3.3.50 \(\int \frac {\tanh ^2(x)}{(a+b \tanh ^2(x))^{5/2}} \, dx\) [250]
Optimal. Leaf size=88 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}-\frac {\tanh (x)}{3 (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac {(2 a-b) \tanh (x)}{3 a (a+b)^2 \sqrt {a+b \tanh ^2(x)}} \]
[Out]
arctanh((a+b)^(1/2)*tanh(x)/(a+b*tanh(x)^2)^(1/2))/(a+b)^(5/2)-1/3*(2*a-b)*tanh(x)/a/(a+b)^2/(a+b*tanh(x)^2)^(
1/2)-1/3*tanh(x)/(a+b)/(a+b*tanh(x)^2)^(3/2)
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Rubi [A]
time = 0.09, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3751, 482, 541,
12, 385, 212} \begin {gather*} -\frac {(2 a-b) \tanh (x)}{3 a (a+b)^2 \sqrt {a+b \tanh ^2(x)}}-\frac {\tanh (x)}{3 (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Tanh[x]^2/(a + b*Tanh[x]^2)^(5/2),x]
[Out]
ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(5/2) - Tanh[x]/(3*(a + b)*(a + b*Tanh[x]^2)^(3/2
)) - ((2*a - b)*Tanh[x])/(3*a*(a + b)^2*Sqrt[a + b*Tanh[x]^2])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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 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 482
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 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] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, 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 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 ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx &=\text {Subst}\left (\int \frac {x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=-\frac {\tanh (x)}{3 (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {1+2 x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )}{3 (a+b)}\\ &=-\frac {\tanh (x)}{3 (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac {(2 a-b) \tanh (x)}{3 a (a+b)^2 \sqrt {a+b \tanh ^2(x)}}-\frac {\text {Subst}\left (\int -\frac {3 a}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )}{3 a (a+b)^2}\\ &=-\frac {\tanh (x)}{3 (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac {(2 a-b) \tanh (x)}{3 a (a+b)^2 \sqrt {a+b \tanh ^2(x)}}+\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 {\tanh (x)}{3 (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac {(2 a-b) \tanh (x)}{3 a (a+b)^2 \sqrt {a+b \tanh ^2(x)}}+\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 {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}-\frac {\tanh (x)}{3 (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac {(2 a-b) \tanh (x)}{3 a (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 5 vs. order 3 in
optimal.
time = 5.56, size = 193, normalized size = 2.19 \begin {gather*} \frac {\coth (x) \left (-12 (a+b)^3 \, _2F_1\left (2,2;\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \sinh ^4(x) \tanh ^2(x) \left (a+b \tanh ^2(x)\right )-\frac {35 a \cosh ^2(x) \left (-5 a-2 b \tanh ^2(x)\right ) \left (3 \text {ArcSin}\left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \left (a+b \tanh ^2(x)\right )^2-a \text {sech}^2(x) \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}} \left (3 a+(a+4 b) \tanh ^2(x)\right )\right )}{\sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}}\right )}{315 a^3 (a+b)^2 \left (a+b \tanh ^2(x)\right )^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[Tanh[x]^2/(a + b*Tanh[x]^2)^(5/2),x]
[Out]
(Coth[x]*(-12*(a + b)^3*Hypergeometric2F1[2, 2, 9/2, -(((a + b)*Sinh[x]^2)/a)]*Sinh[x]^4*Tanh[x]^2*(a + b*Tanh
[x]^2) - (35*a*Cosh[x]^2*(-5*a - 2*b*Tanh[x]^2)*(3*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*(a + b*Tanh[x]^2)^2
- a*Sech[x]^2*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)]*(3*a + (a + 4*b)*Tanh[x]^2)))/Sqrt[
-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)]))/(315*a^3*(a + b)^2*(a + b*Tanh[x]^2)^(3/2))
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(583\) vs.
\(2(74)=148\).
time = 0.66, size = 584, normalized size = 6.64 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(x)^2/(a+b*tanh(x)^2)^(5/2),x,method=_RETURNVERBOSE)
[Out]
-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*(1+tanh(x))^2-2*b*(1+t
anh(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))))-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*(tanh(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*(tanh(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)))
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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)^2/(a+b*tanh(x)^2)^(5/2),x, algorithm="maxima")
[Out]
integrate(tanh(x)^2/(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. 2939 vs.
