3.1.12 \(\int \frac {\text {csch}^5(x)}{a+b \cosh ^2(x)} \, dx\) [12]
Optimal. Leaf size=94 \[ -\frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \cosh (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^3}-\frac {\left (3 a^2+10 a b+15 b^2\right ) \tanh ^{-1}(\cosh (x))}{8 (a+b)^3}+\frac {(3 a+7 b) \coth (x) \text {csch}(x)}{8 (a+b)^2}-\frac {\coth (x) \text {csch}^3(x)}{4 (a+b)} \]
[Out]
-1/8*(3*a^2+10*a*b+15*b^2)*arctanh(cosh(x))/(a+b)^3+1/8*(3*a+7*b)*coth(x)*csch(x)/(a+b)^2-1/4*coth(x)*csch(x)^
3/(a+b)-b^(5/2)*arctan(cosh(x)*b^(1/2)/a^(1/2))/(a+b)^3/a^(1/2)
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Rubi [A]
time = 0.12, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3269, 425, 541,
536, 212, 211} \begin {gather*} -\frac {\left (3 a^2+10 a b+15 b^2\right ) \tanh ^{-1}(\cosh (x))}{8 (a+b)^3}-\frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \cosh (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^3}-\frac {\coth (x) \text {csch}^3(x)}{4 (a+b)}+\frac {(3 a+7 b) \coth (x) \text {csch}(x)}{8 (a+b)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Csch[x]^5/(a + b*Cosh[x]^2),x]
[Out]
-((b^(5/2)*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^3)) - ((3*a^2 + 10*a*b + 15*b^2)*ArcTanh[Cosh[x
]])/(8*(a + b)^3) + ((3*a + 7*b)*Coth[x]*Csch[x])/(8*(a + b)^2) - (Coth[x]*Csch[x]^3)/(4*(a + b))
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 425
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]
Rule 536
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]
Rule 541
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
Rule 3269
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Rubi steps
\begin {align*} \int \frac {\text {csch}^5(x)}{a+b \cosh ^2(x)} \, dx &=-\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^3 \left (a+b x^2\right )} \, dx,x,\cosh (x)\right )\\ &=-\frac {\coth (x) \text {csch}^3(x)}{4 (a+b)}-\frac {\text {Subst}\left (\int \frac {3 a+4 b+3 b x^2}{\left (1-x^2\right )^2 \left (a+b x^2\right )} \, dx,x,\cosh (x)\right )}{4 (a+b)}\\ &=\frac {(3 a+7 b) \coth (x) \text {csch}(x)}{8 (a+b)^2}-\frac {\coth (x) \text {csch}^3(x)}{4 (a+b)}-\frac {\text {Subst}\left (\int \frac {3 a^2+7 a b+8 b^2+b (3 a+7 b) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\cosh (x)\right )}{8 (a+b)^2}\\ &=\frac {(3 a+7 b) \coth (x) \text {csch}(x)}{8 (a+b)^2}-\frac {\coth (x) \text {csch}^3(x)}{4 (a+b)}-\frac {b^3 \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\cosh (x)\right )}{(a+b)^3}-\frac {\left (3 a^2+10 a b+15 b^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (x)\right )}{8 (a+b)^3}\\ &=-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \cosh (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^3}-\frac {\left (3 a^2+10 a b+15 b^2\right ) \tanh ^{-1}(\cosh (x))}{8 (a+b)^3}+\frac {(3 a+7 b) \coth (x) \text {csch}(x)}{8 (a+b)^2}-\frac {\coth (x) \text {csch}^3(x)}{4 (a+b)}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.43, size = 219, normalized size = 2.33 \begin {gather*} \frac {2 \sqrt {a} \left (3 a^2+10 a b+7 b^2\right ) \text {csch}^2\left (\frac {x}{2}\right )-\sqrt {a} (a+b)^2 \text {csch}^4\left (\frac {x}{2}\right )+8 \left (-8 b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b}-i \sqrt {a+b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a}}\right )-8 b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b}+i \sqrt {a+b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a}}\right )+\sqrt {a} \left (3 a^2+10 a b+15 b^2\right ) \log \left (\tanh \left (\frac {x}{2}\right )\right )\right )+2 \sqrt {a} \left (3 a^2+10 a b+7 b^2\right ) \text {sech}^2\left (\frac {x}{2}\right )+\sqrt {a} (a+b)^2 \text {sech}^4\left (\frac {x}{2}\right )}{64 \sqrt {a} (a+b)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Csch[x]^5/(a + b*Cosh[x]^2),x]
