3.215 \(\int \frac {\tanh ^2(x)}{(a+b \text {sech}^2(x))^{5/2}} \, dx\)
Optimal. Leaf size=88 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{a^{5/2}}-\frac {(2 a+3 b) \tanh (x)}{3 a^2 (a+b) \sqrt {a-b \tanh ^2(x)+b}}-\frac {\tanh (x)}{3 a \left (a-b \tanh ^2(x)+b\right )^{3/2}} \]
[Out]
arctanh(a^(1/2)*tanh(x)/(a+b-b*tanh(x)^2)^(1/2))/a^(5/2)-1/3*(2*a+3*b)*tanh(x)/a^2/(a+b)/(a+b-b*tanh(x)^2)^(1/
2)-1/3*tanh(x)/a/(a+b-b*tanh(x)^2)^(3/2)
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Rubi [A] time = 0.25, antiderivative size = 88, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used =
{4141, 1975, 471, 527, 12, 377, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{a^{5/2}}-\frac {(2 a+3 b) \tanh (x)}{3 a^2 (a+b) \sqrt {a-b \tanh ^2(x)+b}}-\frac {\tanh (x)}{3 a \left (a-b \tanh ^2(x)+b\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[Tanh[x]^2/(a + b*Sech[x]^2)^(5/2),x]
[Out]
ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(5/2) - Tanh[x]/(3*a*(a + b - b*Tanh[x]^2)^(3/2)) - ((2
*a + 3*b)*Tanh[x])/(3*a^2*(a + b)*Sqrt[a + b - 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 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 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 471
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 527
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 1975
Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q
, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0]
&& !BinomialMatchQ[{u, v}, x]
Rule 4141
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[((d*ff*x)^m*(a + b*(1 + ff^2*x^2)^(n/2))^p)/(1 + ff^
2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] ||
EqQ[n, 2])
Rubi steps
\begin {align*} \int \frac {\tanh ^2(x)}{\left (a+b \text {sech}^2(x)\right )^{5/2}} \, dx &=\operatorname {Subst}\left (\int \frac {x^2}{\left (1-x^2\right ) \left (a+b \left (1-x^2\right )\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=-\frac {\tanh (x)}{3 a \left (a+b-b \tanh ^2(x)\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {1+2 x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )}{3 a}\\ &=-\frac {\tanh (x)}{3 a \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {(2 a+3 b) \tanh (x)}{3 a^2 (a+b) \sqrt {a+b-b \tanh ^2(x)}}-\frac {\operatorname {Subst}\left (\int -\frac {3 (a+b)}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\tanh (x)\right )}{3 a^2 (a+b)}\\ &=-\frac {\tanh (x)}{3 a \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {(2 a+3 b) \tanh (x)}{3 a^2 (a+b) \sqrt {a+b-b \tanh ^2(x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\tanh (x)\right )}{a^2}\\ &=-\frac {\tanh (x)}{3 a \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {(2 a+3 b) \tanh (x)}{3 a^2 (a+b) \sqrt {a+b-b \tanh ^2(x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )}{a^2}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )}{a^{5/2}}-\frac {\tanh (x)}{3 a \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {(2 a+3 b) \tanh (x)}{3 a^2 (a+b) \sqrt {a+b-b \tanh ^2(x)}}\\ \end {align*}
