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

Optimal. Leaf size=134 \[ -\frac{b (5 a-2 b) \sinh (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}-\frac{b \sinh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{f (a-b)^{5/2}} \]

[Out]

ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]]/((a - b)^(5/2)*f) - (b*Sinh[e + f*x])/(3*a*(a
- b)*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - ((5*a - 2*b)*b*Sinh[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a + b*Sinh[e + f
*x]^2])

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Rubi [A]  time = 0.148909, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3190, 414, 527, 12, 377, 203} \[ -\frac{b (5 a-2 b) \sinh (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}-\frac{b \sinh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{f (a-b)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]]/((a - b)^(5/2)*f) - (b*Sinh[e + f*x])/(3*a*(a
- b)*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - ((5*a - 2*b)*b*Sinh[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a + b*Sinh[e + f
*x]^2])

Rule 3190

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, 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 - 1)/2]

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [C]  time = 9.30254, size = 1331, normalized size = 9.93 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sech[e + f*x]*Tanh[e + f*x]*(1575*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]] + (2100*b*ArcSin[Sqrt[((a - b)*Ta
nh[e + f*x]^2)/a]]*Sinh[e + f*x]^2)/a + (840*b^2*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^4)/a^
2 - (3150*(a - b)*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Tanh[e + f*x]^2)/a - (4200*(a - b)*b*ArcSin[Sqrt[(
(a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^2*Tanh[e + f*x]^2)/a^2 - (1680*(a - b)*b^2*ArcSin[Sqrt[((a - b)*Tan
h[e + f*x]^2)/a]]*Sinh[e + f*x]^4*Tanh[e + f*x]^2)/a^3 + (1575*(a - b)^2*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)
/a]]*Tanh[e + f*x]^4)/a^2 + (2100*(a - b)^2*b*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^2*Tanh[e
+ f*x]^4)/a^3 + (840*(a - b)^2*b^2*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^4*Tanh[e + f*x]^4)
/a^4 + 2100*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2) + (2800*b*Si
nh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2))/a + (1120
*b^2*Sinh[e + f*x]^4*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2))/a^
2 + 96*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)
)/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2) + 24*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Tanh[e + f*x]^2)
/a]*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2) + (168*b*Hypergeomet
ric2F1[2, 2, 9/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/
a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2))/a + (48*b*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Tanh[e + f*x
]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2))
/a + (72*b^2*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^4*Sqrt[(Sech[e + f*x]^2*(
a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2))/a^2 + (24*b^2*HypergeometricPFQ[{2, 2, 2}, {1,
9/2}, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^4*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b
)*Tanh[e + f*x]^2)/a)^(7/2))/a^2 - 1575*Sqrt[((a - b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)
/a^2] - (2100*b*Sinh[e + f*x]^2*Sqrt[((a - b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)/a^2])/a
- (840*b^2*Sinh[e + f*x]^4*Sqrt[((a - b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)/a^2])/a^2))
/(315*a^2*f*Sqrt[a + b*Sinh[e + f*x]^2]*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(1 + (b*Sinh[e + f*x
]^2)/a)*(((a - b)*Tanh[e + f*x]^2)/a)^(5/2))

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Maple [C]  time = 0.133, size = 169, normalized size = 1.3 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ({\frac{-{b}^{2} \left ( \sinh \left ( fx+e \right ) \right ) ^{4}-2\,ab \left ( \sinh \left ( fx+e \right ) \right ) ^{2}-{a}^{2}}{-{b}^{4} \left ( \sinh \left ( fx+e \right ) \right ) ^{10}+ \left ( -4\,a{b}^{3}-{b}^{4} \right ) \left ( \sinh \left ( fx+e \right ) \right ) ^{8}+ \left ( -6\,{a}^{2}{b}^{2}-4\,a{b}^{3} \right ) \left ( \sinh \left ( fx+e \right ) \right ) ^{6}+ \left ( -4\,{a}^{3}b-6\,{a}^{2}{b}^{2} \right ) \left ( \sinh \left ( fx+e \right ) \right ) ^{4}+ \left ( -{a}^{4}-4\,{a}^{3}b \right ) \left ( \sinh \left ( fx+e \right ) \right ) ^{2}-{a}^{4}}{\frac{1}{\sqrt{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}}}},\sinh \left ( fx+e \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

