3.3.46 \(\int \frac {\tanh ^6(x)}{(a+b \tanh ^2(x))^{5/2}} \, dx\) [246]

Optimal. Leaf size=118 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{b^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {a \tanh ^3(x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {a (a+2 b) \tanh (x)}{b^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}} \]

[Out]

-arctanh(b^(1/2)*tanh(x)/(a+b*tanh(x)^2)^(1/2))/b^(5/2)+arctanh((a+b)^(1/2)*tanh(x)/(a+b*tanh(x)^2)^(1/2))/(a+
b)^(5/2)+a*(a+2*b)*tanh(x)/b^2/(a+b)^2/(a+b*tanh(x)^2)^(1/2)+1/3*a*tanh(x)^3/b/(a+b)/(a+b*tanh(x)^2)^(3/2)

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Rubi [A]
time = 0.15, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3751, 481, 592, 537, 223, 212, 385} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{b^{5/2}}+\frac {a (a+2 b) \tanh (x)}{b^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {a \tanh ^3(x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^6/(a + b*Tanh[x]^2)^(5/2),x]

[Out]

-(ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/b^(5/2)) + ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]
^2]]/(a + b)^(5/2) + (a*Tanh[x]^3)/(3*b*(a + b)*(a + b*Tanh[x]^2)^(3/2)) + (a*(a + 2*b)*Tanh[x])/(b^2*(a + b)^
2*Sqrt[a + b*Tanh[x]^2])

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 223

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 385

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 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 537

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 592

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[g^(n - 1)*(b*e - a*f)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c -
a*d)*(p + 1))), x] - Dist[g^n/(b*n*(b*c - a*d)*(p + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*S
imp[c*(b*e - a*f)*(m - n + 1) + (d*(b*e - a*f)*(m + n*q + 1) - b*n*(c*f - d*e)*(p + 1))*x^n, x], x], x] /; Fre
eQ[{a, b, c, d, e, f, g, q}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, 0]

Rule 3751

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 ^6(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx &=\text {Subst}\left (\int \frac {x^6}{\left (1-x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=\frac {a \tanh ^3(x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {x^2 \left (3 a-3 (a+b) x^2\right )}{\left (1-x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )}{3 b (a+b)}\\ &=\frac {a \tanh ^3(x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {a (a+2 b) \tanh (x)}{b^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}-\frac {\text {Subst}\left (\int \frac {3 a (a+2 b)-3 (a+b)^2 x^2}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )}{3 b^2 (a+b)^2}\\ &=\frac {a \tanh ^3(x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {a (a+2 b) \tanh (x)}{b^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )}{b^2}+\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )}{(a+b)^2}\\ &=\frac {a \tanh ^3(x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {a (a+2 b) \tanh (x)}{b^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}-\frac {\text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{b^2}+\frac {\text {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^2}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{b^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {a \tanh ^3(x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {a (a+2 b) \tanh (x)}{b^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 1.45, size = 231, normalized size = 1.96 \begin {gather*} \frac {\sqrt {(a-b+(a+b) \cosh (2 x)) \text {sech}^2(x)} \left (-\frac {3 \sqrt {2} a \coth (x) \left (\left (a^2+3 a b+2 b^2\right ) F\left (\left .\text {ArcSin}\left (\frac {\sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}{\sqrt {2}}\right )\right |1\right )+b^2 \Pi \left (\frac {b}{a+b};\left .\text {ArcSin}\left (\frac {\sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}{\sqrt {2}}\right )\right |1\right )\right )}{b \sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}+\frac {a (a+b) \left (3 a^2+2 a b-7 b^2+\left (3 a^2+10 a b+7 b^2\right ) \cosh (2 x)\right ) \sinh (2 x)}{(a-b+(a+b) \cosh (2 x))^2}\right )}{3 \sqrt {2} b^2 (a+b)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]^6/(a + b*Tanh[x]^2)^(5/2),x]

[Out]

