3.230 \(\int \frac {\tanh ^4(x)}{\sqrt {a+b \tanh ^2(x)}} \, dx\)
Optimal. Leaf size=88 \[ \frac {(a-2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{2 b^{3/2}}-\frac {\tanh (x) \sqrt {a+b \tanh ^2(x)}}{2 b}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{\sqrt {a+b}} \]
[Out]
1/2*(a-2*b)*arctanh(b^(1/2)*tanh(x)/(a+b*tanh(x)^2)^(1/2))/b^(3/2)+arctanh((a+b)^(1/2)*tanh(x)/(a+b*tanh(x)^2)
^(1/2))/(a+b)^(1/2)-1/2*(a+b*tanh(x)^2)^(1/2)*tanh(x)/b
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Rubi [A] time = 0.12, antiderivative size = 88, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used =
{3670, 479, 523, 217, 206, 377} \[ \frac {(a-2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{2 b^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{\sqrt {a+b}}-\frac {\tanh (x) \sqrt {a+b \tanh ^2(x)}}{2 b} \]
Antiderivative was successfully verified.
[In]
Int[Tanh[x]^4/Sqrt[a + b*Tanh[x]^2],x]
[Out]
((a - 2*b)*ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]])/(2*b^(3/2)) + ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[
a + b*Tanh[x]^2]]/Sqrt[a + b] - (Tanh[x]*Sqrt[a + b*Tanh[x]^2])/(2*b)
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 217
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 0]
Rule 377
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Rule 479
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(2*n
- 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*d*(m + n*(p + q) + 1)), x] - Dist[e^(2*n)
/(b*d*(m + n*(p + q) + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m +
n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d
, 0] && IGtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 523
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]
Rule 3670
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))
Rubi steps
\begin {align*} \int \frac {\tanh ^4(x)}{\sqrt {a+b \tanh ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )\\ &=-\frac {\tanh (x) \sqrt {a+b \tanh ^2(x)}}{2 b}+\frac {\operatorname {Subst}\left (\int \frac {a+(-a+2 b) x^2}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )}{2 b}\\ &=-\frac {\tanh (x) \sqrt {a+b \tanh ^2(x)}}{2 b}+\frac {(a-2 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )}{2 b}+\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )\\ &=-\frac {\tanh (x) \sqrt {a+b \tanh ^2(x)}}{2 b}+\frac {(a-2 b) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{2 b}+\operatorname {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )\\ &=\frac {(a-2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{2 b^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{\sqrt {a+b}}-\frac {\tanh (x) \sqrt {a+b \tanh ^2(x)}}{2 b}\\ \end {align*}