[Out]
(2*Sqrt[a]*(3*a^2 + 10*a*b + 7*b^2)*Csch[x/2]^2 - Sqrt[a]*(a + b)^2*Csch[x/2]^4 + 8*(-8*b^(5/2)*ArcTan[(Sqrt[b
] - I*Sqrt[a + b]*Tanh[x/2])/Sqrt[a]] - 8*b^(5/2)*ArcTan[(Sqrt[b] + I*Sqrt[a + b]*Tanh[x/2])/Sqrt[a]] + Sqrt[a
]*(3*a^2 + 10*a*b + 15*b^2)*Log[Tanh[x/2]]) + 2*Sqrt[a]*(3*a^2 + 10*a*b + 7*b^2)*Sech[x/2]^2 + Sqrt[a]*(a + b)
^2*Sech[x/2]^4)/(64*Sqrt[a]*(a + b)^3)
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Maple [A]
time = 1.08, size = 135, normalized size = 1.44
| | |
| method |
result |
size |
| | |
| default |
\(\frac {\left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-4 a -8 b \right )^{2}}{64 \left (a +b \right )^{3}}-\frac {1}{64 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{4}}-\frac {-4 a -8 b}{32 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (6 a^{2}+20 a b +30 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{16 \left (a +b \right )^{3}}-\frac {b^{3} \arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 a +2 b}{4 \sqrt {a b}}\right )}{\left (a +b \right )^{3} \sqrt {a b}}\) |
\(135\) |
| risch |
\(\frac {{\mathrm e}^{x} \left (3 a \,{\mathrm e}^{6 x}+7 \,{\mathrm e}^{6 x} b -11 a \,{\mathrm e}^{4 x}-15 \,{\mathrm e}^{4 x} b -11 \,{\mathrm e}^{2 x} a -15 b \,{\mathrm e}^{2 x}+3 a +7 b \right )}{4 \left ({\mathrm e}^{2 x}-1\right )^{4} \left (a +b \right )^{2}}-\frac {3 \ln \left ({\mathrm e}^{x}+1\right ) a^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {5 \ln \left ({\mathrm e}^{x}+1\right ) a b}{4 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {15 \ln \left ({\mathrm e}^{x}+1\right ) b^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {3 \ln \left ({\mathrm e}^{x}-1\right ) a^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {5 \ln \left ({\mathrm e}^{x}-1\right ) a b}{4 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {15 \ln \left ({\mathrm e}^{x}-1\right ) b^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 x}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{x}}{b}+1\right )}{2 a \left (a +b \right )^{3}}-\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 x}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{x}}{b}+1\right )}{2 a \left (a +b \right )^{3}}\) |
\(330\) |
| | |
|
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(x)^5/(a+b*cosh(x)^2),x,method=_RETURNVERBOSE)
[Out]
1/64*(a*tanh(1/2*x)^2+b*tanh(1/2*x)^2-4*a-8*b)^2/(a+b)^3-1/64/(a+b)/tanh(1/2*x)^4-1/32*(-4*a-8*b)/(a+b)^2/tanh
(1/2*x)^2+1/16/(a+b)^3*(6*a^2+20*a*b+30*b^2)*ln(tanh(1/2*x))-b^3/(a+b)^3/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(
1/2*x)^2-2*a+2*b)/(a*b)^(1/2))
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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(csch(x)^5/(a+b*cosh(x)^2),x, algorithm="maxima")
[Out]
-1/8*(3*a^2 + 10*a*b + 15*b^2)*log(e^x + 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 1/8*(3*a^2 + 10*a*b + 15*b^2)*lo
g(e^x - 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 1/4*((3*a + 7*b)*e^(7*x) - (11*a + 15*b)*e^(5*x) - (11*a + 15*b)*
e^(3*x) + (3*a + 7*b)*e^x)/(a^2 + 2*a*b + b^2 + (a^2 + 2*a*b + b^2)*e^(8*x) - 4*(a^2 + 2*a*b + b^2)*e^(6*x) +
6*(a^2 + 2*a*b + b^2)*e^(4*x) - 4*(a^2 + 2*a*b + b^2)*e^(2*x)) - 32*integrate(1/16*(b^3*e^(3*x) - b^3*e^x)/(a^
3*b + 3*a^2*b^2 + 3*a*b^3 + b^4 + (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*e^(4*x) + 2*(2*a^4 + 7*a^3*b + 9*a^2*b^2
+ 5*a*b^3 + b^4)*e^(2*x)), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2724 vs.
