3.4.61 \(\int \text {sech}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx\) [361]
Optimal. Leaf size=206 \[ \frac {(2 a-b) E\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {b F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f} \]
[Out]
1/3*(2*a-b)*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticE(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-b)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)-1/3*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-b)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+1/3*sech(f*x+e)^
2*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/f
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Rubi [A]
time = 0.12, antiderivative size = 206, 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, 423, 539,
429, 422} \begin {gather*} -\frac {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 )}{3 f (a-b) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(2 a-b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} E\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{3 f (a-b) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\tanh (e+f x) \text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sech[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2],x]
[Out]
((2*a - b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)*f*S
qrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]
*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (Sech[e + f*x]
^2*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*f)
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 423
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-x)*(a + b*x^n)^(p + 1)*((
c + d*x^n)^q/(a*n*(p + 1))), x] + Dist[1/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(n*
(p + 1) + 1) + d*(n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[
p, -1] && LtQ[0, q, 1] && IntBinomialQ[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 \text {sech}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx &=\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\left (1+x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac {\text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f}-\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {-2 a-b x^2}{\left (1+x^2\right )^{3/2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 f}\\ &=\frac {\text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f}+\frac {\left ((2 a-b) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\left (1+x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 (a-b) f}-\frac {\left (a 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 )}{3 (a-b) f}\\ &=\frac {(2 a-b) E\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)}}{3 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {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)}}{3 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.42, size = 204, normalized size = 0.99 \begin {gather*} \frac {8 i a (2 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 )-16 i 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 )+\sqrt {2} \left (\left (8 a^2-4 b^2\right ) \cosh (2 (e+f x))+(2 a-b) (8 a-5 b+b \cosh (4 (e+f x)))\right ) \text {sech}^2(e+f x) \tanh (e+f x)}{24 (a-b) f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sech[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2],x]
[Out]
((8*I)*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (16*I)*a*(a - b)*Sqrt
[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*((8*a^2 - 4*b^2)*Cosh[2*(e + f*x)] +
(2*a - b)*(8*a - 5*b + b*Cosh[4*(e + f*x)]))*Sech[e + f*x]^2*Tanh[e + f*x])/(24*(a - b)*f*Sqrt[2*a - b + b*Co
sh[2*(e + f*x)]])
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Maple [A]
time = 3.15, size = 318, normalized size = 1.54
| | |
| method |
result |
size |
| | |
| default |
\(\frac {\left (2 \sqrt {-\frac {b}{a}}\, a b -\sqrt {-\frac {b}{a}}\, b^{2}\right ) \left (\cosh ^{4}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+\left (2 \sqrt {-\frac {b}{a}}\, a^{2}-2 \sqrt {-\frac {b}{a}}\, a b \right ) \left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+\sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, b \left (a \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-b \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-2 a \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+b \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )+\left (\sqrt {-\frac {b}{a}}\, a^{2}-2 \sqrt {-\frac {b}{a}}\, a b +\sqrt {-\frac {b}{a}}\, b^{2}\right ) \sinh \left (f x +e \right )}{3 \cosh \left (f x +e \right )^{3} \left (a -b \right ) \sqrt {-\frac {b}{a}}\, \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\) |
\(318\) |
| risch |
\(\text {Expression too large to display}\) |
\(243643\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/3*((2*(-1/a*b)^(1/2)*a*b-(-1/a*b)^(1/2)*b^2)*cosh(f*x+e)^4*sinh(f*x+e)+(2*(-1/a*b)^(1/2)*a^2-2*(-1/a*b)^(1/2
)*a*b)*cosh(f*x+e)^2*sinh(f*x+e)+(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*b*(a*EllipticF(sinh(f
*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))-b*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))-2*a*EllipticE(sinh(f*x+e
)*(-1/a*b)^(1/2),(a/b)^(1/2))+b*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2)))*cosh(f*x+e)^2+((-1/a*b)^(1/
2)*a^2-2*(-1/a*b)^(1/2)*a*b+(-1/a*b)^(1/2)*b^2)*sinh(f*x+e))/cosh(f*x+e)^3/(a-b)/(-1/a*b)^(1/2)/(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)^4*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(b*sinh(f*x + e)^2 + a)*sech(f*x + e)^4, x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2257 vs.
\(2 (218) = 436\).
