3.4.99 \(\int \frac {\text {sech}^2(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\) [399]

Optimal. Leaf size=292 \[ \frac {b (3 a+b) \cosh (e+f x) \sinh (e+f x)}{3 a (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\sqrt {b} \left (3 a^2+7 a b-2 b^2\right ) \cosh (e+f x) E\left (\text {ArcTan}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 a^{3/2} (a-b)^3 f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(9 a-b) b F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 (a-b)^3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\tanh (e+f x)}{(a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]

[Out]

1/3*b*(3*a+b)*cosh(f*x+e)*sinh(f*x+e)/a/(a-b)^2/f/(a+b*sinh(f*x+e)^2)^(3/2)+1/3*(3*a^2+7*a*b-2*b^2)*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^(3/2)/(a-b)^3/f/(a*cosh(f*x+e)^2/(a+b*sinh(f*x+e)^2))^(1/2)/(a+b*si
nh(f*x+e)^2)^(1/2)-1/3*(9*a-b)*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^2/(a-b)^3/f/(sech(f*x+e)^2*(a+b*si
nh(f*x+e)^2)/a)^(1/2)+tanh(f*x+e)/(a-b)/f/(a+b*sinh(f*x+e)^2)^(3/2)

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Rubi [A]
time = 0.22, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3271, 425, 541, 539, 429, 422} \begin {gather*} -\frac {b (9 a-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 a^2 f (a-b)^3 \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\sqrt {b} \left (3 a^2+7 a b-2 b^2\right ) \cosh (e+f x) E\left (\text {ArcTan}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 a^{3/2} f (a-b)^3 \sqrt {a+b \sinh ^2(e+f x)} \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}+\frac {\tanh (e+f x)}{f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {b (3 a+b) \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b)^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2),x]

[Out]

(b*(3*a + b)*Cosh[e + f*x]*Sinh[e + f*x])/(3*a*(a - b)^2*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + (Sqrt[b]*(3*a^2 +
7*a*b - 2*b^2)*Cosh[e + f*x]*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b])/(3*a^(3/2)*(a - b)^3
*f*Sqrt[(a*Cosh[e + f*x]^2)/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2]) - ((9*a - b)*b*EllipticF[Arc
Tan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^2*(a - b)^3*f*Sqrt[(Sech[e + f*x]
^2*(a + b*Sinh[e + f*x]^2))/a]) + Tanh[e + f*x]/((a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2))

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 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 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 )^{5/2}} \, 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 )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac {\tanh (e+f x)}{(a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {b-3 b x^2}{\sqrt {1+x^2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{(-a+b) f}\\ &=\frac {b (3 a+b) \cosh (e+f x) \sinh (e+f x)}{3 a (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\tanh (e+f x)}{(a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {2 (3 a-b) b-b (3 a+b) x^2}{\sqrt {1+x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 a (a-b) (-a+b) f}\\ &=\frac {b (3 a+b) \cosh (e+f x) \sinh (e+f x)}{3 a (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\tanh (e+f x)}{(a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\left ((9 a-b) 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 (a-b)^2 (-a+b) f}-\frac {\left (b \left (3 a^2+7 a b-2 b^2\right ) \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 )}{3 a (a-b)^2 (-a+b) f}\\ &=\frac {b (3 a+b) \cosh (e+f x) \sinh (e+f x)}{3 a (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\sqrt {b} \left (3 a^2+7 a b-2 b^2\right ) \cosh (e+f x) E\left (\tan ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 a^{3/2} (a-b)^3 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 {(9 a-b) 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^2 (a-b)^3 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 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.76, size = 260, normalized size = 0.89 \begin {gather*} \frac {2 i a^2 \left (3 a^2+7 a b-2 b^2\right ) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-2 i a^2 \left (3 a^2-2 a b-b^2\right ) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )+\frac {\left (24 a^4-24 a^3 b+41 a^2 b^2-19 a b^3+2 b^4+4 a b \left (6 a^2+5 a b-3 b^2\right ) \cosh (2 (e+f x))+b^2 \left (3 a^2+7 a b-2 b^2\right ) \cosh (4 (e+f x))\right ) \tanh (e+f x)}{\sqrt {2}}}{6 a^2 (a-b)^3 f (2 a-b+b \cosh (2 (e+f x)))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2),x]

[Out]

