3.217 \(\int \frac {1}{(a+b \text {sech}^2(x))^{5/2}} \, dx\)
Optimal. Leaf size=95 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{a^{5/2}}-\frac {b (5 a+3 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a-b \tanh ^2(x)+b}}-\frac {b \tanh (x)}{3 a (a+b) \left (a-b \tanh ^2(x)+b\right )^{3/2}} \]
[Out]
arctanh(a^(1/2)*tanh(x)/(a+b-b*tanh(x)^2)^(1/2))/a^(5/2)-1/3*b*(5*a+3*b)*tanh(x)/a^2/(a+b)^2/(a+b-b*tanh(x)^2)
^(1/2)-1/3*b*tanh(x)/a/(a+b)/(a+b-b*tanh(x)^2)^(3/2)
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Rubi [A] time = 0.09, antiderivative size = 95, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used =
{4128, 414, 527, 12, 377, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{a^{5/2}}-\frac {b (5 a+3 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a-b \tanh ^2(x)+b}}-\frac {b \tanh (x)}{3 a (a+b) \left (a-b \tanh ^2(x)+b\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Sech[x]^2)^(-5/2),x]
[Out]
ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(5/2) - (b*Tanh[x])/(3*a*(a + b)*(a + b - b*Tanh[x]^2)^
(3/2)) - (b*(5*a + 3*b)*Tanh[x])/(3*a^2*(a + b)^2*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 414
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, 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 4128
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \text {sech}^2(x)\right )^{5/2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a+b-b x^2\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=-\frac {b \tanh (x)}{3 a (a+b) \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {-3 a-b-2 b x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )}{3 a (a+b)}\\ &=-\frac {b \tanh (x)}{3 a (a+b) \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {b (5 a+3 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b-b \tanh ^2(x)}}+\frac {\operatorname {Subst}\left (\int \frac {3 (a+b)^2}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\tanh (x)\right )}{3 a^2 (a+b)^2}\\ &=-\frac {b \tanh (x)}{3 a (a+b) \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {b (5 a+3 b) \tanh (x)}{3 a^2 (a+b)^2 \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 {b \tanh (x)}{3 a (a+b) \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {b (5 a+3 b) \tanh (x)}{3 a^2 (a+b)^2 \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 {b \tanh (x)}{3 a (a+b) \left (a+b-b \tanh ^2(x)\right )^{3/2}}-\frac {b (5 a+3 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b-b \tanh ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 130, normalized size = 1.37 \[ \frac {\text {sech}^5(x) \left (\frac {\sqrt {2} (a \cosh (2 x)+a+2 b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sinh (x)}{\sqrt {a \cosh (2 x)+a+2 b}}\right )}{a^{5/2}}-\frac {4 b \sinh (x) (a \cosh (2 x)+a+2 b) \left (3 a^2+a (3 a+2 b) \cosh (2 x)+7 a b+3 b^2\right )}{3 a^2 (a+b)^2}\right )}{8 \left (a+b \text {sech}^2(x)\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Sech[x]^2)^(-5/2),x]
[Out]
(Sech[x]^5*((Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*(a + 2*b + a*Cosh[2*x])^(5
/2))/a^(5/2) - (4*b*(a + 2*b + a*Cosh[2*x])*(3*a^2 + 7*a*b + 3*b^2 + a*(3*a + 2*b)*Cosh[2*x])*Sinh[x])/(3*a^2*
(a + b)^2)))/(8*(a + b*Sech[x]^2)^(5/2))
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fricas [B] time = 0.80, size = 6299, normalized size = 66.31 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*sech(x)^2)^(5/2),x, algorithm="fricas")
[Out]
[1/12*(3*((a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^8 + 8*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)*sinh(x)^7 + (a^4 + 2*a^3*b
+ a^2*b^2)*sinh(x)^8 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^6 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a
*b^3 + 7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^3 + 3*(a^4 +