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Mathematica [B] time = 1.20, size = 290, normalized size = 3.30 \[ \frac {\text {sech}^4(x) \left (\frac {\sqrt {2} \text {csch}(x) \text {sech}(x) \left (-\frac {16 \left (a \sinh ^2(x)+a+b\right ) \left (\frac {a \sinh ^2(x)}{a+b}+1\right ) \left (\frac {a^2 (a+b) \sinh ^4(x)}{\left (a \sinh ^2(x)+a+b\right )^2}+\frac {3 a (a+b) \sinh ^2(x)}{a \sinh ^2(x)+a+b}-\frac {3 \sqrt {a} \sqrt {a+b} \sinh (x) \sinh ^{-1}\left (\frac {\sqrt {a} \sinh (x)}{\sqrt {a+b}}\right )}{\sqrt {\frac {a \sinh ^2(x)+a+b}{a+b}}}\right )}{a^3}+\frac {12 \sinh ^4(x)}{a+b}+\frac {2 \sinh ^2(x) \left (a \sinh ^2(x)+a+b\right )}{(a+b)^2}+\frac {\sinh ^2(x)}{a+b}\right ) (a \cosh (2 x)+a+2 b)^{5/2}}{\left (a \sinh ^2(x)+a+b\right )^{3/2}}-\frac {8 \tanh (x) (a \cosh (2 x)+2 a+3 b) (a \cosh (2 x)+a+2 b)}{(a+b)^2}+\frac {4 \tanh (x) ((3 a+2 b) \cosh (2 x)+b) (a \cosh (2 x)+a+2 b)}{(a+b)^2}\right )}{384 \left (a+b \text {sech}^2(x)\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[Tanh[x]^2/(a + b*Sech[x]^2)^(5/2),x]
[Out]
(Sech[x]^4*((Sqrt[2]*(a + 2*b + a*Cosh[2*x])^(5/2)*Csch[x]*Sech[x]*(Sinh[x]^2/(a + b) + (12*Sinh[x]^4)/(a + b)
+ (2*Sinh[x]^2*(a + b + a*Sinh[x]^2))/(a + b)^2 - (16*(a + b + a*Sinh[x]^2)*(1 + (a*Sinh[x]^2)/(a + b))*((a^2
*(a + b)*Sinh[x]^4)/(a + b + a*Sinh[x]^2)^2 + (3*a*(a + b)*Sinh[x]^2)/(a + b + a*Sinh[x]^2) - (3*Sqrt[a]*Sqrt[
a + b]*ArcSinh[(Sqrt[a]*Sinh[x])/Sqrt[a + b]]*Sinh[x])/Sqrt[(a + b + a*Sinh[x]^2)/(a + b)]))/a^3))/(a + b + a*
Sinh[x]^2)^(3/2) - (8*(a + 2*b + a*Cosh[2*x])*(2*a + 3*b + a*Cosh[2*x])*Tanh[x])/(a + b)^2 + (4*(a + 2*b + a*C
osh[2*x])*(b + (3*a + 2*b)*Cosh[2*x])*Tanh[x])/(a + b)^2))/(384*(a + b*Sech[x]^2)^(5/2))
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fricas [B] time = 0.77, size = 4989, normalized size = 56.69 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^2/(a+b*sech(x)^2)^(5/2),x, algorithm="fricas")
[Out]
[1/12*(3*((a^3 + a^2*b)*cosh(x)^8 + 8*(a^3 + a^2*b)*cosh(x)*sinh(x)^7 + (a^3 + a^2*b)*sinh(x)^8 + 4*(a^3 + 3*a
^2*b + 2*a*b^2)*cosh(x)^6 + 4*(a^3 + 3*a^2*b + 2*a*b^2 + 7*(a^3 + a^2*b)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^3 + a^
2*b)*cosh(x)^3 + 3*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x))*sinh(x)^5 + 2*(3*a^3 + 11*a^2*b + 16*a*b^2 + 8*b^3)*cosh
