3.4.94 \(\int \frac {\text {sech}(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\) [394]

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

[Out]

arctan(sinh(f*x+e)*(a-b)^(1/2)/(a+b*sinh(f*x+e)^2)^(1/2))/(a-b)^(5/2)/f-1/3*b*sinh(f*x+e)/a/(a-b)/f/(a+b*sinh(
f*x+e)^2)^(3/2)-1/3*(5*a-2*b)*b*sinh(f*x+e)/a^2/(a-b)^2/f/(a+b*sinh(f*x+e)^2)^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 134, normalized size of antiderivative = 1.00, 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 = {3269, 425, 541, 12, 385, 209} \begin {gather*} -\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 {\text {ArcTan}\left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{f (a-b)^{5/2}}-\frac {b \sinh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}

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 12

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

Rule 209

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

Rule 385

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 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 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 3269

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]

Rubi steps

\begin {align*} \int \frac {\text {sech}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\text {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 {\text {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 {\text {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 {\text {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 {\text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 8.50, size = 1331, normalized size = 9.93 \begin {gather*} \frac {\text {sech}(e+f x) \tanh (e+f x) \left (1575 \text {ArcSin}\left (\sqrt {\frac {(a-b) \tanh ^2(e+f x)}{a}}\right )+\frac {2100 b \text {ArcSin}\left (\sqrt {\frac {(a-b) \tanh ^2(e+f x)}{a}}\right ) \sinh ^2(e+f x)}{a}+\frac {840 b^2 \text {ArcSin}\left (\sqrt {\frac {(a-b) \tanh ^2(e+f x)}{a}}\right ) \sinh ^4(e+f x)}{a^2}-\frac {3150 (a-b) \text {ArcSin}\left (\sqrt {\frac {(a-b) \tanh ^2(e+f x)}{a}}\right ) \tanh ^2(e+f x)}{a}-\frac {4200 (a-b) b \text {ArcSin}\left (\sqrt {\frac {(a-b) \tanh ^2(e+f x)}{a}}\right ) \sinh ^2(e+f x) \tanh ^2(e+f x)}{a^2}-\frac {1680 (a-b) b^2 \text {ArcSin}\left (\sqrt {\frac {(a-b) \tanh ^2(e+f x)}{a}}\right ) \sinh ^4(e+f x) \tanh ^2(e+f x)}{a^3}+\frac {1575 (a-b)^2 \text {ArcSin}\left (\sqrt {\frac {(a-b) \tanh ^2(e+f x)}{a}}\right ) \tanh ^4(e+f x)}{a^2}+\frac {2100 (a-b)^2 b \text {ArcSin}\left (\sqrt {\frac {(a-b) \tanh ^2(e+f x)}{a}}\right ) \sinh ^2(e+f x) \tanh ^4(e+f x)}{a^3}+\frac {840 (a-b)^2 b^2 \text {ArcSin}\left (\sqrt {\frac {(a-b) \tanh ^2(e+f x)}{a}}\right ) \sinh ^4(e+f x) \tanh ^4(e+f x)}{a^4}+2100 \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{3/2}+\frac {2800 b \sinh ^2(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{3/2}}{a}+\frac {1120 b^2 \sinh ^4(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{3/2}}{a^2}+96 \, _2F_1\left (2,2;\frac {9}{2};\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{7/2}+24 \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{7/2}+\frac {168 b \, _2F_1\left (2,2;\frac {9}{2};\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sinh ^2(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{7/2}}{a}+\frac {48 b \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sinh ^2(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{7/2}}{a}+\frac {72 b^2 \, _2F_1\left (2,2;\frac {9}{2};\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sinh ^4(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{7/2}}{a^2}+\frac {24 b^2 \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sinh ^4(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{7/2}}{a^2}-1575 \sqrt {\frac {(a-b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right ) \tanh ^2(e+f x)}{a^2}}-\frac {2100 b \sinh ^2(e+f x) \sqrt {\frac {(a-b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right ) \tanh ^2(e+f x)}{a^2}}}{a}-\frac {840 b^2 \sinh ^4(e+f x) \sqrt {\frac {(a-b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right ) \tanh ^2(e+f x)}{a^2}}}{a^2}\right )}{315 a^2 f \sqrt {a+b \sinh ^2(e+f x)} \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (1+\frac {b \sinh ^2(e+f x)}{a}\right ) \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{5/2}} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.93, size = 169, normalized size = 1.26

