sinh(f*x+e)^4-(-1/a*b)^(1/2)*b^2*s
inh(f*x+e)^4-5*(-1/a*b)^(1/2)*a^2*sinh(f*x+e)^2-2*(-1/a*b)^(1/2)*a*b*sinh(f*x+e)^2-(-1/a*b)^(1/2)*a^2)/a/sinh(
f*x+e)^3/(-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 \[ \int \sqrt {b \sinh \left (f x + e\right )^{2} + a} \coth \left (f x + e\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*sinh(f*x + e)^2 + a)*coth(f*x + e)^4, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {coth}\left (e+f\,x\right )}^4\,\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(e + f*x)^4*(a + b*sinh(e + f*x)^2)^(1/2),x)

[Out]

int(coth(e + f*x)^4*(a + b*sinh(e + f*x)^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \sinh ^{2}{\left (e + f x \right )}} \coth ^{4}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)**4*(a+b*sinh(f*x+e)**2)**(1/2),x)

[Out]

Integral(sqrt(a + b*sinh(e + f*x)**2)*coth(e + f*x)**4, x)

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