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Mathematica [C] time = 4.90, size = 208, normalized size = 2.36 \[ \frac {\tanh (x) \left (-\left ((a+b) \text {sech}^2(x) ((a+b) \cosh (2 x)+a-b)\right )+\sqrt {2} a (a+b) \sqrt {\frac {\text {csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}{\sqrt {2}}\right )\right |1\right )-2 \sqrt {2} a b \sqrt {\frac {\text {csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} \Pi \left (\frac {b}{a+b};\left .\sin ^{-1}\left (\frac {\sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}{\sqrt {2}}\right )\right |1\right )\right )}{2 \sqrt {2} b (a+b) \sqrt {\text {sech}^2(x) ((a+b) \cosh (2 x)+a-b)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Tanh[x]^4/Sqrt[a + b*Tanh[x]^2],x]
[Out]
((Sqrt[2]*a*(a + b)*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Co
sh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1] - 2*Sqrt[2]*a*b*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*EllipticPi
[b/(a + b), ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1] - (a + b)*(a - b + (a + b)*Cos
h[2*x])*Sech[x]^2)*Tanh[x])/(2*Sqrt[2]*b*(a + b)*Sqrt[(a - b + (a + b)*Cosh[2*x])*Sech[x]^2])
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fricas [B] time = 0.77, size = 5494, normalized size = 62.43 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^4/(a+b*tanh(x)^2)^(1/2),x, algorithm="fricas")
[Out]
[1/4*((b^2*cosh(x)^4 + 4*b^2*cosh(x)*sinh(x)^3 + b^2*sinh(x)^4 + 2*b^2*cosh(x)^2 + 2*(3*b^2*cosh(x)^2 + b^2)*s
inh(x)^2 + b^2 + 4*(b^2*cosh(x)^3 + b^2*cosh(x))*sinh(x))*sqrt(a + b)*log(-((a*b^2 + b^3)*cosh(x)^8 + 8*(a*b^2
+ b^3)*cosh(x)*sinh(x)^7 + (a*b^2 + b^3)*sinh(x)^8 - 2*(a*b^2 + 2*b^3)*cosh(x)^6 - 2*(a*b^2 + 2*b^3 - 14*(a*b
^2 + b^3)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a*b^2 + b^3)*cosh(x)^3 - 3*(a*b^2 + 2*b^3)*cosh(x))*sinh(x)^5 + (a^3 -
a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^4 + (70*(a*b^2 + b^3)*cosh(x)^4 + a^3 - a^2*b + 4*a*b^2 + 6*b^3 - 30*(a*b^2
+ 2*b^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a*b^2 + b^3)*cosh(x)^5 - 10*(a*b^2 + 2*b^3)*cosh(x)^3 + (a^3 - a^2*b +
4*a*b^2 + 6*b^3)*cosh(x))*sinh(x)^3 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 + 2*(a^3 - 3*a*b^2 - 2*b^3)*cosh(x)^2 + 2*
(14*(a*b^2 + b^3)*cosh(x)^6 - 15*(a*b^2 + 2*b^3)*cosh(x)^4 + a^3 - 3*a*b^2 - 2*b^3 + 3*(a^3 - a^2*b + 4*a*b^2
+ 6*b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(b^2*cosh(x)^6 + 6*b^2*cosh(x)*sinh(x)^5 + b^2*sinh(x)^6 - 3*b^2*cosh(
x)^4 + 3*(5*b^2*cosh(x)^2 - b^2)*sinh(x)^4 + 4*(5*b^2*cosh(x)^3 - 3*b^2*cosh(x))*sinh(x)^3 - (a^2 - 2*a*b - 3*
b^2)*cosh(x)^2 + (15*b^2*cosh(x)^4 - 18*b^2*cosh(x)^2 - a^2 + 2*a*b + 3*b^2)*sinh(x)^2 - a^2 - 2*a*b - b^2 + 2
*(3*b^2*cosh(x)^5 - 6*b^2*cosh(x)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(x))*sinh(x))*sqrt(a + b)*sqrt(((a + b)*cosh(x
)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*(2*(a*b^2 + b^3)*cosh(x)^7 -
3*(a*b^2 + 2*b^3)*cosh(x)^5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^3 + (a^3 - 3*a*b^2 - 2*b^3)*cosh(x))*si
nh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x)^4*sinh(x)^2 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(