4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x))*sinh(x)^5 + 2*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(
x)^4 + 2*(35*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^4 + 3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4 + 30*(a^4
+ 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^2)*sinh(x)^4 + a^4 + 2*a^3*b + a^2*b^2 + 8*(7*(a^4 + 2*a^3*b + a^2*b^
2)*cosh(x)^5 + 10*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^3 + (3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3
+ 8*b^4)*cosh(x))*sinh(x)^3 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^2 + 4*(7*(a^4 + 2*a^3*b + a^2*b^
2)*cosh(x)^6 + 15*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^4 + a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3 + 3*(3
*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(x)^2)*sinh(x)^2 + 8*((a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^7
+ 3*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^5 + (3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh
(x)^3 + (a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x))*sinh(x))*sqrt(a)*log((a*b^2*cosh(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*cos
h(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)*sinh(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*sinh(x)^3 + 15*cosh(x)^2*si
nh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) + 3*((a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^8 + 8*(a^4 + 2*a^3*b + a^2*
b^2)*cosh(x)*sinh(x)^7 + (a^4 + 2*a^3*b + a^2*b^2)*sinh(x)^8 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)
^6 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3 + 7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^4 + 2
*a^3*b + a^2*b^2)*cosh(x)^3 + 3*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x))*sinh(x)^5 + 2*(3*a^4 + 14*a^3*b
+ 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(x)^4 + 2*(35*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^4 + 3*a^4 + 14*a^3*b + 2
7*a^2*b^2 + 24*a*b^3 + 8*b^4 + 30*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^2)*sinh(x)^4 + a^4 + 2*a^3*b +
a^2*b^2 + 8*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^5 + 10*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^3 + (3*
a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(x))*sinh(x)^3 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*c
osh(x)^2 + 4*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^6 + 15*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^4 + a^4
+ 4*a^3*b + 5*a^2*b^2 + 2*a*b^3 + 3*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(x)^2)*sinh(x)^2 +
8*((a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^7 + 3*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^5 + (3*a^4 + 14*a^3*
b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(x)^3 + (a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*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)*sin
h(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*cosh(x)^3 + (a + b)*cosh(x))*sinh(x) + a)/(cosh(x)^2
+ 2*cosh(x)*sinh(x) + sinh(x)^2)) - 8*sqrt(2)*((3*a^3*b + 2*a^2*b^2)*cosh(x)^6 + 6*(3*a^3*b + 2*a^2*b^2)*cosh
(x)*sinh(x)^5 + (3*a^3*b + 2*a^2*b^2)*sinh(x)^6 + 3*(a^3*b + 4*a^2*b^2 + 2*a*b^3)*cosh(x)^4 + 3*(a^3*b + 4*a^2
*b^2 + 2*a*b^3 + 5*(3*a^3*b + 2*a^2*b^2)*cosh(x)^2)*sinh(x)^4 - 3*a^3*b - 2*a^2*b^2 + 4*(5*(3*a^3*b + 2*a^2*b^
2)*cosh(x)^3 + 3*(a^3*b + 4*a^2*b^2 + 2*a*b^3)*cosh(x))*sinh(x)^3 - 3*(a^3*b + 4*a^2*b^2 + 2*a*b^3)*cosh(x)^2
+ 3*(5*(3*a^3*b + 2*a^2*b^2)*cosh(x)^4 - a^3*b - 4*a^2*b^2 - 2*a*b^3 + 6*(a^3*b + 4*a^2*b^2 + 2*a*b^3)*cosh(x)
^2)*sinh(x)^2 + 6*((3*a^3*b + 2*a^2*b^2)*cosh(x)^5 + 2*(a^3*b + 4*a^2*b^2 + 2*a*b^3)*cosh(x)^3 - (a^3*b + 4*a^
2*b^2 + 2*a*b^3)*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^7 + 2*a^6*b + a^5*b^2)*cosh(x)^8 + 8*(a^7 + 2*a^6*b + a^5*b^2)*cosh(x)*sinh(x)^7 + (a^7 + 2*
