3.4.90 \(\int \frac {\text {sech}^2(e+f x)}{(a+b \sinh ^2(e+f x))^{3/2}} \, dx\) [390]
Optimal. Leaf size=217 \[ \frac {\sqrt {b} (a+b) \cosh (e+f x) E\left (\text {ArcTan}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} (a-b)^2 f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 b F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a (a-b)^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\tanh (e+f x)}{(a-b) f \sqrt {a+b \sinh ^2(e+f x)}} \]
[Out]
(a+b)*cosh(f*x+e)*(1/(1+b*sinh(f*x+e)^2/a))^(1/2)*(1+b*sinh(f*x+e)^2/a)^(1/2)*EllipticE(sinh(f*x+e)*b^(1/2)/a^
(1/2)/(1+b*sinh(f*x+e)^2/a)^(1/2),(1-a/b)^(1/2))*b^(1/2)/(a-b)^2/f/a^(1/2)/(a*cosh(f*x+e)^2/(a+b*sinh(f*x+e)^2
))^(1/2)/(a+b*sinh(f*x+e)^2)^(1/2)-2*b*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+
e)/(1+sinh(f*x+e)^2)^(1/2),(1-b/a)^(1/2))*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a/(a-b)^2/f/(sech(f*x+e)^2*(a+
b*sinh(f*x+e)^2)/a)^(1/2)+tanh(f*x+e)/(a-b)/f/(a+b*sinh(f*x+e)^2)^(1/2)
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Rubi [A]
time = 0.14, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3271, 425, 539,
429, 422} \begin {gather*} \frac {\sqrt {b} (a+b) \cosh (e+f x) E\left (\text {ArcTan}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} f (a-b)^2 \sqrt {a+b \sinh ^2(e+f x)} \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}-\frac {2 b \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{a f (a-b)^2 \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\tanh (e+f x)}{f (a-b) \sqrt {a+b \sinh ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sech[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(3/2),x]
[Out]
(Sqrt[b]*(a + b)*Cosh[e + f*x]*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b])/(Sqrt[a]*(a - b)^2
*f*Sqrt[(a*Cosh[e + f*x]^2)/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2]) - (2*b*EllipticF[ArcTan[Sinh
[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*(a - b)^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Si
nh[e + f*x]^2))/a]) + Tanh[e + f*x]/((a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])
Rule 422
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Rule 425
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]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]
Rule 429
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*
Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Rule 539
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^(3/2)), x_Symbol] :> Dist[(b*e - a*
f)/(b*c - a*d), Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[Sqrt[a + b
*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] && PosQ[d/c]
Rule 3271
Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Sin[e + f*x], x]}, Dist[ff*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])), Subst[Int[(1 - ff^2*x^2)^((m - 1)/2
)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && !IntegerQ
[p]
Rubi steps
\begin {align*} \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac {\tanh (e+f x)}{(a-b) f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {b-b x^2}{\sqrt {1+x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{(-a+b) f}\\ &=\frac {\tanh (e+f x)}{(a-b) f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\left (2 b \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{(a-b) (-a+b) f}-\frac {\left (b (a+b) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{(a-b) (-a+b) f}\\ &=\frac {\sqrt {b} (a+b) \cosh (e+f x) E\left (\tan ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} (a-b)^2 f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 b F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a (a-b)^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\tanh (e+f x)}{(a-b) f \sqrt {a+b \sinh ^2(e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.06, size = 178, normalized size = 0.82 \begin {gather*} \frac {i \sqrt {2} a (a+b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-i \sqrt {2} a (a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )+\left (2 a^2-a b+b^2+b (a+b) \cosh (2 (e+f x))\right ) \tanh (e+f x)}{a (a-b)^2 f \sqrt {4 a-2 b+2 b \cosh (2 (e+f x))}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sech[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(3/2),x]
[Out]
(I*Sqrt[2]*a*(a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - I*Sqrt[2]*a*(a - b)
*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + (2*a^2 - a*b + b^2 + b*(a + b)*Cosh[2*(
e + f*x)])*Tanh[e + f*x])/(a*(a - b)^2*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])
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Maple [A]
time = 2.34, size = 342, normalized size = 1.58
| | |
| method |
result |
size |
| | |
| default |
\(\frac {\sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{3}\left (f x +e \right )\right )+\sqrt {-\frac {b}{a}}\, b^{2} \left (\sinh ^{3}\left (f x +e \right )\right )-a b \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+b^{2} \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b -\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}+\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}\, a^{2}+\sqrt {-\frac {b}{a}}\, b^{2} \sinh \left (f x +e \right )}{\left (a -b \right )^{2} a \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\) |
\(342\) |
| risch |
\(\text {Expression too large to display}\) |
\(54607\) |
| | |
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|
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sech(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
[Out]
((-1/a*b)^(1/2)*a*b*sinh(f*x+e)^3+(-1/a*b)^(1/2)*b^2*sinh(f*x+e)^3-a*b*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x
+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))+b^2*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^
2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))-((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)
*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*b-((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*Ell
ipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^2+sinh(f*x+e)*(-1/a*b)^(1/2)*a^2+(-1/a*b)^(1/2)*b^2*sinh(f*x+
e))/(a-b)^2/a/(-1/a*b)^(1/2)/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f
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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(sech(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")
[Out]
integrate(sech(f*x + e)^2/(b*sinh(f*x + e)^2 + a)^(3/2), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2786 vs.