\(2 (80) = 160\).
time = 0.49, size = 5326, normalized size = 56.66 \begin {gather*} \text {too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)^5/(a+b*cosh(x)^2),x, algorithm="fricas")
[Out]
[1/8*(2*(3*a^2 + 10*a*b + 7*b^2)*cosh(x)^7 + 14*(3*a^2 + 10*a*b + 7*b^2)*cosh(x)*sinh(x)^6 + 2*(3*a^2 + 10*a*b
+ 7*b^2)*sinh(x)^7 - 2*(11*a^2 + 26*a*b + 15*b^2)*cosh(x)^5 + 2*(21*(3*a^2 + 10*a*b + 7*b^2)*cosh(x)^2 - 11*a
^2 - 26*a*b - 15*b^2)*sinh(x)^5 + 10*(7*(3*a^2 + 10*a*b + 7*b^2)*cosh(x)^3 - (11*a^2 + 26*a*b + 15*b^2)*cosh(x
))*sinh(x)^4 - 2*(11*a^2 + 26*a*b + 15*b^2)*cosh(x)^3 + 2*(35*(3*a^2 + 10*a*b + 7*b^2)*cosh(x)^4 - 10*(11*a^2
+ 26*a*b + 15*b^2)*cosh(x)^2 - 11*a^2 - 26*a*b - 15*b^2)*sinh(x)^3 + 2*(21*(3*a^2 + 10*a*b + 7*b^2)*cosh(x)^5
- 10*(11*a^2 + 26*a*b + 15*b^2)*cosh(x)^3 - 3*(11*a^2 + 26*a*b + 15*b^2)*cosh(x))*sinh(x)^2 + 4*(b^2*cosh(x)^8
+ 8*b^2*cosh(x)*sinh(x)^7 + b^2*sinh(x)^8 - 4*b^2*cosh(x)^6 + 4*(7*b^2*cosh(x)^2 - b^2)*sinh(x)^6 + 6*b^2*cos
h(x)^4 + 8*(7*b^2*cosh(x)^3 - 3*b^2*cosh(x))*sinh(x)^5 + 2*(35*b^2*cosh(x)^4 - 30*b^2*cosh(x)^2 + 3*b^2)*sinh(
x)^4 - 4*b^2*cosh(x)^2 + 8*(7*b^2*cosh(x)^5 - 10*b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^3 + 4*(7*b^2*cosh(x)^6
- 15*b^2*cosh(x)^4 + 9*b^2*cosh(x)^2 - b^2)*sinh(x)^2 + b^2 + 8*(b^2*cosh(x)^7 - 3*b^2*cosh(x)^5 + 3*b^2*cosh
(x)^3 - b^2*cosh(x))*sinh(x))*sqrt(-b/a)*log((b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 - 2*(2*a - b)*
cosh(x)^2 + 2*(3*b*cosh(x)^2 - 2*a + b)*sinh(x)^2 + 4*(b*cosh(x)^3 - (2*a - b)*cosh(x))*sinh(x) - 4*(a*cosh(x)
^3 + 3*a*cosh(x)*sinh(x)^2 + a*sinh(x)^3 + a*cosh(x) + (3*a*cosh(x)^2 + a)*sinh(x))*sqrt(-b/a) + b)/(b*cosh(x)
^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*(2*a + b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 + 2*a + b)*sinh(x)^2 + 4*(
b*cosh(x)^3 + (2*a + b)*cosh(x))*sinh(x) + b)) + 2*(3*a^2 + 10*a*b + 7*b^2)*cosh(x) - ((3*a^2 + 10*a*b + 15*b^
2)*cosh(x)^8 + 8*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)*sinh(x)^7 + (3*a^2 + 10*a*b + 15*b^2)*sinh(x)^8 - 4*(3*a^2
+ 10*a*b + 15*b^2)*cosh(x)^6 + 4*(7*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^2 - 3*a^2 - 10*a*b - 15*b^2)*sinh(x)^6 +
8*(7*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^3 - 3*(3*a^2 + 10*a*b + 15*b^2)*cosh(x))*sinh(x)^5 + 6*(3*a^2 + 10*a*b