method	result	size
default	\(\frac {\mathit {`\,int/indef0`\,}\left (\frac {-b^{2} \left (\sinh ^{4}\left (f x +e \right )\right )-2 a b \left (\sinh ^{2}\left (f x +e \right )\right )-a^{2}}{\left (-b^{4} \left (\sinh ^{10}\left (f x +e \right )\right )+\left (-4 a \,b^{3}-b^{4}\right ) \left (\sinh ^{8}\left (f x +e \right )\right )+\left (-6 a^{2} b^{2}-4 a \,b^{3}\right ) \left (\sinh ^{6}\left (f x +e \right )\right )+\left (-4 a^{3} b -6 a^{2} b^{2}\right ) \left (\sinh ^{4}\left (f x +e \right )\right )+\left (-a^{4}-4 a^{3} b \right ) \left (\sinh ^{2}\left (f x +e \right )\right )-a^{4}\right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\)	\(169\)
risch	\(\text {Expression too large to display}\)	\(1155822\)

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,method=_RETURNVERBOSE)

[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 = 0.00, 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)/(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] Leaf count of result is larger than twice the leaf count of optimal. 2640 vs. \(2 (120) = 240\).
time = 0.64, size = 5396, normalized size = 40.27 \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)/(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^...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}{\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)/(a+b*sinh(f*x+e)**2)**(5/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1177 vs. \(2 (120) = 240\).
time = 0.73, size = 1177, normalized size = 8.78 \begin {gather*} -\frac {{\left (\frac {{\left ({\left (\frac {{\left (5 \, a^{9} b^{2} e^{\left (12 \, e\right )} - 42 \, a^{8} b^{3} e^{\left (12 \, e\right )} + 156 \, a^{7} b^{4} e^{\left (12 \, e\right )} - 336 \, a^{6} b^{5} e^{\left (12 \, e\right )} + 462 \, a^{5} b^{6} e^{\left (12 \, e\right )} - 420 \, a^{4} b^{7} e^{\left (12 \, e\right )} + 252 \, a^{3} b^{8} e^{\left (12 \, e\right )} - 96 \, a^{2} b^{9} e^{\left (12 \, e\right )} + 21 \, a b^{10} e^{\left (12 \, e\right )} - 2 \, b^{11} e^{\left (12 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a^{12} e^{\left (12 \, e\right )} - 10 \, a^{11} b e^{\left (12 \, e\right )} + 45 \, a^{10} b^{2} e^{\left (12 \, e\right )} - 120 \, a^{9} b^{3} e^{\left (12 \, e\right )} + 210 \, a^{8} b^{4} e^{\left (12 \, e\right )} - 252 \, a^{7} b^{5} e^{\left (12 \, e\right )} + 210 \, a^{6} b^{6} e^{\left (12 \, e\right )} - 120 \, a^{5} b^{7} e^{\left (12 \, e\right )} + 45 \, a^{4} b^{8} e^{\left (12 \, e\right )} - 10 \, a^{3} b^{9} e^{\left (12 \, e\right )} + a^{2} b^{10} e^{\left (12 \, e\right )}} + \frac {3 \, {\left (8 \, a^{10} b e^{\left (10 \, e\right )} - 73 \, a^{9} b^{2} e^{\left (10 \, e\right )} + 298 \, a^{8} b^{3} e^{\left (10 \, e\right )} - 716 \, a^{7} b^{4} e^{\left (10 \, e\right )} + 1120 \, a^{6} b^{5} e^{\left (10 \, e\right )} - 1190 \, a^{5} b^{6} e^{\left (10 \, e\right )} + 868 \, a^{4} b^{7} e^{\left (10 \, e\right )} - 428 \, a^{3} b^{8} e^{\left (10 \, e\right )} + 136 \, a^{2} b^{9} e^{\left (10 \, e\right )} - 25 \, a