((2*I)*a^2*(3*a^2 + 7*a*b - 2*b^2)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] - (2*
I)*a^2*(3*a^2 - 2*a*b - b^2)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] + ((24*a^4
- 24*a^3*b + 41*a^2*b^2 - 19*a*b^3 + 2*b^4 + 4*a*b*(6*a^2 + 5*a*b - 3*b^2)*Cosh[2*(e + f*x)] + b^2*(3*a^2 + 7*
a*b - 2*b^2)*Cosh[4*(e + f*x)])*Tanh[e + f*x])/Sqrt[2])/(6*a^2*(a - b)^3*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/
2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1001\) vs. \(2(360)=720\).
time = 2.02, size = 1002, normalized size = 3.43

method	result	size
default	\(\text {Expression too large to display}\)	\(1002\)
risch	\(\text {Expression too large to display}\)	\(88750\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(3*(-1/a*b)^(1/2)*a^2*b^2*sinh(f*x+e)^5+7*(-1/a*b)^(1/2)*a*b^3*sinh(f*x+e)^5-2*(-1/a*b)^(1/2)*b^4*sinh(f*x
+e)^5-6*((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^2*b^2*sinh(f*x+e)^2+8*((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^3*sinh(f*x+e)^2-2*((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))*b^4*sinh(f*x+e)^2-3*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*Elliptic
E(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^2*b^2*sinh(f*x+e)^2-7*((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^3*sinh(f*x+e)^2+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))*b^4*sinh(f*x+e)^2+6*(-1/a*b)^(1/2)*a^3
*b*sinh(f*x+e)^3+8*(-1/a*b)^(1/2)*a^2*b^2*sinh(f*x+e)^3+4*(-1/a*b)^(1/2)*a*b^3*sinh(f*x+e)^3-2*(-1/a*b)^(1/2)*
b^4*sinh(f*x+e)^3-6*((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^3*b+8*((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^2*b^2-2*((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^3-3*((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^3*b-7*((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^2*b^2+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^3+3*(-1/a*b)^(1/2)*a^4*sinh(f*x+e)+8*(-1/a*b)^(1/2)*a^2*b^2*sinh(f*x+e)-3*(-1/a*b)^(1/2)*a
*b^3*sinh(f*x+e))/(-1/a*b)^(1/2)/(a+b*sinh(f*x+e)^2)^(3/2)/a^2/(a-b)^3/cosh(f*x+e)/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)^(5/2),x, algorithm="maxima")

[Out]

