3.484 \(\int \frac {\coth ^5(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx\)
Optimal. Leaf size=126 \[ -\frac {(8 a-3 b) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{8 a^2 f}-\frac {\left (8 a^2-8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{5/2} f}-\frac {\text {csch}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{4 a f} \]
[Out]
-1/8*(8*a^2-8*a*b+3*b^2)*arctanh((a+b*sinh(f*x+e)^2)^(1/2)/a^(1/2))/a^(5/2)/f-1/8*(8*a-3*b)*csch(f*x+e)^2*(a+b
*sinh(f*x+e)^2)^(1/2)/a^2/f-1/4*csch(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(1/2)/a/f
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Rubi [A] time = 0.15, antiderivative size = 126, 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 =
{3194, 89, 78, 63, 208} \[ -\frac {\left (8 a^2-8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{5/2} f}-\frac {(8 a-3 b) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{8 a^2 f}-\frac {\text {csch}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{4 a f} \]
Antiderivative was successfully verified.
[In]
Int[Coth[e + f*x]^5/Sqrt[a + b*Sinh[e + f*x]^2],x]
[Out]
-((8*a^2 - 8*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(8*a^(5/2)*f) - ((8*a - 3*b)*Csch[e +
f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])/(8*a^2*f) - (Csch[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2])/(4*a*f)
Rule 63
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,
b, c, d, m, n, x]
Rule 78
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] || LtQ
[p, n]))))
Rule 89
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 3194
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]
Rubi steps
\begin {align*} \int \frac {\coth ^5(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(1+x)^2}{x^3 \sqrt {a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=-\frac {\text {csch}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{4 a f}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (8 a-3 b)+2 a x}{x^2 \sqrt {a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{4 a f}\\ &=-\frac {(8 a-3 b) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{8 a^2 f}-\frac {\text {csch}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{4 a f}+\frac {\left (8 a^2-8 a b+3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{16 a^2 f}\\ &=-\frac {(8 a-3 b) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{8 a^2 f}-\frac {\text {csch}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{4 a f}+\frac {\left (8 a^2-8 a b+3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^2(e+f x)}\right )}{8 a^2 b f}\\ &=-\frac {\left (8 a^2-8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{5/2} f}-\frac {(8 a-3 b) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{8 a^2 f}-\frac {\text {csch}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{4 a f}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 100, normalized size = 0.79 \[ \frac {\left (-8 a^2+8 a b-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a} \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \left (-2 a \text {csch}^2(e+f x)-8 a+3 b\right )}{8 a^{5/2} f} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[e + f*x]^5/Sqrt[a + b*Sinh[e + f*x]^2],x]
[Out]
((-8*a^2 + 8*a*b - 3*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]] + Sqrt[a]*Csch[e + f*x]^2*(-8*a + 3*b -
2*a*Csch[e + f*x]^2)*Sqrt[a + b*Sinh[e + f*x]^2])/(8*a^(5/2)*f)
________________________________________________________________________________________
fricas [B] time = 0.77, size = 3086, normalized size = 24.49 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(f*x+e)^5/(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")