b^{10} e^{\left (10 \, e\right )} + 2 \, b^{11} e^{\left (10 \, e\right )}\right )}}{a^{12} e^{\left (12 \, e\right )} - 10 \, a^{11} b e^{\left (12 \, e\right )} + 45 \, a^{10} b^{2} e^{\left (12 \, e\right )} - 120 \, a^{9} b^{3} e^{\left (12 \, e\right )} + 210 \, a^{8} b^{4} e^{\left (12 \, e\right )} - 252 \, a^{7} b^{5} e^{\left (12 \, e\right )} + 210 \, a^{6} b^{6} e^{\left (12 \, e\right )} - 120 \, a^{5} b^{7} e^{\left (12 \, e\right )} + 45 \, a^{4} b^{8} e^{\left (12 \, e\right )} - 10 \, a^{3} b^{9} e^{\left (12 \, e\right )} + a^{2} b^{10} e^{\left (12 \, e\right )}}\right )} e^{\left (2 \, f x\right )} - \frac {3 \, {\left (8 \, a^{10} b e^{\left (8 \, e\right )} - 73 \, a^{9} b^{2} e^{\left (8 \, e\right )} + 298 \, a^{8} b^{3} e^{\left (8 \, e\right )} - 716 \, a^{7} b^{4} e^{\left (8 \, e\right )} + 1120 \, a^{6} b^{5} e^{\left (8 \, e\right )} - 1190 \, a^{5} b^{6} e^{\left (8 \, e\right )} + 868 \, a^{4} b^{7} e^{\left (8 \, e\right )} - 428 \, a^{3} b^{8} e^{\left (8 \, e\right )} + 136 \, a^{2} b^{9} e^{\left (8 \, e\right )} - 25 \, a b^{10} e^{\left (8 \, e\right )} + 2 \, b^{11} e^{\left (8 \, e\right )}\right )}}{a^{12} e^{\left (12 \, e\right )} - 10 \, a^{11} b e^{\left (12 \, e\right )} + 45 \, a^{10} b^{2} e^{\left (12 \, e\right )} - 120 \, a^{9} b^{3} e^{\left (12 \, e\right )} + 210 \, a^{8} b^{4} e^{\left (12 \, e\right )} - 252 \, a^{7} b^{5} e^{\left (12 \, e\right )} + 210 \, a^{6} b^{6} e^{\left (12 \, e\right )} - 120 \, a^{5} b^{7} e^{\left (12 \, e\right )} + 45 \, a^{4} b^{8} e^{\left (12 \, e\right )} - 10 \, a^{3} b^{9} e^{\left (12 \, e\right )} + a^{2} b^{10} e^{\left (12 \, e\right )}}\right )} e^{\left (2 \, f x\right )} - \frac {5 \, a^{9} b^{2} e^{\left (6 \, e\right )} - 42 \, a^{8} b^{3} e^{\left (6 \, e\right )} + 156 \, a^{7} b^{4} e^{\left (6 \, e\right )} - 336 \, a^{6} b^{5} e^{\left (6 \, e\right )} + 462 \, a^{5} b^{6} e^{\left (6 \, e\right )} - 420 \, a^{4} b^{7} e^{\left (6 \, e\right )} + 252 \, a^{3} b^{8} e^{\left (6 \, e\right )} - 96 \, a^{2} b^{9} e^{\left (6 \, e\right )} + 21 \, a b^{10} e^{\left (6 \, e\right )} - 2 \, b^{11} e^{\left (6 \, e\right )}}{a^{12} e^{\left (12 \, e\right )} - 10 \, a^{11} b e^{\left (12 \, e\right )} + 45 \, a^{10} b^{2} e^{\left (12 \, e\right )} - 120 \, a^{9} b^{3} e^{\left (12 \, e\right )} + 210 \, a^{8} b^{4} e^{\left (12 \, e\right )} - 252 \, a^{7} b^{5} e^{\left (12 \, e\right )} + 210 \, a^{6} b^{6} e^{\left (12 \, e\right )} - 120 \, a^{5} b^{7} e^{\left (12 \, e\right )} + 45 \, a^{4} b^{8} e^{\left (12 \, e\right )} - 10 \, a^{3} b^{9} e^{\left (12 \, e\right )} + a^{2} b^{10} e^{\left (12 \, e\right )}}}{{\left (b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b\right )}^{\frac {3}{2}}} - \frac {6 \, \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b} + \sqrt {b}}{2 \, \sqrt {a - b}}\right )}{{\left (a^{2} e^{\left (6 \, e\right )} - 2 \, a b e^{\left (6 \, e\right )} + b^{2} e^{\left (6 \, e\right )}\right )} \sqrt {a - b}}\right )} e^{\left (6 \, e\right )}}{3 \, f} \end {gather*}

