3.1.83 \(\int \sqrt {a+b \cosh ^3(x)} \tanh (x) \, dx\) [83]
Optimal. Leaf size=45 \[ -\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \cosh ^3(x)}}{\sqrt {a}}\right )+\frac {2}{3} \sqrt {a+b \cosh ^3(x)} \]
[Out]
-2/3*arctanh((a+b*cosh(x)^3)^(1/2)/a^(1/2))*a^(1/2)+2/3*(a+b*cosh(x)^3)^(1/2)
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Rubi [A]
time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3309, 272, 52,
65, 214} \begin {gather*} \frac {2}{3} \sqrt {a+b \cosh ^3(x)}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \cosh ^3(x)}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[a + b*Cosh[x]^3]*Tanh[x],x]
[Out]
(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cosh[x]^3]/Sqrt[a]])/3 + (2*Sqrt[a + b*Cosh[x]^3])/3
Rule 52
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]
Rule 65
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 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 272
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Rule 3309
Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[x^m*((a + b*(c*ff*x)^n)^p/(1 - ff^2*x^2)^((
m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]
Rubi steps
\begin {align*} \int \sqrt {a+b \cosh ^3(x)} \tanh (x) \, dx &=\text {Subst}\left (\int \frac {\sqrt {a+b x^3}}{x} \, dx,x,\cosh (x)\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\cosh ^3(x)\right )\\ &=\frac {2}{3} \sqrt {a+b \cosh ^3(x)}+\frac {1}{3} a \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\cosh ^3(x)\right )\\ &=\frac {2}{3} \sqrt {a+b \cosh ^3(x)}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^3(x)}\right )}{3 b}\\ &=-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \cosh ^3(x)}}{\sqrt {a}}\right )+\frac {2}{3} \sqrt {a+b \cosh ^3(x)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 45, normalized size = 1.00 \begin {gather*} -\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \cosh ^3(x)}}{\sqrt {a}}\right )+\frac {2}{3} \sqrt {a+b \cosh ^3(x)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[a + b*Cosh[x]^3]*Tanh[x],x]
[Out]
(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cosh[x]^3]/Sqrt[a]])/3 + (2*Sqrt[a + b*Cosh[x]^3])/3
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Maple [A]
time = 6.06, size = 34, normalized size = 0.76
| | |
method |
result |
size |
| | |
derivativedivides |
\(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (\cosh ^{3}\left (x \right )\right )}}{\sqrt {a}}\right ) \sqrt {a}}{3}+\frac {2 \sqrt {a +b \left (\cosh ^{3}\left (x \right )\right )}}{3}\) |
\(34\) |
default |
\(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (\cosh ^{3}\left (x \right )\right )}}{\sqrt {a}}\right ) \sqrt {a}}{3}+\frac {2 \sqrt {a +b \left (\cosh ^{3}\left (x \right )\right )}}{3}\) |
\(34\) |
| | |
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*cosh(x)^3)^(1/2)*tanh(x),x,method=_RETURNVERBOSE)
[Out]
-2/3*arctanh((a+b*cosh(x)^3)^(1/2)/a^(1/2))*a^(1/2)+2/3*(a+b*cosh(x)^3)^(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((a+b*cosh(x)^3)^(1/2)*tanh(x),x, algorithm="maxima")
[Out]
integrate(sqrt(b*cosh(x)^3 + a)*tanh(x), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 271 vs.
\(2 (33) = 66\).