`int/indef0`((-b^2*sinh(f*x+e)^4-2*a*b*sinh(f*x+e)^2-a^2)/(-b^4*sinh(f*x+e)^10+(-4*a*b^3-b^4)*sinh(f*x+e)^8+(-
6*a^2*b^2-4*a*b^3)*sinh(f*x+e)^6+(-4*a^3*b-6*a^2*b^2)*sinh(f*x+e)^4+(-a^4-4*a^3*b)*sinh(f*x+e)^2-a^4)/(a+b*sin
h(f*x+e)^2)^(1/2),sinh(f*x+e))/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}\left (f x + e\right )}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 5.13512, size = 12320, normalized size = 91.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*(3*(a^2*b^2*cosh(f*x + e)^8 + 8*a^2*b^2*cosh(f*x + e)*sinh(f*x + e)^7 + a^2*b^2*sinh(f*x + e)^8 + 4*(2*a
^3*b - a^2*b^2)*cosh(f*x + e)^6 + 4*(7*a^2*b^2*cosh(f*x + e)^2 + 2*a^3*b - a^2*b^2)*sinh(f*x + e)^6 + 8*(7*a^2
*b^2*cosh(f*x + e)^3 + 3*(2*a^3*b - a^2*b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(8*a^4 - 8*a^3*b + 3*a^2*b^2)*
cosh(f*x + e)^4 + 2*(35*a^2*b^2*cosh(f*x + e)^4 + 8*a^4 - 8*a^3*b + 3*a^2*b^2 + 30*(2*a^3*b - a^2*b^2)*cosh(f*
x + e)^2)*sinh(f*x + e)^4 + a^2*b^2 + 8*(7*a^2*b^2*cosh(f*x + e)^5 + 10*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^3 +
(8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^2 + 4*(7*a^
2*b^2*cosh(f*x + e)^6 + 15*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^4 + 2*a^3*b - a^2*b^2 + 3*(8*a^4 - 8*a^3*b + 3*a^
2*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*(a^2*b^2*cosh(f*x + e)^7 + 3*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^5 +
(8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(f*x + e)^3 + (2*a^3*b - a^2*b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt(-a + b
)*log(((a - 2*b)*cosh(f*x + e)^4 + 4*(a - 2*b)*cosh(f*x + e)*sinh(f*x + e)^3 + (a - 2*b)*sinh(f*x + e)^4 - 2*(
3*a - 2*b)*cosh(f*x + e)^2 + 2*(3*(a - 2*b)*cosh(f*x + e)^2 - 3*a + 2*b)*sinh(f*x + e)^2 - 2*sqrt(2)*(cosh(f*x
+ e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 - 1)*sqrt(-a + b)*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)) + 4*((a - 2*b)*cosh(
f*x + e)^3 - (3*a - 2*b)*cosh(f*x + e))*sinh(f*x + e) + a - 2*b)/(cosh(f*x + e)^4 + 4*cosh(f*x + e)*sinh(f*x +
e)^3 + sinh(f*x + e)^4 + 2*(3*cosh(f*x + e)^2 + 1)*sinh(f*x + e)^2 + 2*cosh(f*x + e)^2 + 4*(cosh(f*x + e)^3 +
cosh(f*x + e))*sinh(f*x + e) + 1)) + 2*sqrt(2)*((5*a^2*b^2 - 7*a*b^3 + 2*b^4)*cosh(f*x + e)^6 + 6*(5*a^2*b^2
- 7*a*b^3 + 2*b^4)*cosh(f*x + e)*sinh(f*x + e)^5 + (5*a^2*b^2 - 7*a*b^3 + 2*b^4)*sinh(f*x + e)^6 + 3*(8*a^3*b
- 17*a^2*b^2 + 11*a*b^3 - 2*b^4)*cosh(f*x + e)^4 + 3*(8*a^3*b - 17*a^2*b^2 + 11*a*b^3 - 2*b^4 + 5*(5*a^2*b^2 -
7*a*b^3 + 2*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^4 - 5*a^2*b^2 + 7*a*b^3 - 2*b^4 + 4*(5*(5*a^2*b^2 - 7*a*b^3 +
2*b^4)*cosh(f*x + e)^3 + 3*(8*a^3*b - 17*a^2*b^2 + 11*a*b^3 - 2*b^4)*cosh(f*x + e))*sinh(f*x + e)^3 - 3*(8*a^
3*b - 17*a^2*b^2 + 11*a*b^3 - 2*b^4)*cosh(f*x + e)^2 + 3*(5*(5*a^2*b^2 - 7*a*b^3 + 2*b^4)*cosh(f*x + e)^4 - 8*
a^3*b + 17*a^2*b^2 - 11*a*b^3 + 2*b^4 + 6*(8*a^3*b - 17*a^2*b^2 + 11*a*b^3 - 2*b^4)*cosh(f*x + e)^2)*sinh(f*x
+ e)^2 + 6*((5*a^2*b^2 - 7*a*b^3 + 2*b^4)*cosh(f*x + e)^5 + 2*(8*a^3*b - 17*a^2*b^2 + 11*a*b^3 - 2*b^4)*cosh(f