\(2 (74) = 148\).
time = 0.72, size = 6507, normalized size = 73.94 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^2/(a+b*tanh(x)^2)^(5/2),x, algorithm="fricas")
[Out]
[1/12*(3*((a^3 + 2*a^2*b + a*b^2)*cosh(x)^8 + 8*(a^3 + 2*a^2*b + a*b^2)*cosh(x)*sinh(x)^7 + (a^3 + 2*a^2*b + a
*b^2)*sinh(x)^8 + 4*(a^3 - a*b^2)*cosh(x)^6 + 4*(a^3 - a*b^2 + 7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^2)*sinh(x)^6
+ 8*(7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^3 + 3*(a^3 - a*b^2)*cosh(x))*sinh(x)^5 + 2*(3*a^3 - 2*a^2*b + 3*a*b^2)*
cosh(x)^4 + 2*(35*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^4 + 3*a^3 - 2*a^2*b + 3*a*b^2 + 30*(a^3 - a*b^2)*cosh(x)^2)*
sinh(x)^4 + 8*(7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^5 + 10*(a^3 - a*b^2)*cosh(x)^3 + (3*a^3 - 2*a^2*b + 3*a*b^2)*
cosh(x))*sinh(x)^3 + a^3 + 2*a^2*b + a*b^2 + 4*(a^3 - a*b^2)*cosh(x)^2 + 4*(7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^
6 + 15*(a^3 - a*b^2)*cosh(x)^4 + a^3 - a*b^2 + 3*(3*a^3 - 2*a^2*b + 3*a*b^2)*cosh(x)^2)*sinh(x)^2 + 8*((a^3 +
2*a^2*b + a*b^2)*cosh(x)^7 + 3*(a^3 - a*b^2)*cosh(x)^5 + (3*a^3 - 2*a^2*b + 3*a*b^2)*cosh(x)^3 + (a^3 - a*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))*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)
) + 3*((a^3 + 2*a^2*b + a*b^2)*cosh(x)^8 + 8*(a^3 + 2*a^2*b + a*b^2)*cosh(x)*sinh(x)^7 + (a^3 + 2*a^2*b + a*b^
2)*sinh(x)^8 + 4*(a^3 - a*b^2)*cosh(x)^6 + 4*(a^3 - a*b^2 + 7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^2)*sinh(x)^6 + 8
*(7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^3 + 3*(a^3 - a*b^2)*cosh(x))*sinh(x)^5 + 2*(3*a^3 - 2*a^2*b + 3*a*b^2)*cos
h(x)^4 + 2*(35*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^4 + 3*a^3 - 2*a^2*b + 3*a*b^2 + 30*(a^3 - a*b^2)*cosh(x)^2)*sin
h(x)^4 + 8*(7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^5 + 10*(a^3 - a*b^2)*cosh(x)^3 + (3*a^3 - 2*a^2*b + 3*a*b^2)*cos
h(x))*sinh(x)^3 + a^3 + 2*a^2*b + a*b^2 + 4*(a^3 - a*b^2)*cosh(x)^2 + 4*(7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^6 +
15*(a^3 - a*b^2)*cosh(x)^4 + a^3 - a*b^2 + 3*(3*a^3 - 2*a^2*b + 3*a*b^2)*cosh(x)^2)*sinh(x)^2 + 8*((a^3 + 2*a
^2*b + a*b^2)*cosh(x)^7 + 3*(a^3 - a*b^2)*cosh(x)^5 + (3*a^3 - 2*a^2*b + 3*a*b^2)*cosh(x)^3 + (a^3 - a*b^2)*co
sh(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*cos
h(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)) - 4*sqrt(2)*((3*a
^3 + 5*a^2*b + a*b^2 - b^3)*cosh(x)^6 + 6*(3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(x)*sinh(x)^5 + (3*a^3 + 5*a^2*b
+ a*b^2 - b^3)*sinh(x)^6 + 3*(a^3 - a^2*b - a*b^2 + b^3)*cosh(x)^4 + 3*(a^3 - a^2*b - a*b^2 + b^3 + 5*(3*a^3
+ 5*a^2*b + a*b^2 - b^3)*cosh(x)^2)*sinh(x)^4 + 4*(5*(3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(x)^3 + 3*(a^3 - a^2*
b - a*b^2 + b^3)*cosh(x))*sinh(x)^3 - 3*a^3 - 5*a^2*b - a*b^2 + b^3 - 3*(a^3 - a^2*b - a*b^2 + b^3)*cosh(x)^2
+ 3*(5*(3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(x)^4 - a^3 + a^2*b + a*b^2 - b^3 + 6*(a^3 - a^2*b - a*b^2 + b^3)*c
osh(x)^2)*sinh(x)^2 + 6*((3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(x)^5 + 2*(a^3 - a^2*b - a*b^2 + b^3)*cosh(x)^3 -
(a^3 - a^2*b - a*b^2 + b^3)*cosh(x))*sinh(x))*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^6 + 5*a^5*b + 10*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*cosh(x)^8 +
8*(a^6 + 5*a^5*b + 10*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*cosh(x)*sinh(x)^7 + (a^6 + 5*a^5*b + 10*a^4*b
^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*sinh(x)^8 + 4*(a^6 + 3*a^5*b + 2*a^4*b^2 - 2*a^3*b^3 - 3*a^2*b^4 - a*b^5)
*cosh(x)^6 + 4*(a^6 + 3*a^5*b + 2*a^4*b^2 - 2*a^3*b^3 - 3*a^2*b^4 - a*b^5 + 7*(a^6 + 5*a^5*b + 10*a^4*b^2 + 10
*a^3*b^3 + 5*a^2*b^4 + a*b^5)*cosh(x)^2)*sinh(x...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{2}{\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)**2/(a+b*tanh(x)**2)**(5/2),x)
[Out]
Integral(tanh(x)**2/(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. 752 vs.