(Sqrt[(a - b + (a + b)*Cosh[2*x])*Sech[x]^2]*((-3*Sqrt[2]*a*Coth[x]*((a^2 + 3*a*b + 2*b^2)*EllipticF[ArcSin[Sq
rt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1] + b^2*EllipticPi[b/(a + b), ArcSin[Sqrt[((a - b + (
a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1]))/(b*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]) + (a*(a + b
)*(3*a^2 + 2*a*b - 7*b^2 + (3*a^2 + 10*a*b + 7*b^2)*Cosh[2*x])*Sinh[2*x])/(a - b + (a + b)*Cosh[2*x])^2))/(3*S
qrt[2]*b^2*(a + b)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(705\) vs. \(2(100)=200\).
time = 0.73, size = 706, normalized size = 5.98 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^6/(a+b*tanh(x)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/3*tanh(x)^3/b/(a+b*tanh(x)^2)^(3/2)-1/b*(-tanh(x)/b/(a+b*tanh(x)^2)^(1/2)+1/b^(3/2)*ln(b^(1/2)*tanh(x)+(a+b*
tanh(x)^2)^(1/2)))+1/2*tanh(x)/b/(a+b*tanh(x)^2)^(3/2)-1/2*a/b*(1/3*tanh(x)/a/(a+b*tanh(x)^2)^(3/2)+2/3/a^2*ta
nh(x)/(a+b*tanh(x)^2)^(1/2))-1/3*tanh(x)/a/(a+b*tanh(x)^2)^(3/2)-2/3/a^2*tanh(x)/(a+b*tanh(x)^2)^(1/2)-1/6/(a+
b)/(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(3/2)+1/2*b/(a+b)*(2/3*(2*b*(tanh(x)-1)+2*b)/(4*b*(a+b)-4*b^2)/(b*(ta
nh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(3/2)+16/3*b/(4*b*(a+b)-4*b^2)^2*(2*b*(tanh(x)-1)+2*b)/(b*(tanh(x)-1)^2+2*b*(t
anh(x)-1)+a+b)^(1/2))-1/2/(a+b)*(1/(a+b)/(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(1/2)-2*b/(a+b)*(2*b*(tanh(x)-1
)+2*b)/(4*b*(a+b)-4*b^2)/(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(1/2)-1/(a+b)^(3/2)*ln((2*a+2*b+2*b*(tanh(x)-1)
+2*(a+b)^(1/2)*(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(1/2))/(tanh(x)-1)))+1/6/(a+b)/(b*(1+tanh(x))^2-2*b*(1+ta
nh(x))+a+b)^(3/2)+1/2*b/(a+b)*(2/3*(2*b*(1+tanh(x))-2*b)/(4*b*(a+b)-4*b^2)/(b*(1+tanh(x))^2-2*b*(1+tanh(x))+a+
b)^(3/2)+16/3*b/(4*b*(a+b)-4*b^2)^2*(2*b*(1+tanh(x))-2*b)/(b*(1+tanh(x))^2-2*b*(1+tanh(x))+a+b)^(1/2))+1/2/(a+
b)*(1/(a+b)/(b*(1+tanh(x))^2-2*b*(1+tanh(x))+a+b)^(1/2)+2*b/(a+b)*(2*b*(1+tanh(x))-2*b)/(4*b*(a+b)-4*b^2)/(b*(
1+tanh(x))^2-2*b*(1+tanh(x))+a+b)^(1/2)-1/(a+b)^(3/2)*ln((2*a+2*b-2*b*(1+tanh(x))+2*(a+b)^(1/2)*(b*(1+tanh(x))
^2-2*b*(1+tanh(x))+a+b)^(1/2))/(1+tanh(x))))