*x + e)^3 - (8*a^3*b - 17*a^2*b^2 + 11*a*b^3 - 2*b^4)*cosh(f*x + e))*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^5*b^2 -
3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^8 + 8*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*cosh(f*x
+ e)*sinh(f*x + e)^7 + (a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*sinh(f*x + e)^8 + 4*(2*a^6*b - 7*a^5*b^2
+ 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*f*cosh(f*x + e)^6 + 4*(7*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*cos
h(f*x + e)^2 + (2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*f)*sinh(f*x + e)^6 + 2*(8*a^7 - 32*a^6*
b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5)*f*cosh(f*x + e)^4 + 8*(7*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^
4 - a^2*b^5)*f*cosh(f*x + e)^3 + 3*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*f*cosh(f*x + e))*si
nh(f*x + e)^5 + 2*(35*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^4 + 30*(2*a^6*b - 7*a^5*b^2
+ 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*f*cosh(f*x + e)^2 + (8*a^7 - 32*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^
4 - 3*a^2*b^5)*f)*sinh(f*x + e)^4 + 4*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*f*cosh(f*x + e)^
2 + 8*(7*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^5 + 10*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 -
5*a^3*b^4 + a^2*b^5)*f*cosh(f*x + e)^3 + (8*a^7 - 32*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5
)*f*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(7*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^6 + 15*(
2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*f*cosh(f*x + e)^4 + 3*(8*a^7 - 32*a^6*b + 51*a^5*b^2 -
41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5)*f*cosh(f*x + e)^2 + (2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^
5)*f)*sinh(f*x + e)^2 + (a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f + 8*((a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 -
a^2*b^5)*f*cosh(f*x + e)^7 + 3*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*f*cosh(f*x + e)^5 + (8*
a^7 - 32*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5)*f*cosh(f*x + e)^3 + (2*a^6*b - 7*a^5*b^2 +
9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*f*cosh(f*x + e))*sinh(f*x + e)), 1/3*(3*(a^2*b^2*cosh(f*x + e)^8 + 8*a^2*b^2*
cosh(f*x + e)*sinh(f*x + e)^7 + a^2*b^2*sinh(f*x + e)^8 + 4*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^6 + 4*(7*a^2*b^2
*cosh(f*x + e)^2 + 2*a^3*b - a^2*b^2)*sinh(f*x + e)^6 + 8*(7*a^2*b^2*cosh(f*x + e)^3 + 3*(2*a^3*b - a^2*b^2)*c
osh(f*x + e))*sinh(f*x + e)^5 + 2*(8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(f*x + e)^4 + 2*(35*a^2*b^2*cosh(f*x + e)^
4 + 8*a^4 - 8*a^3*b + 3*a^2*b^2 + 30*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + a^2*b^2 + 8*(7*a^2
*b^2*cosh(f*x + e)^5 + 10*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^3 + (8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(f*x + e))*s
inh(f*x + e)^3 + 4*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^2 + 4*(7*a^2*b^2*cosh(f*x + e)^6 + 15*(2*a^3*b - a^2*b^2)