integrate(sech(f*x + e)^2/(b*sinh(f*x + e)^2 + a)^(5/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 8928 vs. \(2 (298) = 596\).
time = 0.29, size = 8928, normalized size = 30.58 \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)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(((6*a^3*b^2 + 11*a^2*b^3 - 11*a*b^4 + 2*b^5)*cosh(f*x + e)^10 + 10*(6*a^3*b^2 + 11*a^2*b^3 - 11*a*b^4 +
2*b^5)*cosh(f*x + e)*sinh(f*x + e)^9 + (6*a^3*b^2 + 11*a^2*b^3 - 11*a*b^4 + 2*b^5)*sinh(f*x + e)^10 + (48*a^4*
b + 70*a^3*b^2 - 121*a^2*b^3 + 49*a*b^4 - 6*b^5)*cosh(f*x + e)^8 + (48*a^4*b + 70*a^3*b^2 - 121*a^2*b^3 + 49*a
*b^4 - 6*b^5 + 45*(6*a^3*b^2 + 11*a^2*b^3 - 11*a*b^4 + 2*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^8 + 8*(15*(6*a^3*
b^2 + 11*a^2*b^3 - 11*a*b^4 + 2*b^5)*cosh(f*x + e)^3 + (48*a^4*b + 70*a^3*b^2 - 121*a^2*b^3 + 49*a*b^4 - 6*b^5
)*cosh(f*x + e))*sinh(f*x + e)^7 + 2*(48*a^5 + 64*a^4*b - 126*a^3*b^2 + 71*a^2*b^3 - 19*a*b^4 + 2*b^5)*cosh(f*
x + e)^6 + 2*(48*a^5 + 64*a^4*b - 126*a^3*b^2 + 71*a^2*b^3 - 19*a*b^4 + 2*b^5 + 105*(6*a^3*b^2 + 11*a^2*b^3 -
11*a*b^4 + 2*b^5)*cosh(f*x + e)^4 + 14*(48*a^4*b + 70*a^3*b^2 - 121*a^2*b^3 + 49*a*b^4 - 6*b^5)*cosh(f*x + e)^
2)*sinh(f*x + e)^6 + 4*(63*(6*a^3*b^2 + 11*a^2*b^3 - 11*a*b^4 + 2*b^5)*cosh(f*x + e)^5 + 14*(48*a^4*b + 70*a^3
*b^2 - 121*a^2*b^3 + 49*a*b^4 - 6*b^5)*cosh(f*x + e)^3 + 3*(48*a^5 + 64*a^4*b - 126*a^3*b^2 + 71*a^2*b^3 - 19*
a*b^4 + 2*b^5)*cosh(f*x + e))*sinh(f*x + e)^5 + 6*a^3*b^2 + 11*a^2*b^3 - 11*a*b^4 + 2*b^5 + 2*(48*a^5 + 64*a^4
*b - 126*a^3*b^2 + 71*a^2*b^3 - 19*a*b^4 + 2*b^5)*cosh(f*x + e)^4 + 2*(105*(6*a^3*b^2 + 11*a^2*b^3 - 11*a*b^4
+ 2*b^5)*cosh(f*x + e)^6 + 48*a^5 + 64*a^4*b - 126*a^3*b^2 + 71*a^2*b^3 - 19*a*b^4 + 2*b^5 + 35*(48*a^4*b + 70
*a^3*b^2 - 121*a^2*b^3 + 49*a*b^4 - 6*b^5)*cosh(f*x + e)^4 + 15*(48*a^5 + 64*a^4*b - 126*a^3*b^2 + 71*a^2*b^3
- 19*a*b^4 + 2*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 8*(15*(6*a^3*b^2 + 11*a^2*b^3 - 11*a*b^4 + 2*b^5)*cosh(
f*x + e)^7 + 7*(48*a^4*b + 70*a^3*b^2 - 121*a^2*b^3 + 49*a*b^4 - 6*b^5)*cosh(f*x + e)^5 + 5*(48*a^5 + 64*a^4*b
- 126*a^3*b^2 + 71*a^2*b^3 - 19*a*b^4 + 2*b^5)*cosh(f*x + e)^3 + (48*a^5 + 64*a^4*b - 126*a^3*b^2 + 71*a^2*b^
3 - 19*a*b^4 + 2*b^5)*cosh(f*x + e))*sinh(f*x + e)^3 + (48*a^4*b + 70*a^3*b^2 - 121*a^2*b^3 + 49*a*b^4 - 6*b^5
)*cosh(f*x + e)^2 + (45*(6*a^3*b^2 + 11*a^2*b^3 - 11*a*b^4 + 2*b^5)*cosh(f*x + e)^8 + 28*(48*a^4*b + 70*a^3*b^
2 - 121*a^2*b^3 + 49*a*b^4 - 6*b^5)*cosh(f*x + e)^6 + 48*a^4*b + 70*a^3*b^2 - 121*a^2*b^3 + 49*a*b^4 - 6*b^5 +
30*(48*a^5 + 64*a^4*b - 126*a^3*b^2 + 71*a^2*b^3 - 19*a*b^4 + 2*b^5)*cosh(f*x + e)^4 + 12*(48*a^5 + 64*a^4*b
- 126*a^3*b^2 + 71*a^2*b^3 - 19*a*b^4 + 2*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(5*(6*a^3*b^2 + 11*a^2*b^3
- 11*a*b^4 + 2*b^5)*cosh(f*x + e)^9 + 4*(48*a^4*b + 70*a^3*b^2 - 121*a^2*b^3 + 49*a*b^4 - 6*b^5)*cosh(f*x + e
)^7 + 6*(48*a^5 + 64*a^4*b - 126*a^3*b^2 + 71*a^2*b^3 - 19*a*b^4 + 2*b^5)*cosh(f*x + e)^5 + 4*(48*a^5 + 64*a^4