x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) - ((a^2 - a*b - 2*b^2)*cosh(x)^4 + 4*(a^2 - a*b - 2*b^2)*cosh(x)*sinh
(x)^3 + (a^2 - a*b - 2*b^2)*sinh(x)^4 + 2*(a^2 - a*b - 2*b^2)*cosh(x)^2 + 2*(3*(a^2 - a*b - 2*b^2)*cosh(x)^2 +
a^2 - a*b - 2*b^2)*sinh(x)^2 + a^2 - a*b - 2*b^2 + 4*((a^2 - a*b - 2*b^2)*cosh(x)^3 + (a^2 - a*b - 2*b^2)*cos
h(x))*sinh(x))*sqrt(b)*log(-((a + 2*b)*cosh(x)^4 + 4*(a + 2*b)*cosh(x)*sinh(x)^3 + (a + 2*b)*sinh(x)^4 + 2*(a
- 2*b)*cosh(x)^2 + 2*(3*(a + 2*b)*cosh(x)^2 + a - 2*b)*sinh(x)^2 - 2*sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) +
sinh(x)^2 - 1)*sqrt(b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + s
inh(x)^2)) + 4*((a + 2*b)*cosh(x)^3 + (a - 2*b)*cosh(x))*sinh(x) + a + 2*b)/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 +
sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(x) + 1)) + (b^2*cosh(x
)^4 + 4*b^2*cosh(x)*sinh(x)^3 + b^2*sinh(x)^4 + 2*b^2*cosh(x)^2 + 2*(3*b^2*cosh(x)^2 + b^2)*sinh(x)^2 + b^2 +
4*(b^2*cosh(x)^3 + b^2*cosh(x))*sinh(x))*sqrt(a + b)*log(((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a
+ b)*sinh(x)^4 + 2*a*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 + a)*sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(
x) + sinh(x)^2 + 1)*sqrt(a + b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*si
nh(x) + sinh(x)^2)) + 4*((a + b)*cosh(x)^3 + a*cosh(x))*sinh(x) + a + b)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh
(x)^2)) - 2*sqrt(2)*((a*b + b^2)*cosh(x)^2 + 2*(a*b + b^2)*cosh(x)*sinh(x) + (a*b + b^2)*sinh(x)^2 - a*b - b^2
)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/((a*b^2 +
b^3)*cosh(x)^4 + 4*(a*b^2 + b^3)*cosh(x)*sinh(x)^3 + (a*b^2 + b^3)*sinh(x)^4 + a*b^2 + b^3 + 2*(a*b^2 + b^3)*
cosh(x)^2 + 2*(a*b^2 + b^3 + 3*(a*b^2 + b^3)*cosh(x)^2)*sinh(x)^2 + 4*((a*b^2 + b^3)*cosh(x)^3 + (a*b^2 + b^3)
*cosh(x))*sinh(x)), -1/4*(2*((a^2 - a*b - 2*b^2)*cosh(x)^4 + 4*(a^2 - a*b - 2*b^2)*cosh(x)*sinh(x)^3 + (a^2 -
a*b - 2*b^2)*sinh(x)^4 + 2*(a^2 - a*b - 2*b^2)*cosh(x)^2 + 2*(3*(a^2 - a*b - 2*b^2)*cosh(x)^2 + a^2 - a*b - 2*
b^2)*sinh(x)^2 + a^2 - a*b - 2*b^2 + 4*((a^2 - a*b - 2*b^2)*cosh(x)^3 + (a^2 - a*b - 2*b^2)*cosh(x))*sinh(x))*
sqrt(-b)*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-b)*sqrt(((a + b)*cosh(x)^2 + (a
+ b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*si
nh(x)^3 + (a + b)*sinh(x)^4 + 2*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 + a - b)*sinh(x)^2 + 4*((a + b)*cos
h(x)^3 + (a - b)*cosh(x))*sinh(x) + a + b)) - (b^2*cosh(x)^4 + 4*b^2*cosh(x)*sinh(x)^3 + b^2*sinh(x)^4 + 2*b^2
*cosh(x)^2 + 2*(3*b^2*cosh(x)^2 + b^2)*sinh(x)^2 + b^2 + 4*(b^2*cosh(x)^3 + b^2*cosh(x))*sinh(x))*sqrt(a + b)*
log(-((a*b^2 + b^3)*cosh(x)^8 + 8*(a*b^2 + b^3)*cosh(x)*sinh(x)^7 + (a*b^2 + b^3)*sinh(x)^8 - 2*(a*b^2 + 2*b^3
)*cosh(x)^6 - 2*(a*b^2 + 2*b^3 - 14*(a*b^2 + b^3)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a*b^2 + b^3)*cosh(x)^3 - 3*(a*