*cosh(f*x + e)^4 + 2*a^3*b - a^2*b^2 + 3*(8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*(a
^2*b^2*cosh(f*x + e)^7 + 3*(2*a^3*b - a^2*b^2)*cosh(f*x + e)^5 + (8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(f*x + e)^3
+ (2*a^3*b - a^2*b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt(a - b)*arctan(sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x
+ e)*sinh(f*x + e) + sinh(f*x + e)^2 - 1)*sqrt(a - b)*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))/(b*cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*sin
h(f*x + e)^3 + b*sinh(f*x + e)^4 + 2*(2*a - b)*cosh(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2 + 2*a - b)*sinh(f*x +
e)^2 + 4*(b*cosh(f*x + e)^3 + (2*a - b)*cosh(f*x + e))*sinh(f*x + e) + b)) - sqrt(2)*((5*a^2*b^2 - 7*a*b^3 + 2
*b^4)*cosh(f*x + e)^6 + 6*(5*a^2*b^2 - 7*a*b^3 + 2*b^4)*cosh(f*x + e)*sinh(f*x + e)^5 + (5*a^2*b^2 - 7*a*b^3 +
2*b^4)*sinh(f*x + e)^6 + 3*(8*a^3*b - 17*a^2*b^2 + 11*a*b^3 - 2*b^4)*cosh(f*x + e)^4 + 3*(8*a^3*b - 17*a^2*b^
2 + 11*a*b^3 - 2*b^4 + 5*(5*a^2*b^2 - 7*a*b^3 + 2*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^4 - 5*a^2*b^2 + 7*a*b^3
- 2*b^4 + 4*(5*(5*a^2*b^2 - 7*a*b^3 + 2*b^4)*cosh(f*x + e)^3 + 3*(8*a^3*b - 17*a^2*b^2 + 11*a*b^3 - 2*b^4)*cos
h(f*x + e))*sinh(f*x + e)^3 - 3*(8*a^3*b - 17*a^2*b^2 + 11*a*b^3 - 2*b^4)*cosh(f*x + e)^2 + 3*(5*(5*a^2*b^2 -
7*a*b^3 + 2*b^4)*cosh(f*x + e)^4 - 8*a^3*b + 17*a^2*b^2 - 11*a*b^3 + 2*b^4 + 6*(8*a^3*b - 17*a^2*b^2 + 11*a*b^
3 - 2*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 6*((5*a^2*b^2 - 7*a*b^3 + 2*b^4)*cosh(f*x + e)^5 + 2*(8*a^3*b -
17*a^2*b^2 + 11*a*b^3 - 2*b^4)*cosh(f*x + e)^3 - (8*a^3*b - 17*a^2*b^2 + 11*a*b^3 - 2*b^4)*cosh(f*x + e))*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^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^8 + 8*(a^5*b^2 - 3*a^4*
b^3 + 3*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)*sinh(f*x + e)^7 + (a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*sin
h(f*x + e)^8 + 4*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*f*cosh(f*x + e)^6 + 4*(7*(a^5*b^2 - 3
*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^2 + (2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*f)
*sinh(f*x + e)^6 + 2*(8*a^7 - 32*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5)*f*cosh(f*x + e)^4 +
8*(7*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^3 + 3*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a
^3*b^4 + a^2*b^5)*f*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*cosh(
f*x + e)^4 + 30*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*f*cosh(f*x + e)^2 + (8*a^7 - 32*a^6*b
+ 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5)*f)*sinh(f*x + e)^4 + 4*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 -
5*a^3*b^4 + a^2*b^5)*f*cosh(f*x + e)^2 + 8*(7*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^5 +
10*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*f*cosh(f*x + e)^3 + (8*a^7 - 32*a^6*b + 51*a^5*b^2
- 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5)*f*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(7*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^