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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(tanh(x)^6/(a+b*tanh(x)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(tanh(x)^6/(b*tanh(x)^2 + a)^(5/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4472 vs. \(2 (100) = 200\).
time = 1.13, size = 19265, normalized size = 163.26 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^6/(a+b*tanh(x)^2)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(3*((a^2*b^3 + 2*a*b^4 + b^5)*cosh(x)^8 + 8*(a^2*b^3 + 2*a*b^4 + b^5)*cosh(x)*sinh(x)^7 + (a^2*b^3 + 2*a
*b^4 + b^5)*sinh(x)^8 + 4*(a^2*b^3 - b^5)*cosh(x)^6 + 4*(a^2*b^3 - b^5 + 7*(a^2*b^3 + 2*a*b^4 + b^5)*cosh(x)^2
)*sinh(x)^6 + 8*(7*(a^2*b^3 + 2*a*b^4 + b^5)*cosh(x)^3 + 3*(a^2*b^3 - b^5)*cosh(x))*sinh(x)^5 + a^2*b^3 + 2*a*
b^4 + b^5 + 2*(3*a^2*b^3 - 2*a*b^4 + 3*b^5)*cosh(x)^4 + 2*(3*a^2*b^3 - 2*a*b^4 + 3*b^5 + 35*(a^2*b^3 + 2*a*b^4
+ b^5)*cosh(x)^4 + 30*(a^2*b^3 - b^5)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^2*b^3 + 2*a*b^4 + b^5)*cosh(x)^5 + 10*(a
^2*b^3 - b^5)*cosh(x)^3 + (3*a^2*b^3 - 2*a*b^4 + 3*b^5)*cosh(x))*sinh(x)^3 + 4*(a^2*b^3 - b^5)*cosh(x)^2 + 4*(
7*(a^2*b^3 + 2*a*b^4 + b^5)*cosh(x)^6 + a^2*b^3 - b^5 + 15*(a^2*b^3 - b^5)*cosh(x)^4 + 3*(3*a^2*b^3 - 2*a*b^4
+ 3*b^5)*cosh(x)^2)*sinh(x)^2 + 8*((a^2*b^3 + 2*a*b^4 + b^5)*cosh(x)^7 + 3*(a^2*b^3 - b^5)*cosh(x)^5 + (3*a^2*
b^3 - 2*a*b^4 + 3*b^5)*cosh(x)^3 + (a^2*b^3 - b^5)*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)*cos
h(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))*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)) + 6*((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 +
b^5)*cosh(x)^8 + 8*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(x)*sinh(x)^7 + (a^5 + 5*a^4*
b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*sinh(x)^8 + 4*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 -
b^5)*cosh(x)^6 + 4*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5 + 7*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10
*a^2*b^3 + 5*a*b^4 + b^5)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5
)*cosh(x)^3 + 3*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x))*sinh(x)^5 + a^5 + 5*a^4*b + 1
0*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5 + 2*(3*a^5 + 7*a^4*b + 6*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(x)
^4 + 2*(3*a^5 + 7*a^4*b + 6*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*b^5 + 35*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^
3 + 5*a*b^4 + b^5)*cosh(x)^4 + 30*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x)^2)*sinh(x)^4
+ 8*(7*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(x)^5 + 10*(a^5 + 3*a^4*b + 2*a^3*b^2 -
2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x)^3 + (3*a^5 + 7*a^4*b + 6*a^3*b^2 + 6*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(x))*si
nh(x)^3 + 4*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x)^2 + 4*(7*(a^5 + 5*a^4*b + 10*a^3*b
^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(x)^6 + a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5 + 15*(a^5 +
3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x)^4 + 3*(3*a^5 + 7*a^4*b + 6*a^3*b^2 + 6*a^2*b^3 + 7*a
*b^4 + 3*b^5)*cosh(x)^2)*sinh(x)^2 + 8*((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*cosh(x)^7 +
3*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(x)^5 + (3*a^5 + 7*a^4*b + 6*a^3*b^2 + 6*a^2*b^3
+ 7*a*b^4 + 3*b^5)*cosh(x)^3 + (a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*cosh(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) + 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*c
osh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(x) + 1)) + 3*((a^2*b^3 + 2*a*b^4 + b^5)*c
osh(x)^8 + 8*(a^2*b^3 + 2*a*b^4 + b^5)*cosh(x)*sinh(x)^7 + (a^2*b^3 + 2*a*b^4 + b^5)*sinh(x)^8 + 4*(a^2*b^3 -
b^5)*cosh(x)^6 + 4*(a^2*b^3 - b^5 + 7*(a^2*b^3 ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{6}{\left (x \right )}}{\left (a + b \tanh ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)**6/(a+b*tanh(x)**2)**(5/2),x)

[Out]