b^2 + 2*b^3)*cosh(x))*sinh(x)^5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^4 + (70*(a*b^2 + b^3)*cosh(x)^4 + a^
3 - a^2*b + 4*a*b^2 + 6*b^3 - 30*(a*b^2 + 2*b^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a*b^2 + b^3)*cosh(x)^5 - 10*(a*
b^2 + 2*b^3)*cosh(x)^3 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x))*sinh(x)^3 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 +
2*(a^3 - 3*a*b^2 - 2*b^3)*cosh(x)^2 + 2*(14*(a*b^2 + b^3)*cosh(x)^6 - 15*(a*b^2 + 2*b^3)*cosh(x)^4 + a^3 - 3*a
*b^2 - 2*b^3 + 3*(a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(b^2*cosh(x)^6 + 6*b^2*cosh(x)
*sinh(x)^5 + b^2*sinh(x)^6 - 3*b^2*cosh(x)^4 + 3*(5*b^2*cosh(x)^2 - b^2)*sinh(x)^4 + 4*(5*b^2*cosh(x)^3 - 3*b^
2*cosh(x))*sinh(x)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(x)^2 + (15*b^2*cosh(x)^4 - 18*b^2*cosh(x)^2 - a^2 + 2*a*b +
3*b^2)*sinh(x)^2 - a^2 - 2*a*b - b^2 + 2*(3*b^2*cosh(x)^5 - 6*b^2*cosh(x)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(x))*s
inh(x))*sqrt(a + b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh
(x)^2)) + 4*(2*(a*b^2 + b^3)*cosh(x)^7 - 3*(a*b^2 + 2*b^3)*cosh(x)^5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)
^3 + (a^3 - 3*a*b^2 - 2*b^3)*cosh(x))*sinh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x)^4*sinh(x)^2 + 20*
cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) - (b^2*cosh(x)^4 + 4*b^2*cosh
(x)*sinh(x)^3 + b^2*sinh(x)^4 + 2*b^2*cosh(x)^2 + 2*(3*b^2*cosh(x)^2 + b^2)*sinh(x)^2 + b^2 + 4*(b^2*cosh(x)^3
+ b^2*cosh(x))*sinh(x))*sqrt(a + b)*log(((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4
+ 2*a*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 + a)*sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 +
1)*sqrt(a + b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^
2)) + 4*((a + b)*cosh(x)^3 + a*cosh(x))*sinh(x) + a + b)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) + 2*sqrt
(2)*((a*b + b^2)*cosh(x)^2 + 2*(a*b + b^2)*cosh(x)*sinh(x) + (a*b + b^2)*sinh(x)^2 - a*b - b^2)*sqrt(((a + b)*
cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/((a*b^2 + b^3)*cosh(x)^4
+ 4*(a*b^2 + b^3)*cosh(x)*sinh(x)^3 + (a*b^2 + b^3)*sinh(x)^4 + a*b^2 + b^3 + 2*(a*b^2 + b^3)*cosh(x)^2 + 2*(a
*b^2 + b^3 + 3*(a*b^2 + b^3)*cosh(x)^2)*sinh(x)^2 + 4*((a*b^2 + b^3)*cosh(x)^3 + (a*b^2 + b^3)*cosh(x))*sinh(x
)), -1/4*(2*(b^2*cosh(x)^4 + 4*b^2*cosh(x)*sinh(x)^3 + b^2*sinh(x)^4 + 2*b^2*cosh(x)^2 + 2*(3*b^2*cosh(x)^2 +
b^2)*sinh(x)^2 + b^2 + 4*(b^2*cosh(x)^3 + b^2*cosh(x))*sinh(x))*sqrt(-a - b)*arctan(sqrt(2)*(b*cosh(x)^2 + 2*b
*cosh(x)*sinh(x) + b*sinh(x)^2 - a - b)*sqrt(-a - b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cos
h(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/((a*b + b^2)*cosh(x)^4 + 4*(a*b + b^2)*cosh(x)*sinh(x)^3 + (a*b + b^2
)*sinh(x)^4 + (a^2 - a*b - 2*b^2)*cosh(x)^2 + (6*(a*b + b^2)*cosh(x)^2 + a^2 - a*b - 2*b^2)*sinh(x)^2 + a^2 +
2*a*b + b^2 + 2*(2*(a*b + b^2)*cosh(x)^3 + (a^2 - a*b - 2*b^2)*cosh(x))*sinh(x))) + 2*(b^2*cosh(x)^4 + 4*b^2*c