\(2 (74) = 148\).
time = 0.63, size = 752, normalized size = 8.55 \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^{7} b^{2} + 14 \, a^{6} b^{3} + 25 \, a^{5} b^{4} + 20 \, a^{4} b^{5} + 5 \, a^{3} b^{6} - 2 \, a^{2} b^{7} - a b^{8}\right )} e^{\left (2 \, x\right )}}{a^{8} b^{2} + 6 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 20 \, a^{5} b^{5} + 15 \, a^{4} b^{6} + 6 \, a^{3} b^{7} + a^{2} b^{8}} + \frac {3 \, {\left (a^{7} b^{2} + 2 \, a^{6} b^{3} - a^{5} b^{4} - 4 \, a^{4} b^{5} - a^{3} b^{6} + 2 \, a^{2} b^{7} + a b^{8}\right )}}{a^{8} b^{2} + 6 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 20 \, a^{5} b^{5} + 15 \, a^{4} b^{6} + 6 \, a^{3} b^{7} + a^{2} b^{8}}\right )} e^{\left (2 \, x\right )} - \frac {3 \, {\left (a^{7} b^{2} + 2 \, a^{6} b^{3} - a^{5} b^{4} - 4 \, a^{4} b^{5} - a^{3} b^{6} + 2 \, a^{2} b^{7} + a b^{8}\right )}}{a^{8} b^{2} + 6 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 20 \, a^{5} b^{5} + 15 \, a^{4} b^{6} + 6 \, a^{3} b^{7} + a^{2} b^{8}}\right )} e^{\left (2 \, x\right )} - \frac {3 \, a^{7} b^{2} + 14 \, a^{6} b^{3} + 25 \, a^{5} b^{4} + 20 \, a^{4} b^{5} + 5 \, a^{3} b^{6} - 2 \, a^{2} b^{7} - a b^{8}}{a^{8} b^{2} + 6 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 20 \, a^{5} b^{5} + 15 \, a^{4} b^{6} + 6 \, a^{3} b^{7} + a^{2} b^{8}}}{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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^2/(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^7*b^2 + 14*a^6*b^3 + 25*a^5*b^4 + 20
*a^4*b^5 + 5*a^3*b^6 - 2*a^2*b^7 - a*b^8)*e^(2*x)/(a^8*b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6
+ 6*a^3*b^7 + a^2*b^8) + 3*(a^7*b^2 + 2*a^6*b^3 - a^5*b^4 - 4*a^4*b^5 - a^3*b^6 + 2*a^2*b^7 + a*b^8)/(a^8*b^2
+ 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 6*a^3*b^7 + a^2*b^8))*e^(2*x) - 3*(a^7*b^2 + 2*a^6*b^3 -
a^5*b^4 - 4*a^4*b^5 - a^3*b^6 + 2*a^2*b^7 + a*b^8)/(a^8*b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6
+ 6*a^3*b^7 + a^2*b^8))*e^(2*x) - (3*a^7*b^2 + 14*a^6*b^3 + 25*a^5*b^4 + 20*a^4*b^5 + 5*a^3*b^6 - 2*a^2*b^7 -
a*b^8)/(a^8*b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 6*a^3*b^7 + a^2*b^8))/(a*e^(4*x) + b*e^(
4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b)^(3/2)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {tanh}\left (x\right )}^2}{{\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)^2/(a + b*tanh(x)^2)^(5/2),x)
[Out]
int(tanh(x)^2/(a + b*tanh(x)^2)^(5/2), x)
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