time = 0.94, size = 1648, normalized size = 36.62 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*cosh(x)^3)^(1/2)*tanh(x),x, algorithm="fricas")
[Out]
[1/6*(sqrt(a)*(cosh(x) + sinh(x))*log(-(b^2*cosh(x)^12 + 12*b^2*cosh(x)*sinh(x)^11 + b^2*sinh(x)^12 + 6*b^2*co
sh(x)^10 + 64*a*b*cosh(x)^9 + 6*(11*b^2*cosh(x)^2 + b^2)*sinh(x)^10 + 15*b^2*cosh(x)^8 + 4*(55*b^2*cosh(x)^3 +
15*b^2*cosh(x) + 16*a*b)*sinh(x)^9 + 192*a*b*cosh(x)^7 + 3*(165*b^2*cosh(x)^4 + 90*b^2*cosh(x)^2 + 192*a*b*co
sh(x) + 5*b^2)*sinh(x)^8 + 24*(33*b^2*cosh(x)^5 + 30*b^2*cosh(x)^3 + 96*a*b*cosh(x)^2 + 5*b^2*cosh(x) + 8*a*b)
*sinh(x)^7 + 192*a*b*cosh(x)^5 + 4*(128*a^2 + 5*b^2)*cosh(x)^6 + 4*(231*b^2*cosh(x)^6 + 315*b^2*cosh(x)^4 + 13
44*a*b*cosh(x)^3 + 105*b^2*cosh(x)^2 + 336*a*b*cosh(x) + 128*a^2 + 5*b^2)*sinh(x)^6 + 15*b^2*cosh(x)^4 + 24*(3
3*b^2*cosh(x)^7 + 63*b^2*cosh(x)^5 + 336*a*b*cosh(x)^4 + 35*b^2*cosh(x)^3 + 168*a*b*cosh(x)^2 + 8*a*b + (128*a
^2 + 5*b^2)*cosh(x))*sinh(x)^5 + 64*a*b*cosh(x)^3 + 3*(165*b^2*cosh(x)^8 + 420*b^2*cosh(x)^6 + 2688*a*b*cosh(x
)^5 + 350*b^2*cosh(x)^4 + 2240*a*b*cosh(x)^3 + 320*a*b*cosh(x) + 20*(128*a^2 + 5*b^2)*cosh(x)^2 + 5*b^2)*sinh(
x)^4 + 6*b^2*cosh(x)^2 + 4*(55*b^2*cosh(x)^9 + 180*b^2*cosh(x)^7 + 1344*a*b*cosh(x)^6 + 210*b^2*cosh(x)^5 + 16
80*a*b*cosh(x)^4 + 480*a*b*cosh(x)^2 + 20*(128*a^2 + 5*b^2)*cosh(x)^3 + 15*b^2*cosh(x) + 16*a*b)*sinh(x)^3 + 6
*(11*b^2*cosh(x)^10 + 45*b^2*cosh(x)^8 + 384*a*b*cosh(x)^7 + 70*b^2*cosh(x)^6 + 672*a*b*cosh(x)^5 + 320*a*b*co
sh(x)^3 + 10*(128*a^2 + 5*b^2)*cosh(x)^4 + 15*b^2*cosh(x)^2 + 32*a*b*cosh(x) + b^2)*sinh(x)^2 + b^2 - 16*(b*co
sh(x)^8 + 8*b*cosh(x)*sinh(x)^7 + b*sinh(x)^8 + 3*b*cosh(x)^6 + (28*b*cosh(x)^2 + 3*b)*sinh(x)^6 + 16*a*cosh(x
)^5 + 2*(28*b*cosh(x)^3 + 9*b*cosh(x) + 8*a)*sinh(x)^5 + 3*b*cosh(x)^4 + (70*b*cosh(x)^4 + 45*b*cosh(x)^2 + 80
*a*cosh(x) + 3*b)*sinh(x)^4 + 4*(14*b*cosh(x)^5 + 15*b*cosh(x)^3 + 40*a*cosh(x)^2 + 3*b*cosh(x))*sinh(x)^3 + b
*cosh(x)^2 + (28*b*cosh(x)^6 + 45*b*cosh(x)^4 + 160*a*cosh(x)^3 + 18*b*cosh(x)^2 + b)*sinh(x)^2 + 2*(4*b*cosh(
x)^7 + 9*b*cosh(x)^5 + 40*a*cosh(x)^4 + 6*b*cosh(x)^3 + b*cosh(x))*sinh(x))*sqrt(a)*sqrt((b*cosh(x)^3 + 3*b*co
sh(x)*sinh(x)^2 + 3*b*cosh(x) + 4*a)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 12*(b^2*cosh(x)^11 + 5*b^2
*cosh(x)^9 + 48*a*b*cosh(x)^8 + 10*b^2*cosh(x)^7 + 112*a*b*cosh(x)^6 + 80*a*b*cosh(x)^4 + 2*(128*a^2 + 5*b^2)*