+ 15*b^2)*cosh(x)^4 + 2*(35*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^4 - 30*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^2 + 9*
a^2 + 30*a*b + 45*b^2)*sinh(x)^4 + 8*(7*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^5 - 10*(3*a^2 + 10*a*b + 15*b^2)*cos
h(x)^3 + 3*(3*a^2 + 10*a*b + 15*b^2)*cosh(x))*sinh(x)^3 - 4*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^2 + 4*(7*(3*a^2
+ 10*a*b + 15*b^2)*cosh(x)^6 - 15*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^4 + 9*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^2
- 3*a^2 - 10*a*b - 15*b^2)*sinh(x)^2 + 3*a^2 + 10*a*b + 15*b^2 + 8*((3*a^2 + 10*a*b + 15*b^2)*cosh(x)^7 - 3*(3
*a^2 + 10*a*b + 15*b^2)*cosh(x)^5 + 3*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^3 - (3*a^2 + 10*a*b + 15*b^2)*cosh(x))
*sinh(x))*log(cosh(x) + sinh(x) + 1) + ((3*a^2 + 10*a*b + 15*b^2)*cosh(x)^8 + 8*(3*a^2 + 10*a*b + 15*b^2)*cosh
(x)*sinh(x)^7 + (3*a^2 + 10*a*b + 15*b^2)*sinh(x)^8 - 4*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^6 + 4*(7*(3*a^2 + 10
*a*b + 15*b^2)*cosh(x)^2 - 3*a^2 - 10*a*b - 15*b^2)*sinh(x)^6 + 8*(7*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^3 - 3*(
3*a^2 + 10*a*b + 15*b^2)*cosh(x))*sinh(x)^5 + 6*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^4 + 2*(35*(3*a^2 + 10*a*b +
15*b^2)*cosh(x)^4 - 30*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^2 + 9*a^2 + 30*a*b + 45*b^2)*sinh(x)^4 + 8*(7*(3*a^2
+ 10*a*b + 15*b^2)*cosh(x)^5 - 10*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^3 + 3*(3*a^2 + 10*a*b + 15*b^2)*cosh(x))*s
inh(x)^3 - 4*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^2 + 4*(7*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^6 - 15*(3*a^2 + 10*a
*b + 15*b^2)*cosh(x)^4 + 9*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^2 - 3*a^2 - 10*a*b - 15*b^2)*sinh(x)^2 + 3*a^2 +
10*a*b + 15*b^2 + 8*((3*a^2 + 10*a*b + 15*b^2)*cosh(x)^7 - 3*(3*a^2 + 10*a*b + 15*b^2)*cosh(x)^5 + 3*(3*a^2 +
10*a*b + 15*b^2)*cosh(x)^3 - (3*a^2 + 10*a*b + 15*b^2)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) - 1) + 2*(7*(3*
a^2 + 10*a*b + 7*b^2)*cosh(x)^6 - 5*(11*a^2 + 26*a*b + 15*b^2)*cosh(x)^4 - 3*(11*a^2 + 26*a*b + 15*b^2)*cosh(x
)^2 + 3*a^2 + 10*a*b + 7*b^2)*sinh(x))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^8 + 8*(a^3 + 3*a^2*b + 3*a*b^2