4 - a^2*b^5)*f*cosh(f*x + e)^6 + 15*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*f*cosh(f*x + e)^4
+ 3*(8*a^7 - 32*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5)*f*cosh(f*x + e)^2 + (2*a^6*b - 7*a^5
*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*f)*sinh(f*x + e)^2 + (a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f + 8
*((a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*cosh(f*x + e)^7 + 3*(2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b
^4 + a^2*b^5)*f*cosh(f*x + e)^5 + (8*a^7 - 32*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5)*f*cosh
(f*x + e)^3 + (2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*f*cosh(f*x + e))*sinh(f*x + e))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 3.25782, size = 1831, normalized size = 13.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(5*a*b - 2*b^2)/(a^4*sqrt(b)*f - 2*a^3*b^(3/2)*f + a^2*b^(5/2)*f) - 1/3*((((5*a^13*b^4*f^3 - 52*a^12*b^5*
f^3 + 245*a^11*b^6*f^3 - 690*a^10*b^7*f^3 + 1290*a^9*b^8*f^3 - 1680*a^8*b^9*f^3 + 1554*a^7*b^10*f^3 - 1020*a^6
*b^11*f^3 + 465*a^5*b^12*f^3 - 140*a^4*b^13*f^3 + 25*a^3*b^14*f^3 - 2*a^2*b^15*f^3)*e^(2*f*x + 2*e)/(a^16*b^2*
f^4 - 12*a^15*b^3*f^4 + 66*a^14*b^4*f^4 - 220*a^13*b^5*f^4 + 495*a^12*b^6*f^4 - 792*a^11*b^7*f^4 + 924*a^10*b^
8*f^4 - 792*a^9*b^9*f^4 + 495*a^8*b^10*f^4 - 220*a^7*b^11*f^4 + 66*a^6*b^12*f^4 - 12*a^5*b^13*f^4 + a^4*b^14*f
^4) + 3*(8*a^14*b^3*f^3 - 89*a^13*b^4*f^3 + 452*a^12*b^5*f^3 - 1385*a^11*b^6*f^3 + 2850*a^10*b^7*f^3 - 4146*a^
9*b^8*f^3 + 4368*a^8*b^9*f^3 - 3354*a^7*b^10*f^3 + 1860*a^6*b^11*f^3 - 725*a^5*b^12*f^3 + 188*a^4*b^13*f^3 - 2
9*a^3*b^14*f^3 + 2*a^2*b^15*f^3)/(a^16*b^2*f^4 - 12*a^15*b^3*f^4 + 66*a^14*b^4*f^4 - 220*a^13*b^5*f^4 + 495*a^
12*b^6*f^4 - 792*a^11*b^7*f^4 + 924*a^10*b^8*f^4 - 792*a^9*b^9*f^4 + 495*a^8*b^10*f^4 - 220*a^7*b^11*f^4 + 66*
a^6*b^12*f^4 - 12*a^5*b^13*f^4 + a^4*b^14*f^4))*e^(2*f*x + 2*e) - 3*(8*a^14*b^3*f^3 - 89*a^13*b^4*f^3 + 452*a^
12*b^5*f^3 - 1385*a^11*b^6*f^3 + 2850*a^10*b^7*f^3 - 4146*a^9*b^8*f^3 + 4368*a^8*b^9*f^3 - 3354*a^7*b^10*f^3 +
1860*a^6*b^11*f^3 - 725*a^5*b^12*f^3 + 188*a^4*b^13*f^3 - 29*a^3*b^14*f^3 + 2*a^2*b^15*f^3)/(a^16*b^2*f^4 - 1
2*a^15*b^3*f^4 + 66*a^14*b^4*f^4 - 220*a^13*b^5*f^4 + 495*a^12*b^6*f^4 - 792*a^11*b^7*f^4 + 924*a^10*b^8*f^4 -
792*a^9*b^9*f^4 + 495*a^8*b^10*f^4 - 220*a^7*b^11*f^4 + 66*a^6*b^12*f^4 - 12*a^5*b^13*f^4 + a^4*b^14*f^4))*e^
(2*f*x + 2*e) - (5*a^13*b^4*f^3 - 52*a^12*b^5*f^3 + 245*a^11*b^6*f^3 - 690*a^10*b^7*f^3 + 1290*a^9*b^8*f^3 - 1
680*a^8*b^9*f^3 + 1554*a^7*b^10*f^3 - 1020*a^6*b^11*f^3 + 465*a^5*b^12*f^3 - 140*a^4*b^13*f^3 + 25*a^3*b^14*f^
3 - 2*a^2*b^15*f^3)/(a^16*b^2*f^4 - 12*a^15*b^3*f^4 + 66*a^14*b^4*f^4 - 220*a^13*b^5*f^4 + 495*a^12*b^6*f^4 -
792*a^11*b^7*f^4 + 924*a^10*b^8*f^4 - 792*a^9*b^9*f^4 + 495*a^8*b^10*f^4 - 220*a^7*b^11*f^4 + 66*a^6*b^12*f^4
- 12*a^5*b^13*f^4 + a^4*b^14*f^4))/(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b)^(3/2) +
2*arctan(-1/2*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) +
b) + sqrt(b))/sqrt(a - b))/((a^2*f - 2*a*b*f + b^2*f)*sqrt(a - b))