cosh(x)^5 + 5*b^2*cosh(x)^3 + 16*a*b*cosh(x)^2 + b^2*cosh(x))*sinh(x))/(cosh(x)^12 + 12*cosh(x)*sinh(x)^11 + s
inh(x)^12 + 6*(11*cosh(x)^2 + 1)*sinh(x)^10 + 6*cosh(x)^10 + 20*(11*cosh(x)^3 + 3*cosh(x))*sinh(x)^9 + 15*(33*
cosh(x)^4 + 18*cosh(x)^2 + 1)*sinh(x)^8 + 15*cosh(x)^8 + 24*(33*cosh(x)^5 + 30*cosh(x)^3 + 5*cosh(x))*sinh(x)^
7 + 4*(231*cosh(x)^6 + 315*cosh(x)^4 + 105*cosh(x)^2 + 5)*sinh(x)^6 + 20*cosh(x)^6 + 24*(33*cosh(x)^7 + 63*cos
h(x)^5 + 35*cosh(x)^3 + 5*cosh(x))*sinh(x)^5 + 15*(33*cosh(x)^8 + 84*cosh(x)^6 + 70*cosh(x)^4 + 20*cosh(x)^2 +
1)*sinh(x)^4 + 15*cosh(x)^4 + 20*(11*cosh(x)^9 + 36*cosh(x)^7 + 42*cosh(x)^5 + 20*cosh(x)^3 + 3*cosh(x))*sinh
(x)^3 + 6*(11*cosh(x)^10 + 45*cosh(x)^8 + 70*cosh(x)^6 + 50*cosh(x)^4 + 15*cosh(x)^2 + 1)*sinh(x)^2 + 6*cosh(x
)^2 + 12*(cosh(x)^11 + 5*cosh(x)^9 + 10*cosh(x)^7 + 10*cosh(x)^5 + 5*cosh(x)^3 + cosh(x))*sinh(x) + 1)) + 2*sq
rt((b*cosh(x)^3 + 3*b*cosh(x)*sinh(x)^2 + 3*b*cosh(x) + 4*a)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(co
sh(x) + sinh(x)), 1/3*(sqrt(-a)*(cosh(x) + sinh(x))*arctan(8*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)*sqrt(
-a)*sqrt((b*cosh(x)^3 + 3*b*cosh(x)*sinh(x)^2 + 3*b*cosh(x) + 4*a)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)
)/(b*cosh(x)^6 + 6*b*cosh(x)*sinh(x)^5 + b*sinh(x)^6 + 3*b*cosh(x)^4 + 3*(5*b*cosh(x)^2 + b)*sinh(x)^4 + 16*a*
cosh(x)^3 + 4*(5*b*cosh(x)^3 + 3*b*cosh(x) + 4*a)*sinh(x)^3 + 3*b*cosh(x)^2 + 3*(5*b*cosh(x)^4 + 6*b*cosh(x)^2
+ 16*a*cosh(x) + b)*sinh(x)^2 + 6*(b*cosh(x)^5 + 2*b*cosh(x)^3 + 8*a*cosh(x)^2 + b*cosh(x))*sinh(x) + b)) + s
qrt((b*cosh(x)^3 + 3*b*cosh(x)*sinh(x)^2 + 3*b*cosh(x) + 4*a)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(c
osh(x) + sinh(x))]
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \cosh ^{3}{\left (x \right )}} \tanh {\left (x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*cosh(x)**3)**(1/2)*tanh(x),x)
[Out]
Integral(sqrt(a + b*cosh(x)**3)*tanh(x), x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*cosh(x)^3)^(1/2)*tanh(x),x, algorithm="giac")
[Out]
integrate(sqrt(b*cosh(x)^3 + a)*tanh(x), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \mathrm {tanh}\left (x\right )\,\sqrt {b\,{\mathrm {cosh}\left (x\right )}^3+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(x)*(a + b*cosh(x)^3)^(1/2),x)
[Out]
int(tanh(x)*(a + b*cosh(x)^3)^(1/2), x)
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