\(2 (231) = 462\).
time = 0.14, size = 2786, normalized size = 12.84 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")
[Out]
-(((2*a^2*b + a*b^2 - b^3)*cosh(f*x + e)^6 + 6*(2*a^2*b + a*b^2 - b^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (2*a^2*
b + a*b^2 - b^3)*sinh(f*x + e)^6 + (8*a^3 + 2*a^2*b - 5*a*b^2 + b^3)*cosh(f*x + e)^4 + (8*a^3 + 2*a^2*b - 5*a*
b^2 + b^3 + 15*(2*a^2*b + a*b^2 - b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(5*(2*a^2*b + a*b^2 - b^3)*cosh(f*
x + e)^3 + (8*a^3 + 2*a^2*b - 5*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e)^3 + 2*a^2*b + a*b^2 - b^3 + (8*a^3 +
2*a^2*b - 5*a*b^2 + b^3)*cosh(f*x + e)^2 + (15*(2*a^2*b + a*b^2 - b^3)*cosh(f*x + e)^4 + 8*a^3 + 2*a^2*b - 5*
a*b^2 + b^3 + 6*(8*a^3 + 2*a^2*b - 5*a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(3*(2*a^2*b + a*b^2 - b
^3)*cosh(f*x + e)^5 + 2*(8*a^3 + 2*a^2*b - 5*a*b^2 + b^3)*cosh(f*x + e)^3 + (8*a^3 + 2*a^2*b - 5*a*b^2 + b^3)*
cosh(f*x + e))*sinh(f*x + e) - 2*((a*b^2 + b^3)*cosh(f*x + e)^6 + 6*(a*b^2 + b^3)*cosh(f*x + e)*sinh(f*x + e)^
5 + (a*b^2 + b^3)*sinh(f*x + e)^6 + (4*a^2*b + 3*a*b^2 - b^3)*cosh(f*x + e)^4 + (4*a^2*b + 3*a*b^2 - b^3 + 15*
(a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(5*(a*b^2 + b^3)*cosh(f*x + e)^3 + (4*a^2*b + 3*a*b^2 - b^3
)*cosh(f*x + e))*sinh(f*x + e)^3 + a*b^2 + b^3 + (4*a^2*b + 3*a*b^2 - b^3)*cosh(f*x + e)^2 + (15*(a*b^2 + b^3)
*cosh(f*x + e)^4 + 4*a^2*b + 3*a*b^2 - b^3 + 6*(4*a^2*b + 3*a*b^2 - b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*
(3*(a*b^2 + b^3)*cosh(f*x + e)^5 + 2*(4*a^2*b + 3*a*b^2 - b^3)*cosh(f*x + e)^3 + (4*a^2*b + 3*a*b^2 - b^3)*cos
h(f*x + e))*sinh(f*x + e))*sqrt((a^2 - a*b)/b^2))*sqrt(b)*sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*ellipt
ic_e(arcsin(sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*(cosh(f*x + e) + sinh(f*x + e))), (8*a^2 - 8*a*b + b
^2 + 4*(2*a*b - b^2)*sqrt((a^2 - a*b)/b^2))/b^2) - 4*((2*a^2*b - a*b^2)*cosh(f*x + e)^6 + 6*(2*a^2*b - a*b^2)*
cosh(f*x + e)*sinh(f*x + e)^5 + (2*a^2*b - a*b^2)*sinh(f*x + e)^6 + (8*a^3 - 6*a^2*b + a*b^2)*cosh(f*x + e)^4
+ (8*a^3 - 6*a^2*b + a*b^2 + 15*(2*a^2*b - a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(5*(2*a^2*b - a*b^2)*co
sh(f*x + e)^3 + (8*a^3 - 6*a^2*b + a*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 2*a^2*b - a*b^2 + (8*a^3 - 6*a^2*b
+ a*b^2)*cosh(f*x + e)^2 + (15*(2*a^2*b - a*b^2)*cosh(f*x + e)^4 + 8*a^3 - 6*a^2*b + a*b^2 + 6*(8*a^3 - 6*a^2*
b + a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(3*(2*a^2*b - a*b^2)*cosh(f*x + e)^5 + 2*(8*a^3 - 6*a^2*b + a*
b^2)*cosh(f*x + e)^3 + (8*a^3 - 6*a^2*b + a*b^2)*cosh(f*x + e))*sinh(f*x + e) + ((a*b^2 - b^3)*cosh(f*x + e)^6
+ 6*(a*b^2 - b^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (a*b^2 - b^3)*sinh(f*x + e)^6 + (4*a^2*b - 5*a*b^2 + b^3)*c
osh(f*x + e)^4 + (4*a^2*b - 5*a*b^2 + b^3 + 15*(a*b^2 - b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(5*(a*b^2 -