[Out]
[1/16*(((8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^8 + 8*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + (8
*a^2 - 8*a*b + 3*b^2)*sinh(f*x + e)^8 - 4*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^6 + 4*(7*(8*a^2 - 8*a*b + 3*b^
2)*cosh(f*x + e)^2 - 8*a^2 + 8*a*b - 3*b^2)*sinh(f*x + e)^6 + 8*(7*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^3 - 3
*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 6*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^4 + 2*(35*(8
*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^4 - 30*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^2 + 24*a^2 - 24*a*b + 9*b^2)*
sinh(f*x + e)^4 + 8*(7*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^5 - 10*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^3 +
3*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 - 4*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^2 + 4*(7*(8
*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^6 - 15*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^4 + 9*(8*a^2 - 8*a*b + 3*b^2)
*cosh(f*x + e)^2 - 8*a^2 + 8*a*b - 3*b^2)*sinh(f*x + e)^2 + 8*a^2 - 8*a*b + 3*b^2 + 8*((8*a^2 - 8*a*b + 3*b^2)
*cosh(f*x + e)^7 - 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^5 + 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^3 - (8*
a^2 - 8*a*b + 3*b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt(a)*log((b*cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*sinh(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)) - 4*
sqrt(2)*((8*a^2 - 3*a*b)*cosh(f*x + e)^5 + 5*(8*a^2 - 3*a*b)*cosh(f*x + e)*sinh(f*x + e)^4 + (8*a^2 - 3*a*b)*s
inh(f*x + e)^5 - 2*(4*a^2 - 3*a*b)*cosh(f*x + e)^3 + 2*(5*(8*a^2 - 3*a*b)*cosh(f*x + e)^2 - 4*a^2 + 3*a*b)*sin
h(f*x + e)^3 + 2*(5*(8*a^2 - 3*a*b)*cosh(f*x + e)^3 - 3*(4*a^2 - 3*a*b)*cosh(f*x + e))*sinh(f*x + e)^2 + (8*a^
2 - 3*a*b)*cosh(f*x + e) + (5*(8*a^2 - 3*a*b)*cosh(f*x + e)^4 - 6*(4*a^2 - 3*a*b)*cosh(f*x + e)^2 + 8*a^2 - 3*
a*b)*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^3*f*cosh(f*x + e)^8 + 8*a^3*f*cosh(f*x + e)*sinh(f*x + e)^7 + a^3*f*sinh
(f*x + e)^8 - 4*a^3*f*cosh(f*x + e)^6 + 6*a^3*f*cosh(f*x + e)^4 + 4*(7*a^3*f*cosh(f*x + e)^2 - a^3*f)*sinh(f*x
+ e)^6 - 4*a^3*f*cosh(f*x + e)^2 + 8*(7*a^3*f*cosh(f*x + e)^3 - 3*a^3*f*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(3
5*a^3*f*cosh(f*x + e)^4 - 30*a^3*f*cosh(f*x + e)^2 + 3*a^3*f)*sinh(f*x + e)^4 + a^3*f + 8*(7*a^3*f*cosh(f*x +
e)^5 - 10*a^3*f*cosh(f*x + e)^3 + 3*a^3*f*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(7*a^3*f*cosh(f*x + e)^6 - 15*a^3
*f*cosh(f*x + e)^4 + 9*a^3*f*cosh(f*x + e)^2 - a^3*f)*sinh(f*x + e)^2 + 8*(a^3*f*cosh(f*x + e)^7 - 3*a^3*f*cos
h(f*x + e)^5 + 3*a^3*f*cosh(f*x + e)^3 - a^3*f*cosh(f*x + e))*sinh(f*x + e)), 1/8*(((8*a^2 - 8*a*b + 3*b^2)*co
sh(f*x + e)^8 + 8*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + (8*a^2 - 8*a*b + 3*b^2)*sinh(f*x + e
)^8 - 4*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^6 + 4*(7*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^2 - 8*a^2 + 8*a*b
- 3*b^2)*sinh(f*x + e)^6 + 8*(7*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^3 - 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x
+ e))*sinh(f*x + e)^5 + 6*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^4 + 2*(35*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e
)^4 - 30*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^2 + 24*a^2 - 24*a*b + 9*b^2)*sinh(f*x + e)^4 + 8*(7*(8*a^2 - 8*
a*b + 3*b^2)*cosh(f*x + e)^5 - 10*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^3 + 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x