a^6*b + a^5*b^2)*sinh(x)^8 + a^7 + 2*a^6*b + a^5*b^2 + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*cosh(x)^6 + 4
*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3 + 7*(a^7 + 2*a^6*b + a^5*b^2)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^7 + 2*a^6
*b + a^5*b^2)*cosh(x)^3 + 3*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*cosh(x))*sinh(x)^5 + 2*(3*a^7 + 14*a^6*b +
27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4)*cosh(x)^4 + 2*(3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4 +
35*(a^7 + 2*a^6*b + a^5*b^2)*cosh(x)^4 + 30*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*cosh(x)^2)*sinh(x)^4 + 8*(
7*(a^7 + 2*a^6*b + a^5*b^2)*cosh(x)^5 + 10*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*cosh(x)^3 + (3*a^7 + 14*a^6
*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4)*cosh(x))*sinh(x)^3 + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*cosh(
x)^2 + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3 + 7*(a^7 + 2*a^6*b + a^5*b^2)*cosh(x)^6 + 15*(a^7 + 4*a^6*b +
5*a^5*b^2 + 2*a^4*b^3)*cosh(x)^4 + 3*(3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4)*cosh(x)^2)*sinh(
x)^2 + 8*((a^7 + 2*a^6*b + a^5*b^2)*cosh(x)^7 + 3*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*cosh(x)^5 + (3*a^7 +
14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4)*cosh(x)^3 + (a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*cosh(x))*
sinh(x)), -1/6*(3*((a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^8 + 8*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)*sinh(x)^7 + (a^4
+ 2*a^3*b + a^2*b^2)*sinh(x)^8 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^6 + 4*(a^4 + 4*a^3*b + 5*a^2*
b^2 + 2*a*b^3 + 7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^3 +
3*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x))*sinh(x)^5 + 2*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b
^4)*cosh(x)^4 + 2*(35*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^4 + 3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4 +
30*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^2)*sinh(x)^4 + a^4 + 2*a^3*b + a^2*b^2 + 8*(7*(a^4 + 2*a^3*b
+ a^2*b^2)*cosh(x)^5 + 10*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^3 + (3*a^4 + 14*a^3*b + 27*a^2*b^2 +
24*a*b^3 + 8*b^4)*cosh(x))*sinh(x)^3 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^2 + 4*(7*(a^4 + 2*a^3*b
+ a^2*b^2)*cosh(x)^6 + 15*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^4 + a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b
^3 + 3*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(x)^2)*sinh(x)^2 + 8*((a^4 + 2*a^3*b + a^2*b^2)*
cosh(x)^7 + 3*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^5 + (3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*
b^4)*cosh(x)^3 + (a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x))*sinh(x))*sqrt(-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*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*co
sh(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^4 + 2*a^3*b + a^2*b^2)*cosh(x)^8 + 8*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)*sinh(x)^7 + (a^4 + 2*a^3*b + a^2*b
^2)*sinh(x)^8 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^6 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3 + 7
*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^3 + 3*(a^4 + 4*a^3*b
+ 5*a^2*b^2 + 2*a*b^3)*cosh(x))*sinh(x)^5 + 2*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(x)^4 + 2
*(35*(a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^4 + 3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4 + 30*(a^4 + 4*a^3*
b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^2)*sinh(x)^4 + a^4 + 2*a^3*b + a^2*b^2 + 8*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(
x)^5 + 10*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^3 + (3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)