osh(x)*sinh(x)^3 + b^2*sinh(x)^4 + 2*b^2*cosh(x)^2 + 2*(3*b^2*cosh(x)^2 + b^2)*sinh(x)^2 + b^2 + 4*(b^2*cosh(x
)^3 + b^2*cosh(x))*sinh(x))*sqrt(-a - b)*arctan(sqrt(2)*sqrt(-a - b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)
^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a +
b)*sinh(x)^2 + a + b)) + ((a^2 - a*b - 2*b^2)*cosh(x)^4 + 4*(a^2 - a*b - 2*b^2)*cosh(x)*sinh(x)^3 + (a^2 - a*
b - 2*b^2)*sinh(x)^4 + 2*(a^2 - a*b - 2*b^2)*cosh(x)^2 + 2*(3*(a^2 - a*b - 2*b^2)*cosh(x)^2 + a^2 - a*b - 2*b^
2)*sinh(x)^2 + a^2 - a*b - 2*b^2 + 4*((a^2 - a*b - 2*b^2)*cosh(x)^3 + (a^2 - a*b - 2*b^2)*cosh(x))*sinh(x))*sq
rt(b)*log(-((a + 2*b)*cosh(x)^4 + 4*(a + 2*b)*cosh(x)*sinh(x)^3 + (a + 2*b)*sinh(x)^4 + 2*(a - 2*b)*cosh(x)^2
+ 2*(3*(a + 2*b)*cosh(x)^2 + a - 2*b)*sinh(x)^2 - 2*sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sq
rt(b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*((
a + 2*b)*cosh(x)^3 + (a - 2*b)*cosh(x))*sinh(x) + a + 2*b)/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3
*cosh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(x) + 1)) + 2*sqrt(2)*((a*b + b^2)*cosh(
x)^2 + 2*(a*b + b^2)*cosh(x)*sinh(x) + (a*b + b^2)*sinh(x)^2 - a*b - b^2)*sqrt(((a + b)*cosh(x)^2 + (a + b)*si
nh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/((a*b^2 + b^3)*cosh(x)^4 + 4*(a*b^2 + b^3)*cosh
(x)*sinh(x)^3 + (a*b^2 + b^3)*sinh(x)^4 + a*b^2 + b^3 + 2*(a*b^2 + b^3)*cosh(x)^2 + 2*(a*b^2 + b^3 + 3*(a*b^2
+ b^3)*cosh(x)^2)*sinh(x)^2 + 4*((a*b^2 + b^3)*cosh(x)^3 + (a*b^2 + b^3)*cosh(x))*sinh(x)), -1/2*((b^2*cosh(x)
^4 + 4*b^2*cosh(x)*sinh(x)^3 + b^2*sinh(x)^4 + 2*b^2*cosh(x)^2 + 2*(3*b^2*cosh(x)^2 + b^2)*sinh(x)^2 + b^2 + 4
*(b^2*cosh(x)^3 + b^2*cosh(x))*sinh(x))*sqrt(-a - b)*arctan(sqrt(2)*(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sin
h(x)^2 - a - b)*sqrt(-a - b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(
x) + sinh(x)^2))/((a*b + b^2)*cosh(x)^4 + 4*(a*b + b^2)*cosh(x)*sinh(x)^3 + (a*b + b^2)*sinh(x)^4 + (a^2 - a*b
- 2*b^2)*cosh(x)^2 + (6*(a*b + b^2)*cosh(x)^2 + a^2 - a*b - 2*b^2)*sinh(x)^2 + a^2 + 2*a*b + b^2 + 2*(2*(a*b
+ b^2)*cosh(x)^3 + (a^2 - a*b - 2*b^2)*cosh(x))*sinh(x))) + ((a^2 - a*b - 2*b^2)*cosh(x)^4 + 4*(a^2 - a*b - 2*
b^2)*cosh(x)*sinh(x)^3 + (a^2 - a*b - 2*b^2)*sinh(x)^4 + 2*(a^2 - a*b - 2*b^2)*cosh(x)^2 + 2*(3*(a^2 - a*b - 2
*b^2)*cosh(x)^2 + a^2 - a*b - 2*b^2)*sinh(x)^2 + a^2 - a*b - 2*b^2 + 4*((a^2 - a*b - 2*b^2)*cosh(x)^3 + (a^2 -
a*b - 2*b^2)*cosh(x))*sinh(x))*sqrt(-b)*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-
b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/((a + b)*
cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 + 2*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 + a
- b)*sinh(x)^2 + 4*((a + b)*cosh(x)^3 + (a - b)*cosh(x))*sinh(x) + a + b)) + (b^2*cosh(x)^4 + 4*b^2*cosh(x)*s
inh(x)^3 + b^2*sinh(x)^4 + 2*b^2*cosh(x)^2 + 2*(3*b^2*cosh(x)^2 + b^2)*sinh(x)^2 + b^2 + 4*(b^2*cosh(x)^3 + b^