(x)^4 + 2*(35*(a^3 + a^2*b)*cosh(x)^4 + 3*a^3 + 11*a^2*b + 16*a*b^2 + 8*b^3 + 30*(a^3 + 3*a^2*b + 2*a*b^2)*cos
h(x)^2)*sinh(x)^4 + 8*(7*(a^3 + a^2*b)*cosh(x)^5 + 10*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)^3 + (3*a^3 + 11*a^2*b
+ 16*a*b^2 + 8*b^3)*cosh(x))*sinh(x)^3 + a^3 + a^2*b + 4*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)^2 + 4*(7*(a^3 + a^2
*b)*cosh(x)^6 + 15*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)^4 + a^3 + 3*a^2*b + 2*a*b^2 + 3*(3*a^3 + 11*a^2*b + 16*a*
b^2 + 8*b^3)*cosh(x)^2)*sinh(x)^2 + 8*((a^3 + a^2*b)*cosh(x)^7 + 3*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)^5 + (3*a^
3 + 11*a^2*b + 16*a*b^2 + 8*b^3)*cosh(x)^3 + (a^3 + 3*a^2*b + 2*a*b^2)*cosh(x))*sinh(x))*sqrt(a)*log((a*b^2*co
sh(x)^8 + 8*a*b^2*cosh(x)*sinh(x)^7 + a*b^2*sinh(x)^8 - 2*(a*b^2 - b^3)*cosh(x)^6 + 2*(14*a*b^2*cosh(x)^2 - a*
b^2 + b^3)*sinh(x)^6 + 4*(14*a*b^2*cosh(x)^3 - 3*(a*b^2 - b^3)*cosh(x))*sinh(x)^5 + (a^3 + 4*a^2*b + 9*a*b^2)*
cosh(x)^4 + (70*a*b^2*cosh(x)^4 + a^3 + 4*a^2*b + 9*a*b^2 - 30*(a*b^2 - b^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*a*b^
2*cosh(x)^5 - 10*(a*b^2 - b^3)*cosh(x)^3 + (a^3 + 4*a^2*b + 9*a*b^2)*cosh(x))*sinh(x)^3 + a^3 + 2*(a^3 + 3*a^2
*b)*cosh(x)^2 + 2*(14*a*b^2*cosh(x)^6 - 15*(a*b^2 - b^3)*cosh(x)^4 + a^3 + 3*a^2*b + 3*(a^3 + 4*a^2*b + 9*a*b^
2)*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 + 4*a*b)*cosh(x)^2
+ (15*b^2*cosh(x)^4 - 18*b^2*cosh(x)^2 - a^2 - 4*a*b)*sinh(x)^2 - a^2 + 2*(3*b^2*cosh(x)^5 - 6*b^2*cosh(x)^3
- (a^2 + 4*a*b)*cosh(x))*sinh(x))*sqrt(a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*si
nh(x) + sinh(x)^2)) + 4*(2*a*b^2*cosh(x)^7 - 3*(a*b^2 - b^3)*cosh(x)^5 + (a^3 + 4*a^2*b + 9*a*b^2)*cosh(x)^3 +
(a^3 + 3*a^2*b)*cosh(x))*sinh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x)^4*sinh(x)^2 + 20*cosh(x)^3*si
nh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) + 3*((a^3 + a^2*b)*cosh(x)^8 + 8*(a^3 + a
^2*b)*cosh(x)*sinh(x)^7 + (a^3 + a^2*b)*sinh(x)^8 + 4*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)^6 + 4*(a^3 + 3*a^2*b +
2*a*b^2 + 7*(a^3 + a^2*b)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^3 + a^2*b)*cosh(x)^3 + 3*(a^3 + 3*a^2*b + 2*a*b^2)*c
osh(x))*sinh(x)^5 + 2*(3*a^3 + 11*a^2*b + 16*a*b^2 + 8*b^3)*cosh(x)^4 + 2*(35*(a^3 + a^2*b)*cosh(x)^4 + 3*a^3
+ 11*a^2*b + 16*a*b^2 + 8*b^3 + 30*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^3 + a^2*b)*cosh(x)
^5 + 10*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)^3 + (3*a^3 + 11*a^2*b + 16*a*b^2 + 8*b^3)*cosh(x))*sinh(x)^3 + a^3 +
a^2*b + 4*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)^2 + 4*(7*(a^3 + a^2*b)*cosh(x)^6 + 15*(a^3 + 3*a^2*b + 2*a*b^2)*c