+ e))*sinh(f*x + e)^3 - 4*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^2 + 4*(7*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e
)^6 - 15*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^4 + 9*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^2 - 8*a^2 + 8*a*b -
3*b^2)*sinh(f*x + e)^2 + 8*a^2 - 8*a*b + 3*b^2 + 8*((8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^7 - 3*(8*a^2 - 8*a*
b + 3*b^2)*cosh(f*x + e)^5 + 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^3 - (8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)
)*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*cosh(f*x + e) + a*sinh(f*x + e))) - 2*s
qrt(2)*((8*a^2 - 3*a*b)*cosh(f*x + e)^5 + 5*(8*a^2 - 3*a*b)*cosh(f*x + e)*sinh(f*x + e)^4 + (8*a^2 - 3*a*b)*si
nh(f*x + e)^5 - 2*(4*a^2 - 3*a*b)*cosh(f*x + e)^3 + 2*(5*(8*a^2 - 3*a*b)*cosh(f*x + e)^2 - 4*a^2 + 3*a*b)*sinh
(f*x + e)^3 + 2*(5*(8*a^2 - 3*a*b)*cosh(f*x + e)^3 - 3*(4*a^2 - 3*a*b)*cosh(f*x + e))*sinh(f*x + e)^2 + (8*a^2
- 3*a*b)*cosh(f*x + e) + (5*(8*a^2 - 3*a*b)*cosh(f*x + e)^4 - 6*(4*a^2 - 3*a*b)*cosh(f*x + e)^2 + 8*a^2 - 3*a
*b)*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)*s
inh(f*x + e) + sinh(f*x + e)^2)))/(a^3*f*cosh(f*x + e)^8 + 8*a^3*f*cosh(f*x + e)*sinh(f*x + e)^7 + a^3*f*sinh(
f*x + e)^8 - 4*a^3*f*cosh(f*x + e)^6 + 6*a^3*f*cosh(f*x + e)^4 + 4*(7*a^3*f*cosh(f*x + e)^2 - a^3*f)*sinh(f*x
+ e)^6 - 4*a^3*f*cosh(f*x + e)^2 + 8*(7*a^3*f*cosh(f*x + e)^3 - 3*a^3*f*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35
*a^3*f*cosh(f*x + e)^4 - 30*a^3*f*cosh(f*x + e)^2 + 3*a^3*f)*sinh(f*x + e)^4 + a^3*f + 8*(7*a^3*f*cosh(f*x + e
)^5 - 10*a^3*f*cosh(f*x + e)^3 + 3*a^3*f*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(7*a^3*f*cosh(f*x + e)^6 - 15*a^3*
f*cosh(f*x + e)^4 + 9*a^3*f*cosh(f*x + e)^2 - a^3*f)*sinh(f*x + e)^2 + 8*(a^3*f*cosh(f*x + e)^7 - 3*a^3*f*cosh
(f*x + e)^5 + 3*a^3*f*cosh(f*x + e)^3 - a^3*f*cosh(f*x + e))*sinh(f*x + e))]
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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)^5/(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:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]Evaluation time: 3.83Unable to divide, perhaps due to rounding error%%%{4096,[10,12,10]%%%}+%%%{%%
%{-20480,[1]%%%},[10,12,9]%%%}+%%%{%%%{40960,[2]%%%},[10,12,8]%%%}+%%%{%%%{-40960,[3]%%%},[10,12,7]%%%}+%%%{%%
%{20480,[4]%%%},[10,12,6]%%%}+%%%{%%%{-4096,[5]%%%},[10,12,5]%%%}+%%%{%%{[40960,0]:[1,0,%%%{-1,[1]%%%}]%%},[9,
12,10]%%%}+%%%{%%{[%%%{-204800,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,12,9]%%%}+%%%{%%{[%%%{409600,[2]%%%},0]:[
1,0,%%%{-1,[1]%%%}]%%},[9,12,8]%%%}+%%%{%%{[%%%{-409600,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,12,7]%%%}+%%%{%%
{[%%%{204800,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[9,12,6]%%%}+%%%{%%{[%%%{-40960,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}
]%%},[9,12,5]%%%}+%%%{-81920,[8,12,11]%%%}+%%%{%%%{593920,[1]%%%},[8,12,10]%%%}+%%%{%%%{-1740800,[2]%%%},[8,12
,9]%%%}+%%%{%%%{2662400,[3]%%%},[8,12,8]%%%}+%%%{%%%{-2252800,[4]%%%},[8,12,7]%%%}+%%%{%%%{1003520,[5]%%%},[8,
12,6]%%%}+%%%{%%%{-184320,[6]%%%},[8,12,5]%%%}+%%%{%%{[-655360,0]:[1,0,%%%{-1,[1]%%%}]%%},[7,12,11]%%%}+%%%{%%
{[%%%{3768320,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,12,10]%%%}+%%%{%%{[%%%{-9011200,[2]%%%},0]:[1,0,%%%{-1,[1]
%%%}]%%},[7,12,9]%%%}+%%%{%%{[%%%{11468800,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,12,8]%%%}+%%%{%%{[%%%{-819200
0,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,12,7]%%%}+%%%{%%{[%%%{3112960,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,12