2*cosh(x))*sinh(x))*sqrt(-a - b)*arctan(sqrt(2)*sqrt(-a - b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a -
b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh
(x)^2 + a + b)) + sqrt(2)*((a*b + b^2)*cosh(x)^2 + 2*(a*b + b^2)*cosh(x)*sinh(x) + (a*b + b^2)*sinh(x)^2 - a*b
- b^2)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/((a
*b^2 + b^3)*cosh(x)^4 + 4*(a*b^2 + b^3)*cosh(x)*sinh(x)^3 + (a*b^2 + b^3)*sinh(x)^4 + a*b^2 + b^3 + 2*(a*b^2 +
b^3)*cosh(x)^2 + 2*(a*b^2 + b^3 + 3*(a*b^2 + b^3)*cosh(x)^2)*sinh(x)^2 + 4*((a*b^2 + b^3)*cosh(x)^3 + (a*b^2
+ b^3)*cosh(x))*sinh(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(tanh(x)^4/(a+b*tanh(x)^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+1)]Evaluation time: 0.6Error: Bad Argument Type
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maple [B] time = 0.13, size = 178, normalized size = 2.02 \[ -\frac {\sqrt {a +b \left (\tanh ^{2}\relax (x )\right )}\, \tanh \relax (x )}{2 b}+\frac {a \ln \left (\sqrt {b}\, \tanh \relax (x )+\sqrt {a +b \left (\tanh ^{2}\relax (x )\right )}\right )}{2 b^{\frac {3}{2}}}-\frac {\ln \left (\sqrt {b}\, \tanh \relax (x )+\sqrt {a +b \left (\tanh ^{2}\relax (x )\right )}\right )}{\sqrt {b}}+\frac {\ln \left (\frac {2 a +2 b +2 \left (\tanh \relax (x )-1\right ) b +2 \sqrt {a +b}\, \sqrt {\left (\tanh \relax (x )-1\right )^{2} b +2 \left (\tanh \relax (x )-1\right ) b +a +b}}{\tanh \relax (x )-1}\right )}{2 \sqrt {a +b}}-\frac {\ln \left (\frac {2 a +2 b -2 \left (1+\tanh \relax (x )\right ) b +2 \sqrt {a +b}\, \sqrt {\left (1+\tanh \relax (x )\right )^{2} b -2 \left (1+\tanh \relax (x )\right ) b +a +b}}{1+\tanh \relax (x )}\right )}{2 \sqrt {a +b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(x)^4/(a+b*tanh(x)^2)^(1/2),x)
[Out]
-1/2*(a+b*tanh(x)^2)^(1/2)*tanh(x)/b+1/2*a/b^(3/2)*ln(b^(1/2)*tanh(x)+(a+b*tanh(x)^2)^(1/2))-ln(b^(1/2)*tanh(x
)+(a+b*tanh(x)^2)^(1/2))/b^(1/2)+1/2/(a+b)^(1/2)*ln((2*a+2*b+2*(tanh(x)-1)*b+2*(a+b)^(1/2)*((tanh(x)-1)^2*b+2*
(tanh(x)-1)*b+a+b)^(1/2))/(tanh(x)-1))-1/2/(a+b)^(1/2)*ln((2*a+2*b-2*(1+tanh(x))*b+2*(a+b)^(1/2)*((1+tanh(x))^
2*b-2*(1+tanh(x))*b+a+b)^(1/2))/(1+tanh(x)))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh \relax (x)^{4}}{\sqrt {b \tanh \relax (x)^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^4/(a+b*tanh(x)^2)^(1/2),x, algorithm="maxima")
[Out]
integrate(tanh(x)^4/sqrt(b*tanh(x)^2 + a), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {tanh}\relax (x)}^4}{\sqrt {b\,{\mathrm {tanh}\relax (x)}^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(x)^4/(a + b*tanh(x)^2)^(1/2),x)
[Out]
int(tanh(x)^4/(a + b*tanh(x)^2)^(1/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{4}{\relax (x )}}{\sqrt {a + b \tanh ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)**4/(a+b*tanh(x)**2)**(1/2),x)
[Out]
Integral(tanh(x)**4/sqrt(a + b*tanh(x)**2), x)
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