osh(x)^4 + a^3 + 3*a^2*b + 2*a*b^2 + 3*(3*a^3 + 11*a^2*b + 16*a*b^2 + 8*b^3)*cosh(x)^2)*sinh(x)^2 + 8*((a^3 +
a^2*b)*cosh(x)^7 + 3*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)^5 + (3*a^3 + 11*a^2*b + 16*a*b^2 + 8*b^3)*cosh(x)^3 + (
a^3 + 3*a^2*b + 2*a*b^2)*cosh(x))*sinh(x))*sqrt(a)*log(-(a*cosh(x)^4 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^4 + 2
*(a + b)*cosh(x)^2 + 2*(3*a*cosh(x)^2 + a + b)*sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2
+ 1)*sqrt(a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*(a*co
sh(x)^3 + (a + b)*cosh(x))*sinh(x) + a)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) - 4*sqrt(2)*((3*a^3 + 4*a
^2*b)*cosh(x)^6 + 6*(3*a^3 + 4*a^2*b)*cosh(x)*sinh(x)^5 + (3*a^3 + 4*a^2*b)*sinh(x)^6 + 3*(a^3 + 4*a^2*b + 4*a
*b^2)*cosh(x)^4 + 3*(a^3 + 4*a^2*b + 4*a*b^2 + 5*(3*a^3 + 4*a^2*b)*cosh(x)^2)*sinh(x)^4 + 4*(5*(3*a^3 + 4*a^2*
b)*cosh(x)^3 + 3*(a^3 + 4*a^2*b + 4*a*b^2)*cosh(x))*sinh(x)^3 - 3*a^3 - 4*a^2*b - 3*(a^3 + 4*a^2*b + 4*a*b^2)*
cosh(x)^2 + 3*(5*(3*a^3 + 4*a^2*b)*cosh(x)^4 - a^3 - 4*a^2*b - 4*a*b^2 + 6*(a^3 + 4*a^2*b + 4*a*b^2)*cosh(x)^2
)*sinh(x)^2 + 6*((3*a^3 + 4*a^2*b)*cosh(x)^5 + 2*(a^3 + 4*a^2*b + 4*a*b^2)*cosh(x)^3 - (a^3 + 4*a^2*b + 4*a*b^
2)*cosh(x))*sinh(x))*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/
((a^6 + a^5*b)*cosh(x)^8 + 8*(a^6 + a^5*b)*cosh(x)*sinh(x)^7 + (a^6 + a^5*b)*sinh(x)^8 + 4*(a^6 + 3*a^5*b + 2*
a^4*b^2)*cosh(x)^6 + 4*(a^6 + 3*a^5*b + 2*a^4*b^2 + 7*(a^6 + a^5*b)*cosh(x)^2)*sinh(x)^6 + a^6 + a^5*b + 8*(7*
(a^6 + a^5*b)*cosh(x)^3 + 3*(a^6 + 3*a^5*b + 2*a^4*b^2)*cosh(x))*sinh(x)^5 + 2*(3*a^6 + 11*a^5*b + 16*a^4*b^2
+ 8*a^3*b^3)*cosh(x)^4 + 2*(3*a^6 + 11*a^5*b + 16*a^4*b^2 + 8*a^3*b^3 + 35*(a^6 + a^5*b)*cosh(x)^4 + 30*(a^6 +
3*a^5*b + 2*a^4*b^2)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^6 + a^5*b)*cosh(x)^5 + 10*(a^6 + 3*a^5*b + 2*a^4*b^2)*cos
h(x)^3 + (3*a^6 + 11*a^5*b + 16*a^4*b^2 + 8*a^3*b^3)*cosh(x))*sinh(x)^3 + 4*(a^6 + 3*a^5*b + 2*a^4*b^2)*cosh(x
)^2 + 4*(7*(a^6 + a^5*b)*cosh(x)^6 + a^6 + 3*a^5*b + 2*a^4*b^2 + 15*(a^6 + 3*a^5*b + 2*a^4*b^2)*cosh(x)^4 + 3*
(3*a^6 + 11*a^5*b + 16*a^4*b^2 + 8*a^3*b^3)*cosh(x)^2)*sinh(x)^2 + 8*((a^6 + a^5*b)*cosh(x)^7 + 3*(a^6 + 3*a^5
*b + 2*a^4*b^2)*cosh(x)^5 + (3*a^6 + 11*a^5*b + 16*a^4*b^2 + 8*a^3*b^3)*cosh(x)^3 + (a^6 + 3*a^5*b + 2*a^4*b^2
)*cosh(x))*sinh(x)), -1/6*(3*((a^3 + a^2*b)*cosh(x)^8 + 8*(a^3 + a^2*b)*cosh(x)*sinh(x)^7 + (a^3 + a^2*b)*sinh
(x)^8 + 4*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)^6 + 4*(a^3 + 3*a^2*b + 2*a*b^2 + 7*(a^3 + a^2*b)*cosh(x)^2)*sinh(x