*b - 126*a^3*b^2 + 71*a^2*b^3 - 19*a*b^4 + 2*b^5)*cosh(f*x + e)^3 + (48*a^4*b + 70*a^3*b^2 - 121*a^2*b^3 + 49*
a*b^4 - 6*b^5)*cosh(f*x + e))*sinh(f*x + e) - 2*((3*a^2*b^3 + 7*a*b^4 - 2*b^5)*cosh(f*x + e)^10 + 10*(3*a^2*b^
3 + 7*a*b^4 - 2*b^5)*cosh(f*x + e)*sinh(f*x + e)^9 + (3*a^2*b^3 + 7*a*b^4 - 2*b^5)*sinh(f*x + e)^10 + (24*a^3*
b^2 + 47*a^2*b^3 - 37*a*b^4 + 6*b^5)*cosh(f*x + e)^8 + (24*a^3*b^2 + 47*a^2*b^3 - 37*a*b^4 + 6*b^5 + 45*(3*a^2
*b^3 + 7*a*b^4 - 2*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^8 + 8*(15*(3*a^2*b^3 + 7*a*b^4 - 2*b^5)*cosh(f*x + e)^3
+ (24*a^3*b^2 + 47*a^2*b^3 - 37*a*b^4 + 6*b^5)*cosh(f*x + e))*sinh(f*x + e)^7 + 2*(24*a^4*b + 44*a^3*b^2 - 41
*a^2*b^3 + 15*a*b^4 - 2*b^5)*cosh(f*x + e)^6 + 2*(24*a^4*b + 44*a^3*b^2 - 41*a^2*b^3 + 15*a*b^4 - 2*b^5 + 105*
(3*a^2*b^3 + 7*a*b^4 - 2*b^5)*cosh(f*x + e)^4 + 14*(24*a^3*b^2 + 47*a^2*b^3 - 37*a*b^4 + 6*b^5)*cosh(f*x + e)^
2)*sinh(f*x + e)^6 + 4*(63*(3*a^2*b^3 + 7*a*b^4 - 2*b^5)*cosh(f*x + e)^5 + 14*(24*a^3*b^2 + 47*a^2*b^3 - 37*a*
b^4 + 6*b^5)*cosh(f*x + e)^3 + 3*(24*a^4*b + 44*a^3*b^2 - 41*a^2*b^3 + 15*a*b^4 - 2*b^5)*cosh(f*x + e))*sinh(f
*x + e)^5 + 3*a^2*b^3 + 7*a*b^4 - 2*b^5 + 2*(24*a^4*b + 44*a^3*b^2 - 41*a^2*b^3 + 15*a*b^4 - 2*b^5)*cosh(f*x +
e)^4 + 2*(105*(3*a^2*b^3 + 7*a*b^4 - 2*b^5)*cosh(f*x + e)^6 + 24*a^4*b + 44*a^3*b^2 - 41*a^2*b^3 + 15*a*b^4 -
2*b^5 + 35*(24*a^3*b^2 + 47*a^2*b^3 - 37*a*b^4 + 6*b^5)*cosh(f*x + e)^4 + 15*(24*a^4*b + 44*a^3*b^2 - 41*a^2*
b^3 + 15*a*b^4 - 2*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 8*(15*(3*a^2*b^3 + 7*a*b^4 - 2*b^5)*cosh(f*x + e)^7
+ 7*(24*a^3*b^2 + 47*a^2*b^3 - 37*a*b^4 + 6*b^5)*cosh(f*x + e)^5 + 5*(24*a^4*b + 44*a^3*b^2 - 41*a^2*b^3 + 15
*a*b^4 - 2*b^5)*cosh(f*x + e)^3 + (24*a^4*b + 44*a^3*b^2 - 41*a^2*b^3 + 15*a*b^4 - 2*b^5)*cosh(f*x + e))*sinh(
f*x + e)^3 + (24*a^3*b^2 + 47*a^2*b^3 - 37*a*b^4 + 6*b^5)*cosh(f*x + e)^2 + (45*(3*a^2*b^3 + 7*a*b^4 - 2*b^5)*
cosh(f*x + e)^8 + 28*(24*a^3*b^2 + 47*a^2*b^3 - 37*a*b^4 + 6*b^5)*cosh(f*x + e)^6 + 24*a^3*b^2 + 47*a^2*b^3 -
37*a*b^4 + 6*b^5 + 30*(24*a^4*b + 44*a^3*b^2 - 41*a^2*b^3 + 15*a*b^4 - 2*b^5)*cosh(f*x + e)^4 + 12*(24*a^4*b +
44*a^3*b^2 - 41*a^2*b^3 + 15*a*b^4 - 2*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(5*(3*a^2*b^3 + 7*a*b^4 - 2*
b^5)*cosh(f*x + e)^9 + 4*(24*a^3*b^2 + 47*a^2*b^3 - 37*a*b^4 + 6*b^5)*cosh(f*x + e)^7 + 6*(24*a^4*b + 44*a^3*b
^2 - 41*a^2*b^3 + 15*a*b^4 - 2*b^5)*cosh(f*x + ...

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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 {5}{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)**(5/2),x)

[Out]

Integral(sech(e + f*x)**2/(a + b*sinh(e + f*x)**2)**(5/2), 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)^(5/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Evaluation time:
2.05Error: 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 )}^{5/2}} \,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)^(5/2)),x)

[Out]

int(1/(cosh(e + f*x)^2*(a + b*sinh(e + f*x)^2)^(5/2)), x)

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