time = 0.12, size = 2257, normalized size = 10.96 \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)^4*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")
[Out]
-1/3*(((4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^6 + 6*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (4*a^2
- 4*a*b + b^2)*sinh(f*x + e)^6 + 3*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^4 + 3*(5*(4*a^2 - 4*a*b + b^2)*cosh(f*x
+ e)^2 + 4*a^2 - 4*a*b + b^2)*sinh(f*x + e)^4 + 4*(5*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^3 + 3*(4*a^2 - 4*a*b
+ b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^2 + 3*(5*(4*a^2 - 4*a*b + b^2)*
cosh(f*x + e)^4 + 6*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^2 + 4*a^2 - 4*a*b + b^2)*sinh(f*x + e)^2 + 4*a^2 - 4*a
*b + b^2 + 6*((4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^5 + 2*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^3 + (4*a^2 - 4*a*b
+ b^2)*cosh(f*x + e))*sinh(f*x + e) - 2*((2*a*b - b^2)*cosh(f*x + e)^6 + 6*(2*a*b - b^2)*cosh(f*x + e)*sinh(f
*x + e)^5 + (2*a*b - b^2)*sinh(f*x + e)^6 + 3*(2*a*b - b^2)*cosh(f*x + e)^4 + 3*(5*(2*a*b - b^2)*cosh(f*x + e)
^2 + 2*a*b - b^2)*sinh(f*x + e)^4 + 4*(5*(2*a*b - b^2)*cosh(f*x + e)^3 + 3*(2*a*b - b^2)*cosh(f*x + e))*sinh(f
*x + e)^3 + 3*(2*a*b - b^2)*cosh(f*x + e)^2 + 3*(5*(2*a*b - b^2)*cosh(f*x + e)^4 + 6*(2*a*b - b^2)*cosh(f*x +
e)^2 + 2*a*b - b^2)*sinh(f*x + e)^2 + 2*a*b - b^2 + 6*((2*a*b - b^2)*cosh(f*x + e)^5 + 2*(2*a*b - b^2)*cosh(f*
x + e)^3 + (2*a*b - b^2)*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_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) - 2*((2*a^2 - a*b)*cosh(f*x + e)^
6 + 6*(2*a^2 - a*b)*cosh(f*x + e)*sinh(f*x + e)^5 + (2*a^2 - a*b)*sinh(f*x + e)^6 + 3*(2*a^2 - a*b)*cosh(f*x +
e)^4 + 3*(5*(2*a^2 - a*b)*cosh(f*x + e)^2 + 2*a^2 - a*b)*sinh(f*x + e)^4 + 4*(5*(2*a^2 - a*b)*cosh(f*x + e)^3
+ 3*(2*a^2 - a*b)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(2*a^2 - a*b)*cosh(f*x + e)^2 + 3*(5*(2*a^2 - a*b)*cosh(
f*x + e)^4 + 6*(2*a^2 - a*b)*cosh(f*x + e)^2 + 2*a^2 - a*b)*sinh(f*x + e)^2 + 2*a^2 - a*b + 6*((2*a^2 - a*b)*c
osh(f*x + e)^5 + 2*(2*a^2 - a*b)*cosh(f*x + e)^3 + (2*a^2 - a*b)*cosh(f*x + e))*sinh(f*x + e) - 2*((a*b - b^2)
*cosh(f*x + e)^6 + 6*(a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (a*b - b^2)*sinh(f*x + e)^6 + 3*(a*b - b^2)*c
osh(f*x + e)^4 + 3*(5*(a*b - b^2)*cosh(f*x + e)^2 + a*b - b^2)*sinh(f*x + e)^4 + 4*(5*(a*b - b^2)*cosh(f*x + e
)^3 + 3*(a*b - b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(a*b - b^2)*cosh(f*x + e)^2 + 3*(5*(a*b - b^2)*cosh(f*x
+ e)^4 + 6*(a*b - b^2)*cosh(f*x + e)^2 + a*b - b^2)*sinh(f*x + e)^2 + a*b - b^2 + 6*((a*b - b^2)*cosh(f*x + e
)^5 + 2*(a*b - b^2)*cosh(f*x + e)^3 + (a*b - b^2)*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
)*((2*a*b - b^2)*cosh(f*x + e)^5 + 5*(2*a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^4 + (2*a*b - b^2)*sinh(f*x + e)
^5 + 2*(3*a*b - 2*b^2)*cosh(f*x + e)^3 + 2*(5*(2*a*b - b^2)*cosh(f*x + e)^2 + 3*a*b - 2*b^2)*sinh(f*x + e)^3 +
b^2*cosh(f*x + e) + 2*(5*(2*a*b - b^2)*cosh(f*x + e)^3 + 3*(3*a*b - 2*b^2)*cosh(f*x + e))*sinh(f*x + e)^2 + (
5*(2*a*b - b^2)*cosh(f*x + e)^4 + 6*(3*a*b - 2*b^2)*cosh(f*x + e)^2 + b^2)*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*b
- b^2)*f*cosh(f*x + e)^6 + 6*(a*b - b^2)*f*cosh(f*x + e)*sinh(f*x + e)^5 + (a*b - b^2)*f*sinh(f*x + e)^6 + 3*
(a*b - b^2)*f*cosh(f*x + e)^4 + 3*(5*(a*b - b^2)*f*cosh(f*x + e)^2 + (a*b - b^2)*f)*sinh(f*x + e)^4 + 3*(a*b -
b^2)*f*cosh(f*x + e)^2 + 4*(5*(a*b - b^2)*f*cosh(f*x + e)^3 + 3*(a*b - b^2)*f*cosh(f*x + e))*sinh(f*x + e)^3
+ 3*(5*(a*b - b^2)*f*cosh(f*x + e)^4 + 6*(a*b - b^2)*f*cosh(f*x + e)^2 + (a*b - b^2)*f)*sinh(f*x + e)^2 + (a*b
- b^2)*f + 6*((a*b - b^2)*f*cosh(f*x + e)^5 + 2*(a*b - b^2)*f*cosh(f*x + e)^3 + (a*b - b^2)*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 \sqrt {a + b \sinh ^{2}{\left (e + f x \right )}} \operatorname {sech}^{4}{\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(f*x+e)**4*(a+b*sinh(f*x+e)**2)**(1/2),x)
[Out]
Integral(sqrt(a + b*sinh(e + f*x)**2)*sech(e + f*x)**4, x)
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}}{{\mathrm {cosh}\left (e+f\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*sinh(e + f*x)^2)^(1/2)/cosh(e + f*x)^4,x)
[Out]
int((a + b*sinh(e + f*x)^2)^(1/2)/cosh(e + f*x)^4, x)
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