+ b^3)*cosh(x)*sinh(x)^7 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(x)^8 - 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh
(x)^6 - 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 7*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^3
+ 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^3 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x))*sinh(x)^5 + 6*(a^3 + 3*a^2*b
+ 3*a*b^2 + b^3)*cosh(x)^4 + 2*(35*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^4 + 3*a^3 + 9*a^2*b + 9*a*b^2 + 3*
b^3 - 30*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^5
- 10*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x))*sinh(x)^3 + a^3 +
3*a^2*b + 3*a*b^2 + b^3 - 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^2 + 4*(7*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*
cosh(x)^6 - 15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 9*(a^3 + 3*a^2*b +
3*a*b^2 + b^3)*cosh(x)^2)*sinh(x)^2 + 8*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^7 - 3*(a^3 + 3*a^2*b + 3*a*b^
2 + b^3)*cosh(x)^5 + 3*(a^3 + 3*a^2*b + 3*a*b^2...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}^{5}{\left (x \right )}}{a + b \cosh ^{2}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)**5/(a+b*cosh(x)**2),x)
[Out]
Integral(csch(x)**5/(a + b*cosh(x)**2), x)
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)^5/(a+b*cosh(x)^2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done
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Mupad [B]
time = 14.74, size = 2500, normalized size = 26.60 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(sinh(x)^5*(a + b*cosh(x)^2)),x)
[Out]
(atan((exp(x)*(243*a^12*(- 6*a*b^5 - 6*a^5*b - a^6 - b^6 - 15*a^2*b^4 - 20*a^3*b^3 - 15*a^4*b^2)^(3/2) + 3840*
b^12*(- 6*a*b^5 - 6*a^5*b - a^6 - b^6 - 15*a^2*b^4 - 20*a^3*b^3 - 15*a^4*b^2)^(3/2) + 110560*a*b^11*(- 6*a*b^5
- 6*a^5*b - a^6 - b^6 - 15*a^2*b^4 - 20*a^3*b^3 - 15*a^4*b^2)^(3/2) + 4050*a^11*b*(- 6*a*b^5 - 6*a^5*b - a^6
- b^6 - 15*a^2*b^4 - 20*a^3*b^3 - 15*a^4*b^2)^(3/2) + 976143*a^2*b^10*(- 6*a*b^5 - 6*a^5*b - a^6 - b^6 - 15*a^
2*b^4 - 20*a^3*b^3 - 15*a^4*b^2)^(3/2) + 2740050*a^3*b^9*(- 6*a*b^5 - 6*a^5*b - a^6 - b^6 - 15*a^2*b^4 - 20*a^
3*b^3 - 15*a^4*b^2)^(3/2) + 4252775*a^4*b^8*(- 6*a*b^5 - 6*a^5*b - a^6 - b^6 - 15*a^2*b^4 - 20*a^3*b^3 - 15*a^
4*b^2)^(3/2) + 4316760*a^5*b^7*(- 6*a*b^5 - 6*a^5*b - a^6 - b^6 - 15*a^2*b^4 - 20*a^3*b^3 - 15*a^4*b^2)^(3/2)