)^6 + 8*(7*(a^3 + a^2*b)*cosh(x)^3 + 3*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x))*sinh(x)^5 + 2*(3*a^3 + 11*a^2*b + 16
*a*b^2 + 8*b^3)*cosh(x)^4 + 2*(35*(a^3 + a^2*b)*cosh(x)^4 + 3*a^3 + 11*a^2*b + 16*a*b^2 + 8*b^3 + 30*(a^3 + 3*
a^2*b + 2*a*b^2)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^3 + a^2*b)*cosh(x)^5 + 10*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)^3
+ (3*a^3 + 11*a^2*b + 16*a*b^2 + 8*b^3)*cosh(x))*sinh(x)^3 + a^3 + a^2*b + 4*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)
^2 + 4*(7*(a^3 + a^2*b)*cosh(x)^6 + 15*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)^4 + a^3 + 3*a^2*b + 2*a*b^2 + 3*(3*a^
3 + 11*a^2*b + 16*a*b^2 + 8*b^3)*cosh(x)^2)*sinh(x)^2 + 8*((a^3 + a^2*b)*cosh(x)^7 + 3*(a^3 + 3*a^2*b + 2*a*b^
2)*cosh(x)^5 + (3*a^3 + 11*a^2*b + 16*a*b^2 + 8*b^3)*cosh(x)^3 + (a^3 + 3*a^2*b + 2*a*b^2)*cosh(x))*sinh(x))*s
qrt(-a)*arctan(sqrt(2)*(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + a)*sqrt(-a)*sqrt((a*cosh(x)^2 + a*si
nh(x)^2 + a + 2*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 - (a^2 + 3*a*b)*cosh(x)^2 + (6*a*b*cosh(x)^2 - a^2 - 3*a*b)*sinh(x)^2 - a^2 + 2*(2*a*b*cosh(x)^3 -
(a^2 + 3*a*b)*cosh(x))*sinh(x))) + 3*((a^3 + a^2*b)*cosh(x)^8 + 8*(a^3 + a^2*b)*cosh(x)*sinh(x)^7 + (a^3 + a^2
*b)*sinh(x)^8 + 4*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)^6 + 4*(a^3 + 3*a^2*b + 2*a*b^2 + 7*(a^3 + a^2*b)*cosh(x)^2
)*sinh(x)^6 + 8*(7*(a^3 + a^2*b)*cosh(x)^3 + 3*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x))*sinh(x)^5 + 2*(3*a^3 + 11*a^
2*b + 16*a*b^2 + 8*b^3)*cosh(x)^4 + 2*(35*(a^3 + a^2*b)*cosh(x)^4 + 3*a^3 + 11*a^2*b + 16*a*b^2 + 8*b^3 + 30*(
a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^3 + a^2*b)*cosh(x)^5 + 10*(a^3 + 3*a^2*b + 2*a*b^2)*co
sh(x)^3 + (3*a^3 + 11*a^2*b + 16*a*b^2 + 8*b^3)*cosh(x))*sinh(x)^3 + a^3 + a^2*b + 4*(a^3 + 3*a^2*b + 2*a*b^2)
*cosh(x)^2 + 4*(7*(a^3 + a^2*b)*cosh(x)^6 + 15*(a^3 + 3*a^2*b + 2*a*b^2)*cosh(x)^4 + a^3 + 3*a^2*b + 2*a*b^2 +
3*(3*a^3 + 11*a^2*b + 16*a*b^2 + 8*b^3)*cosh(x)^2)*sinh(x)^2 + 8*((a^3 + a^2*b)*cosh(x)^7 + 3*(a^3 + 3*a^2*b
+ 2*a*b^2)*cosh(x)^5 + (3*a^3 + 11*a^2*b + 16*a*b^2 + 8*b^3)*cosh(x)^3 + (a^3 + 3*a^2*b + 2*a*b^2)*cosh(x))*si
nh(x))*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh
(x) + sinh(x)^2))/(a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 + a)) + 2*sqrt(2)*((3*a^3 + 4*a^2*b)*cosh(x
)^6 + 6*(3*a^3 + 4*a^2*b)*cosh(x)*sinh(x)^5 + (3*a^3 + 4*a^2*b)*sinh(x)^6 + 3*(a^3 + 4*a^2*b + 4*a*b^2)*cosh(x
)^4 + 3*(a^3 + 4*a^2*b + 4*a*b^2 + 5*(3*a^3 + 4*a^2*b)*cosh(x)^2)*sinh(x)^4 + 4*(5*(3*a^3 + 4*a^2*b)*cosh(x)^3
+ 3*(a^3 + 4*a^2*b + 4*a*b^2)*cosh(x))*sinh(x)^3 - 3*a^3 - 4*a^2*b - 3*(a^3 + 4*a^2*b + 4*a*b^2)*cosh(x)^2 +