,6]%%%}+%%%{%%{[%%%{-491520,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,12,5]%%%}+%%%{655360,[6,12,12]%%%}+%%%{%%%{-
5570560,[1]%%%},[6,12,11]%%%}+%%%{%%%{18882560,[2]%%%},[6,12,10]%%%}+%%%{%%%{-33792000,[3]%%%},[6,12,9]%%%}+%%
%{%%%{34816000,[4]%%%},[6,12,8]%%%}+%%%{%%%{-20725760,[5]%%%},[6,12,7]%%%}+%%%{%%%{6594560,[6]%%%},[6,12,6]%%%
}+%%%{%%%{-860160,[7]%%%},[6,12,5]%%%}+%%%{%%{[3932160,0]:[1,0,%%%{-1,[1]%%%}]%%},[5,12,12]%%%}+%%%{%%{[%%%{-2
4248320,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,12,11]%%%}+%%%{%%{[%%%{63291392,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%
%},[5,12,10]%%%}+%%%{%%{[%%%{-90357760,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,12,9]%%%}+%%%{%%{[%%%{75857920,[4
]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,12,8]%%%}+%%%{%%{[%%%{-37191680,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,12,7
]%%%}+%%%{%%{[%%%{9748480,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,12,6]%%%}+%%%{%%{[%%%{-1032192,[7]%%%},0]:[1,0
,%%%{-1,[1]%%%}]%%},[5,12,5]%%%}+%%%{-2621440,[4,12,13]%%%}+%%%{%%%{22937600,[1]%%%},[4,12,12]%%%}+%%%{%%%{-81
100800,[2]%%%},[4,12,11]%%%}+%%%{%%%{154050560,[3]%%%},[4,12,10]%%%}+%%%{%%%{-173056000,[4]%%%},[4,12,9]%%%}+%
%%{%%%{117719040,[5]%%%},[4,12,8]%%%}+%%%{%%%{-47104000,[6]%%%},[4,12,7]%%%}+%%%{%%%{10035200,[7]%%%},[4,12,6]
%%%}+%%%{%%%{-860160,[8]%%%},[4,12,5]%%%}+%%%{%%{[-10485760,0]:[1,0,%%%{-1,[1]%%%}]%%},[3,12,13]%%%}+%%%{%%{[%
%%{65536000,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,12,12]%%%}+%%%{%%{[%%%{-174981120,[2]%%%},0]:[1,0,%%%{-1,[1]
%%%}]%%},[3,12,11]%%%}+%%%{%%{[%%%{259358720,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,12,10]%%%}+%%%{%%{[%%%{-231
833600,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,12,9]%%%}+%%%{%%{[%%%{126812160,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%
},[3,12,8]%%%}+%%%{%%{[%%%{-40960000,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,12,7]%%%}+%%%{%%{[%%%{7045120,[7]%%
%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,12,6]%%%}+%%%{%%{[%%%{-491520,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,12,5]%%%}
+%%%{5242880,[2,12,14]%%%}+%%%{%%%{-41943040,[1]%%%},[2,12,13]%%%}+%%%{%%%{140902400,[2]%%%},[2,12,12]%%%}+%%%
{%%%{-261160960,[3]%%%},[2,12,11]%%%}+%%%{%%%{293457920,[4]%%%},[2,12,10]%%%}+%%%{%%%{-206049280,[5]%%%},[2,12
,9]%%%}+%%%{%%%{89661440,[6]%%%},[2,12,8]%%%}+%%%{%%%{-23142400,[7]%%%},[2,12,7]%%%}+%%%{%%%{3215360,[8]%%%},[
2,12,6]%%%}+%%%{%%%{-184320,[9]%%%},[2,12,5]%%%}+%%%{%%{[10485760,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,12,14]%%%}+%%%
{%%{[%%%{-62914560,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,12,13]%%%}+%%%{%%{[%%%{161218560,[2]%%%},0]:[1,0,%%%{
-1,[1]%%%}]%%},[1,12,12]%%%}+%%%{%%{[%%%{-230031360,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,12,11]%%%}+%%%{%%{[%
%%{199925760,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,12,10]%%%}+%%%{%%{[%%%{-108994560,[5]%%%},0]:[1,0,%%%{-1,[1
]%%%}]%%},[1,12,9]%%%}+%%%{%%{[%%%{37109760,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,12,8]%%%}+%%%{%%{[%%%{-76185
60,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,12,7]%%%}+%%%{%%{[%%%{860160,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,12
,6]%%%}+%%%{%%{[%%%{-40960,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,12,5]%%%}+%%%{-4194304,[0,12,15]%%%}+%%%{%%%{
26214400,[1]%%%},[0,12,14]%%%}+%%%{%%%{-70778880,[2]%%%},[0,12,13]%%%}+%%%{%%%{108134400,[3]%%%},[0,12,12]%%%}
+%%%{%%%{-102973440,[4]%%%},[0,12,11]%%%}+%%%{%%%{63590400,[5]%%%},[0,12,10]%%%}+%%%{%%%{-25743360,[6]%%%},[0,
12,9]%%%}+%%%{%%%{6758400,[7]%%%},[0,12,8]%%%}+%%%{%%%{-1105920,[8]%%%},[0,12,7]%%%}+%%%{%%%{102400,[9]%%%},[0