b^3)*cosh(f*x + e)^3 + (4*a^2*b - 5*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e)^3 + a*b^2 - b^3 + (4*a^2*b - 5*a
*b^2 + b^3)*cosh(f*x + e)^2 + (15*(a*b^2 - b^3)*cosh(f*x + e)^4 + 4*a^2*b - 5*a*b^2 + b^3 + 6*(4*a^2*b - 5*a*b
^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(3*(a*b^2 - b^3)*cosh(f*x + e)^5 + 2*(4*a^2*b - 5*a*b^2 + b^3)*
cosh(f*x + e)^3 + (4*a^2*b - 5*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2 - a*b)/b^2))*sqrt(b)*sqrt(
(2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*(cosh
(f*x + e) + sinh(f*x + e))), (8*a^2 - 8*a*b + b^2 + 4*(2*a*b - b^2)*sqrt((a^2 - a*b)/b^2))/b^2) - sqrt(2)*(4*a
^2*b*cosh(f*x + e)^3 + (a*b^2 + b^3)*cosh(f*x + e)^5 + 5*(a*b^2 + b^3)*cosh(f*x + e)*sinh(f*x + e)^4 + (a*b^2
+ b^3)*sinh(f*x + e)^5 + 2*(2*a^2*b + 5*(a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^3 + 2*(6*a^2*b*cosh(f*x +
e) + 5*(a*b^2 + b^3)*cosh(f*x + e)^3)*sinh(f*x + e)^2 + (3*a*b^2 - b^3)*cosh(f*x + e) + (12*a^2*b*cosh(f*x +
e)^2 + 5*(a*b^2 + b^3)*cosh(f*x + e)^4 + 3*a*b^2 - b^3)*sinh(f*x + e))*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x +
e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/((a^3*b^2 - 2*a^2*b^3 +
a*b^4)*f*cosh(f*x + e)^6 + 6*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*f*cosh(f*x + e)*sinh(f*x + e)^5 + (a^3*b^2 - 2*a^2*
b^3 + a*b^4)*f*sinh(f*x + e)^6 + (4*a^4*b - 9*a^3*b^2 + 6*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^4 + (15*(a^3*b^2 -
2*a^2*b^3 + a*b^4)*f*cosh(f*x + e)^2 + (4*a^4*b - 9*a^3*b^2 + 6*a^2*b^3 - a*b^4)*f)*sinh(f*x + e)^4 + (4*a^4*b
- 9*a^3*b^2 + 6*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^2 + 4*(5*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*f*cosh(f*x + e)^3 + (
4*a^4*b - 9*a^3*b^2 + 6*a^2*b^3 - a*b^4)*f*cosh(f*x + e))*sinh(f*x + e)^3 + (15*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*
f*cosh(f*x + e)^4 + 6*(4*a^4*b - 9*a^3*b^2 + 6*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^2 + (4*a^4*b - 9*a^3*b^2 + 6*a
^2*b^3 - a*b^4)*f)*sinh(f*x + e)^2 + (a^3*b^2 - 2*a^2*b^3 + a*b^4)*f + 2*(3*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*f*co
sh(f*x + e)^5 + 2*(4*a^4*b - 9*a^3*b^2 + 6*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^3 + (4*a^4*b - 9*a^3*b^2 + 6*a^2*b
^3 - a*b^4)*f*cosh(f*x + e))*sinh(f*x + e))
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}^{2}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(f*x+e)**2/(a+b*sinh(f*x+e)**2)**(3/2),x)
[Out]
Integral(sech(e + f*x)**2/(a + b*sinh(e + f*x)**2)**(3/2), x)
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")
[Out]
Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Evaluation time:
0.5Error: Bad Argument Type
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\mathrm {cosh}\left (e+f\,x\right )}^2\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(cosh(e + f*x)^2*(a + b*sinh(e + f*x)^2)^(3/2)),x)
[Out]
int(1/(cosh(e + f*x)^2*(a + b*sinh(e + f*x)^2)^(3/2)), x)
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