3*(5*(3*a^3 + 4*a^2*b)*cosh(x)^4 - a^3 - 4*a^2*b - 4*a*b^2 + 6*(a^3 + 4*a^2*b + 4*a*b^2)*cosh(x)^2)*sinh(x)^2
+ 6*((3*a^3 + 4*a^2*b)*cosh(x)^5 + 2*(a^3 + 4*a^2*b + 4*a*b^2)*cosh(x)^3 - (a^3 + 4*a^2*b + 4*a*b^2)*cosh(x))*
sinh(x))*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/((a^6 + a^5*
b)*cosh(x)^8 + 8*(a^6 + a^5*b)*cosh(x)*sinh(x)^7 + (a^6 + a^5*b)*sinh(x)^8 + 4*(a^6 + 3*a^5*b + 2*a^4*b^2)*cos
h(x)^6 + 4*(a^6 + 3*a^5*b + 2*a^4*b^2 + 7*(a^6 + a^5*b)*cosh(x)^2)*sinh(x)^6 + a^6 + a^5*b + 8*(7*(a^6 + a^5*b
)*cosh(x)^3 + 3*(a^6 + 3*a^5*b + 2*a^4*b^2)*cosh(x))*sinh(x)^5 + 2*(3*a^6 + 11*a^5*b + 16*a^4*b^2 + 8*a^3*b^3)
*cosh(x)^4 + 2*(3*a^6 + 11*a^5*b + 16*a^4*b^2 + 8*a^3*b^3 + 35*(a^6 + a^5*b)*cosh(x)^4 + 30*(a^6 + 3*a^5*b + 2
*a^4*b^2)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^6 + a^5*b)*cosh(x)^5 + 10*(a^6 + 3*a^5*b + 2*a^4*b^2)*cosh(x)^3 + (3*
a^6 + 11*a^5*b + 16*a^4*b^2 + 8*a^3*b^3)*cosh(x))*sinh(x)^3 + 4*(a^6 + 3*a^5*b + 2*a^4*b^2)*cosh(x)^2 + 4*(7*(
a^6 + a^5*b)*cosh(x)^6 + a^6 + 3*a^5*b + 2*a^4*b^2 + 15*(a^6 + 3*a^5*b + 2*a^4*b^2)*cosh(x)^4 + 3*(3*a^6 + 11*
a^5*b + 16*a^4*b^2 + 8*a^3*b^3)*cosh(x)^2)*sinh(x)^2 + 8*((a^6 + a^5*b)*cosh(x)^7 + 3*(a^6 + 3*a^5*b + 2*a^4*b
^2)*cosh(x)^5 + (3*a^6 + 11*a^5*b + 16*a^4*b^2 + 8*a^3*b^3)*cosh(x)^3 + (a^6 + 3*a^5*b + 2*a^4*b^2)*cosh(x))*s
inh(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)^2/(a+b*sech(x)^2)^(5/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a sub
stitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perha
ps be purged.Error: Bad Argument Type
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{2}\relax (x )}{\left (a +b \mathrm {sech}\relax (x )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(x)^2/(a+b*sech(x)^2)^(5/2),x)
[Out]
int(tanh(x)^2/(a+b*sech(x)^2)^(5/2),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh \relax (x)^{2}}{{\left (b \operatorname {sech}\relax (x)^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^2/(a+b*sech(x)^2)^(5/2),x, algorithm="maxima")
[Out]
integrate(tanh(x)^2/(b*sech(x)^2 + a)^(5/2), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {tanh}\relax (x)}^2}{{\left (a+\frac {b}{{\mathrm {cosh}\relax (x)}^2}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(x)^2/(a + b/cosh(x)^2)^(5/2),x)
[Out]
int(tanh(x)^2/(a + b/cosh(x)^2)^(5/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{2}{\relax (x )}}{\left (a + b \operatorname {sech}^{2}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)**2/(a+b*sech(x)**2)**(5/2),x)
[Out]
Integral(tanh(x)**2/(a + b*sech(x)**2)**(5/2), x)
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