+ 3087390*a^6*b^6*(- 6*a*b^5 - 6*a^5*b - a^6 - b^6 - 15*a^2*b^4 - 20*a^3*b^3 - 15*a^4*b^2)^(3/2) + 1608364*a^7
*b^5*(- 6*a*b^5 - 6*a^5*b - a^6 - b^6 - 15*a^2*b^4 - 20*a^3*b^3 - 15*a^4*b^2)^(3/2) + 615750*a^8*b^4*(- 6*a*b^
5 - 6*a^5*b - a^6 - b^6 - 15*a^2*b^4 - 20*a^3*b^3 - 15*a^4*b^2)^(3/2) + 171000*a^9*b^3*(- 6*a*b^5 - 6*a^5*b -
a^6 - b^6 - 15*a^2*b^4 - 20*a^3*b^3 - 15*a^4*b^2)^(3/2) + 33075*a^10*b^2*(- 6*a*b^5 - 6*a^5*b - a^6 - b^6 - 15
*a^2*b^4 - 20*a^3*b^3 - 15*a^4*b^2)^(3/2)))/(81*a^19*(300*a*b^3 + 60*a^3*b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^(1
/2) + 256*b^19*(300*a*b^3 + 60*a^3*b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^(1/2) + 9504*a*b^18*(300*a*b^3 + 60*a^3*
b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^(1/2) + 1809*a^18*b*(300*a*b^3 + 60*a^3*b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^
(1/2) + 134241*a^2*b^17*(300*a*b^3 + 60*a^3*b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^(1/2) + 963809*a^3*b^16*(300*a*
b^3 + 60*a^3*b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^(1/2) + 4252296*a^4*b^15*(300*a*b^3 + 60*a^3*b + 9*a^4 + 225*b
^4 + 190*a^2*b^2)^(1/2) + 12815304*a^5*b^14*(300*a*b^3 + 60*a^3*b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^(1/2) + 281
02636*a^6*b^13*(300*a*b^3 + 60*a^3*b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^(1/2) + 46681644*a^7*b^12*(300*a*b^3 + 6
0*a^3*b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^(1/2) + 60321816*a^8*b^11*(300*a*b^3 + 60*a^3*b + 9*a^4 + 225*b^4 + 1
90*a^2*b^2)^(1/2) + 61717144*a^9*b^10*(300*a*b^3 + 60*a^3*b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^(1/2) + 50559894*
a^10*b^9*(300*a*b^3 + 60*a^3*b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^(1/2) + 33362646*a^11*b^8*(300*a*b^3 + 60*a^3*
b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^(1/2) + 17752184*a^12*b^7*(300*a*b^3 + 60*a^3*b + 9*a^4 + 225*b^4 + 190*a^2
*b^2)^(1/2) + 7586616*a^13*b^6*(300*a*b^3 + 60*a^3*b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^(1/2) + 2577804*a^14*b^5
*(300*a*b^3 + 60*a^3*b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^(1/2) + 683596*a^15*b^4*(300*a*b^3 + 60*a^3*b + 9*a^4
+ 225*b^4 + 190*a^2*b^2)^(1/2) + 137064*a^16*b^3*(300*a*b^3 + 60*a^3*b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^(1/2)