*cosh(x))*sinh(x)^3 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^2 + 4*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(
x)^6 + 15*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^4 + a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3 + 3*(3*a^4 + 1
4*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(x)^2)*sinh(x)^2 + 8*((a^4 + 2*a^3*b + a^2*b^2)*cosh(x)^7 + 3*(a^
4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x)^5 + (3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(x)^3 +
(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(x))*sinh(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)) + 4*sqrt(2)*((3*a^3*b + 2*a^2*b^2)*cosh(x)^6 + 6*(3*a^3*b + 2*a^2*b^2)*cosh(x)*sinh(x)^5 + (3*a^3*
b + 2*a^2*b^2)*sinh(x)^6 + 3*(a^3*b + 4*a^2*b^2 + 2*a*b^3)*cosh(x)^4 + 3*(a^3*b + 4*a^2*b^2 + 2*a*b^3 + 5*(3*a
^3*b + 2*a^2*b^2)*cosh(x)^2)*sinh(x)^4 - 3*a^3*b - 2*a^2*b^2 + 4*(5*(3*a^3*b + 2*a^2*b^2)*cosh(x)^3 + 3*(a^3*b
+ 4*a^2*b^2 + 2*a*b^3)*cosh(x))*sinh(x)^3 - 3*(a^3*b + 4*a^2*b^2 + 2*a*b^3)*cosh(x)^2 + 3*(5*(3*a^3*b + 2*a^2
*b^2)*cosh(x)^4 - a^3*b - 4*a^2*b^2 - 2*a*b^3 + 6*(a^3*b + 4*a^2*b^2 + 2*a*b^3)*cosh(x)^2)*sinh(x)^2 + 6*((3*a
^3*b + 2*a^2*b^2)*cosh(x)^5 + 2*(a^3*b + 4*a^2*b^2 + 2*a*b^3)*cosh(x)^3 - (a^3*b + 4*a^2*b^2 + 2*a*b^3)*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^7 + 2
*a^6*b + a^5*b^2)*cosh(x)^8 + 8*(a^7 + 2*a^6*b + a^5*b^2)*cosh(x)*sinh(x)^7 + (a^7 + 2*a^6*b + a^5*b^2)*sinh(x
)^8 + a^7 + 2*a^6*b + a^5*b^2 + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*cosh(x)^6 + 4*(a^7 + 4*a^6*b + 5*a^5
*b^2 + 2*a^4*b^3 + 7*(a^7 + 2*a^6*b + a^5*b^2)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^7 + 2*a^6*b + a^5*b^2)*cosh(x)^3
+ 3*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*cosh(x))*sinh(x)^5 + 2*(3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^
3 + 8*a^3*b^4)*cosh(x)^4 + 2*(3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4 + 35*(a^7 + 2*a^6*b + a^5
*b^2)*cosh(x)^4 + 30*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^7 + 2*a^6*b + a^5*
b^2)*cosh(x)^5 + 10*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*cosh(x)^3 + (3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^
4*b^3 + 8*a^3*b^4)*cosh(x))*sinh(x)^3 + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*cosh(x)^2 + 4*(a^7 + 4*a^6*b
+ 5*a^5*b^2 + 2*a^4*b^3 + 7*(a^7 + 2*a^6*b + a^5*b^2)*cosh(x)^6 + 15*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*
cosh(x)^4 + 3*(3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4)*cosh(x)^2)*sinh(x)^2 + 8*((a^7 + 2*a^6*
b + a^5*b^2)*cosh(x)^7 + 3*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*cosh(x)^5 + (3*a^7 + 14*a^6*b + 27*a^5*b^2
+ 24*a^4*b^3 + 8*a^3*b^4)*cosh(x)^3 + (a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*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(1/(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 = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a +b \mathrm {sech}\relax (x )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*sech(x)^2)^(5/2),x)
[Out]
int(1/(a+b*sech(x)^2)^(5/2),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \operatorname {sech}\relax (x)^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*sech(x)^2)^(5/2),x, algorithm="maxima")
[Out]
integrate((b*sech(x)^2 + a)^(-5/2), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+\frac {b}{{\mathrm {cosh}\relax (x)}^2}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a + b/cosh(x)^2)^(5/2),x)
[Out]
int(1/(a + b/cosh(x)^2)^(5/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \operatorname {sech}^{2}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*sech(x)**2)**(5/2),x)
[Out]
Integral((a + b*sech(x)**2)**(-5/2), x)
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