Integral(tanh(x)**6/(a + b*tanh(x)**2)**(5/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 829 vs. \(2 (100) = 200\).
time = 0.77, size = 829, normalized size = 7.03 \begin {gather*} -\frac {\sqrt {a + b} \log \left ({\left | -{\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} {\left (a + b\right )} - \sqrt {a + b} {\left (a - b\right )} \right |}\right )}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {\sqrt {a + b} \log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} + \sqrt {a + b} \right |}\right )}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {\sqrt {a + b} \log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} - \sqrt {a + b} \right |}\right )}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {{\left ({\left (\frac {{\left (3 \, a^{9} b^{8} + 22 \, a^{8} b^{9} + 65 \, a^{7} b^{10} + 100 \, a^{6} b^{11} + 85 \, a^{5} b^{12} + 38 \, a^{4} b^{13} + 7 \, a^{3} b^{14}\right )} e^{\left (2 \, x\right )}}{a^{8} b^{10} + 6 \, a^{7} b^{11} + 15 \, a^{6} b^{12} + 20 \, a^{5} b^{13} + 15 \, a^{4} b^{14} + 6 \, a^{3} b^{15} + a^{2} b^{16}} + \frac {3 \, {\left (a^{9} b^{8} + 2 \, a^{8} b^{9} - 9 \, a^{7} b^{10} - 36 \, a^{6} b^{11} - 49 \, a^{5} b^{12} - 30 \, a^{4} b^{13} - 7 \, a^{3} b^{14}\right )}}{a^{8} b^{10} + 6 \, a^{7} b^{11} + 15 \, a^{6} b^{12} + 20 \, a^{5} b^{13} + 15 \, a^{4} b^{14} + 6 \, a^{3} b^{15} + a^{2} b^{16}}\right )} e^{\left (2 \, x\right )} - \frac {3 \, {\left (a^{9} b^{8} + 2 \, a^{8} b^{9} - 9 \, a^{7} b^{10} - 36 \, a^{6} b^{11} - 49 \, a^{5} b^{12} - 30 \, a^{4} b^{13} - 7 \, a^{3} b^{14}\right )}}{a^{8} b^{10} + 6 \, a^{7} b^{11} + 15 \, a^{6} b^{12} + 20 \, a^{5} b^{13} + 15 \, a^{4} b^{14} + 6 \, a^{3} b^{15} + a^{2} b^{16}}\right )} e^{\left (2 \, x\right )} - \frac {3 \, a^{9} b^{8} + 22 \, a^{8} b^{9} + 65 \, a^{7} b^{10} + 100 \, a^{6} b^{11} + 85 \, a^{5} b^{12} + 38 \, a^{4} b^{13} + 7 \, a^{3} b^{14}}{a^{8} b^{10} + 6 \, a^{7} b^{11} + 15 \, a^{6} b^{12} + 20 \, a^{5} b^{13} + 15 \, a^{4} b^{14} + 6 \, a^{3} b^{15} + a^{2} b^{16}}}{3 \, {\left (a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b\right )}^{\frac {3}{2}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} + \sqrt {a + b}}{2 \, \sqrt {-b}}\right )}{\sqrt {-b} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^6/(a+b*tanh(x)^2)^(5/2),x, algorithm="giac")

[Out]

-1/2*sqrt(a + b)*log(abs(-(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a +
b))*(a + b) - sqrt(a + b)*(a - b)))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 1/2*sqrt(a + b)*log(abs(-sqrt(a + b)*e^(
2*x) + sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b) + sqrt(a + b)))/(a^3 + 3*a^2*b + 3*a*b^
2 + b^3) + 1/2*sqrt(a + b)*log(abs(-sqrt(a + b)*e^(2*x) + sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*
x) + a + b) - sqrt(a + b)))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 1/3*((((3*a^9*b^8 + 22*a^8*b^9 + 65*a^7*b^10 + 1
00*a^6*b^11 + 85*a^5*b^12 + 38*a^4*b^13 + 7*a^3*b^14)*e^(2*x)/(a^8*b^10 + 6*a^7*b^11 + 15*a^6*b^12 + 20*a^5*b^
13 + 15*a^4*b^14 + 6*a^3*b^15 + a^2*b^16) + 3*(a^9*b^8 + 2*a^8*b^9 - 9*a^7*b^10 - 36*a^6*b^11 - 49*a^5*b^12 -
30*a^4*b^13 - 7*a^3*b^14)/(a^8*b^10 + 6*a^7*b^11 + 15*a^6*b^12 + 20*a^5*b^13 + 15*a^4*b^14 + 6*a^3*b^15 + a^2*
b^16))*e^(2*x) - 3*(a^9*b^8 + 2*a^8*b^9 - 9*a^7*b^10 - 36*a^6*b^11 - 49*a^5*b^12 - 30*a^4*b^13 - 7*a^3*b^14)/(
a^8*b^10 + 6*a^7*b^11 + 15*a^6*b^12 + 20*a^5*b^13 + 15*a^4*b^14 + 6*a^3*b^15 + a^2*b^16))*e^(2*x) - (3*a^9*b^8
+ 22*a^8*b^9 + 65*a^7*b^10 + 100*a^6*b^11 + 85*a^5*b^12 + 38*a^4*b^13 + 7*a^3*b^14)/(a^8*b^10 + 6*a^7*b^11 +
15*a^6*b^12 + 20*a^5*b^13 + 15*a^4*b^14 + 6*a^3*b^15 + a^2*b^16))/(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e
^(2*x) + a + b)^(3/2) - 2*arctan(-1/2*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^
(2*x) + a + b) + sqrt(a + b))/sqrt(-b))/(sqrt(-b)*b^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {tanh}\left (x\right )}^6}{{\left (b\,{\mathrm {tanh}\left (x\right )}^2+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^6/(a + b*tanh(x)^2)^(5/2),x)

[Out]

int(tanh(x)^6/(a + b*tanh(x)^2)^(5/2), x)

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