,12,6]%%%}+%%%{%%%{-4096,[10]%%%},[0,12,5]%%%} / %%%{%%{poly1[%%%{1,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[10,0,0
]%%%}+%%%{%%%{10,[3]%%%},[9,0,0]%%%}+%%%{%%{[%%%{-20,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,0,1]%%%}+%%%{%%{pol
y1[%%%{45,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,0,0]%%%}+%%%{%%%{-160,[3]%%%},[7,0,1]%%%}+%%%{%%%{120,[4]%%%},
[7,0,0]%%%}+%%%{%%{poly1[%%%{160,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,2]%%%}+%%%{%%{[%%%{-560,[3]%%%},0]:[1
,0,%%%{-1,[1]%%%}]%%},[6,0,1]%%%}+%%%{%%{poly1[%%%{210,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,0]%%%}+%%%{%%%{
960,[3]%%%},[5,0,2]%%%}+%%%{%%%{-1120,[4]%%%},[5,0,1]%%%}+%%%{%%%{252,[5]%%%},[5,0,0]%%%}+%%%{%%{[%%%{-640,[2]
%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,3]%%%}+%%%{%%{poly1[%%%{2400,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,2]%%
%}+%%%{%%{[%%%{-1400,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,1]%%%}+%%%{%%{poly1[%%%{210,[5]%%%},0]:[1,0,%%%{-
1,[1]%%%}]%%},[4,0,0]%%%}+%%%{%%%{-2560,[3]%%%},[3,0,3]%%%}+%%%{%%%{3200,[4]%%%},[3,0,2]%%%}+%%%{%%%{-1120,[5]
%%%},[3,0,1]%%%}+%%%{%%%{120,[6]%%%},[3,0,0]%%%}+%%%{%%{[%%%{1280,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,4]%%
%}+%%%{%%{[%%%{-3840,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,3]%%%}+%%%{%%{poly1[%%%{2400,[4]%%%},0]:[1,0,%%%{
-1,[1]%%%}]%%},[2,0,2]%%%}+%%%{%%{[%%%{-560,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,1]%%%}+%%%{%%{poly1[%%%{45
,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,0]%%%}+%%%{%%%{2560,[3]%%%},[1,0,4]%%%}+%%%{%%%{-2560,[4]%%%},[1,0,3]
%%%}+%%%{%%%{960,[5]%%%},[1,0,2]%%%}+%%%{%%%{-160,[6]%%%},[1,0,1]%%%}+%%%{%%%{10,[7]%%%},[1,0,0]%%%}+%%%{%%{[%
%%{-1024,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,5]%%%}+%%%{%%{[%%%{1280,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0
,0,4]%%%}+%%%{%%{[%%%{-640,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,3]%%%}+%%%{%%{poly1[%%%{160,[5]%%%},0]:[1,0
,%%%{-1,[1]%%%}]%%},[0,0,2]%%%}+%%%{%%{[%%%{-20,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,1]%%%}+%%%{%%{poly1[%%
%{1,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,0]%%%} Error: Bad Argument Value
________________________________________________________________________________________
maple [C] time = 0.22, size = 54, normalized size = 0.43 \[ \frac {\mathit {`\,int/indef0`\,}\left (\frac {\frac {1}{\sinh \left (f x +e \right )}+\frac {2}{\sinh \left (f x +e \right )^{3}}+\frac {1}{\sinh \left (f x +e \right )^{5}}}{\sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(f*x+e)^5/(a+b*sinh(f*x+e)^2)^(1/2),x)
[Out]
`int/indef0`((1/sinh(f*x+e)+2/sinh(f*x+e)^3+1/sinh(f*x+e)^5)/(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 \[ \int \frac {\coth \left (f x + e\right )^{5}}{\sqrt {b \sinh \left (f x + e\right )^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(f*x+e)^5/(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="maxima")
[Out]
integrate(coth(f*x + e)^5/sqrt(b*sinh(f*x + e)^2 + a), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {coth}\left (e+f\,x\right )}^5}{\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(e + f*x)^5/(a + b*sinh(e + f*x)^2)^(1/2),x)
[Out]
int(coth(e + f*x)^5/(a + b*sinh(e + f*x)^2)^(1/2), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{5}{\left (e + f x \right )}}{\sqrt {a + b \sinh ^{2}{\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(f*x+e)**5/(a+b*sinh(f*x+e)**2)**(1/2),x)
[Out]
Integral(coth(e + f*x)**5/sqrt(a + b*sinh(e + f*x)**2), x)
________________________________________________________________________________________