+ 19656*a^17*b^2*(300*a*b^3 + 60*a^3*b + 9*a^4 + 225*b^4 + 190*a^2*b^2)^(1/2)))*(300*a*b^3 + 60*a^3*b + 9*a^4
+ 225*b^4 + 190*a^2*b^2)^(1/2))/(4*(- 6*a*b^5 - 6*a^5*b - a^6 - b^6 - 15*a^2*b^4 - 20*a^3*b^3 - 15*a^4*b^2)^(1
/2)) - (4*exp(x))/((a + b)*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1)) - ((b^5)^(1/2)*(2*atan((b^3*
exp(x)*(a*(a + b)^6)^(1/2))/(2*a*(a + b)^3*(b^5)^(1/2))) - 2*atan((exp(x)*((2*(16*b^14*(a*b^6 + 6*a^6*b + a^7
+ 6*a^2*b^5 + 15*a^3*b^4 + 20*a^4*b^3 + 15*a^5*b^2)^(1/2) + 321*a*b^13*(a*b^6 + 6*a^6*b + a^7 + 6*a^2*b^5 + 15
*a^3*b^4 + 20*a^4*b^3 + 15*a^5*b^2)^(1/2) + 1890*a^2*b^12*(a*b^6 + 6*a^6*b + a^7 + 6*a^2*b^5 + 15*a^3*b^4 + 20
*a^4*b^3 + 15*a^5*b^2)^(1/2) + 5685*a^3*b^11*(a*b^6 + 6*a^6*b + a^7 + 6*a^2*b^5 + 15*a^3*b^4 + 20*a^4*b^3 + 15
*a^5*b^2)^(1/2) + 10440*a^4*b^10*(a*b^6 + 6*a^6*b + a^7 + 6*a^2*b^5 + 15*a^3*b^4 + 20*a^4*b^3 + 15*a^5*b^2)^(1
/2) + 12690*a^5*b^9*(a*b^6 + 6*a^6*b + a^7 + 6*a^2*b^5 + 15*a^3*b^4 + 20*a^4*b^3 + 15*a^5*b^2)^(1/2) + 10620*a
^6*b^8*(a*b^6 + 6*a^6*b + a^7 + 6*a^2*b^5 + 15*a^3*b^4 + 20*a^4*b^3 + 15*a^5*b^2)^(1/2) + 6210*a^7*b^7*(a*b^6
+ 6*a^6*b + a^7 + 6*a^2*b^5 + 15*a^3*b^4 + 20*a^4*b^3 + 15*a^5*b^2)^(1/2) + 2520*a^8*b^6*(a*b^6 + 6*a^6*b + a^
7 + 6*a^2*b^5 + 15*a^3*b^4 + 20*a^4*b^3 + 15*a^5*b^2)^(1/2) + 685*a^9*b^5*(a*b^6 + 6*a^6*b + a^7 + 6*a^2*b^5 +
15*a^3*b^4 + 20*a^4*b^3 + 15*a^5*b^2)^(1/2) + 114*a^10*b^4*(a*b^6 + 6*a^6*b + a^7 + 6*a^2*b^5 + 15*a^3*b^4 +
20*a^4*b^3 + 15*a^5*b^2)^(1/2) + 9*a^11*b^3*(a*b^6 + 6*a^6*b + a^7 + 6*a^2*b^5 + 15*a^3*b^4 + 20*a^4*b^3 + 15*
a^5*b^2)^(1/2)))/(a^2*b*(a + b)^10*(a*b + a^2)*(b^5)^(1/2)*(3*a*b^2 + 3*a^2*b + a^3 + b^3)*(4*a*b^3 + 4*a^3*b
+ a^4 + b^4 + 6*a^2*b^2)*(225*a*b^4 + 60*a^4*b + 9*a^5 + 16*b^5 + 300*a^2*b^3 + 190*a^3*b^2)*(6*a*b^5 + 6*a^5*
b + a^6 + b^6 + 15*a^2*b^4 + 20*a^3*b^3 + 15*a^4*b^2)*(a*b^6 + 6*a^6*b + a^7 + 6*a^2*b^5 + 15*a^3*b^4 + 20*a^4
*b^3 + 15*a^5*b^2)^(1/2)) + (4*(4032*a^5*(b^5)^(5/2) + 74990*a^10*(b^5)^(3/2) + 18*a^15*(b^5)^(1/2) + 288*a^2*
b^8*(b^5)^(3/2) + 1152*a^3*b^7*(b^5)^(3/2) + 2688*a^4*b^6*(b^5)^(3/2) + 4032*a^6*b^4*(b^5)^(3/2) + 2688*a^7*b^
3*(b^5)^(3/2) + 1152*a^8*b^2*(b^5)^(3/2) + 450*a^2*b^13*(b^5)^(1/2) + 4650*a^3*b^12*(b^5)^(1/2) + 21980*a^4*b^
11*(b^5)^(1/2) + 62940*a^5*b^10*(b^5)^(1/2) + 1...
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