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^9*b^2*e^(12*e) - 42*a^8*b^3*e^(12*e) + 156*a^7*b^4*e^(12*e) - 336*a^6*b^5*e^(12*e) + 462*a^5*b^6
*e^(12*e) - 420*a^4*b^7*e^(12*e) + 252*a^3*b^8*e^(12*e) - 96*a^2*b^9*e^(12*e) + 21*a*b^10*e^(12*e) - 2*b^11*e^
(12*e))*e^(2*f*x)/(a^12*e^(12*e) - 10*a^11*b*e^(12*e) + 45*a^10*b^2*e^(12*e) - 120*a^9*b^3*e^(12*e) + 210*a^8*
b^4*e^(12*e) - 252*a^7*b^5*e^(12*e) + 210*a^6*b^6*e^(12*e) - 120*a^5*b^7*e^(12*e) + 45*a^4*b^8*e^(12*e) - 10*a
^3*b^9*e^(12*e) + a^2*b^10*e^(12*e)) + 3*(8*a^10*b*e^(10*e) - 73*a^9*b^2*e^(10*e) + 298*a^8*b^3*e^(10*e) - 716
*a^7*b^4*e^(10*e) + 1120*a^6*b^5*e^(10*e) - 1190*a^5*b^6*e^(10*e) + 868*a^4*b^7*e^(10*e) - 428*a^3*b^8*e^(10*e
) + 136*a^2*b^9*e^(10*e) - 25*a*b^10*e^(10*e) + 2*b^11*e^(10*e))/(a^12*e^(12*e) - 10*a^11*b*e^(12*e) + 45*a^10
*b^2*e^(12*e) - 120*a^9*b^3*e^(12*e) + 210*a^8*b^4*e^(12*e) - 252*a^7*b^5*e^(12*e) + 210*a^6*b^6*e^(12*e) - 12
0*a^5*b^7*e^(12*e) + 45*a^4*b^8*e^(12*e) - 10*a^3*b^9*e^(12*e) + a^2*b^10*e^(12*e)))*e^(2*f*x) - 3*(8*a^10*b*e
^(8*e) - 73*a^9*b^2*e^(8*e) + 298*a^8*b^3*e^(8*e) - 716*a^7*b^4*e^(8*e) + 1120*a^6*b^5*e^(8*e) - 1190*a^5*b^6*
e^(8*e) + 868*a^4*b^7*e^(8*e) - 428*a^3*b^8*e^(8*e) + 136*a^2*b^9*e^(8*e) - 25*a*b^10*e^(8*e) + 2*b^11*e^(8*e)
)/(a^12*e^(12*e) - 10*a^11*b*e^(12*e) + 45*a^10*b^2*e^(12*e) - 120*a^9*b^3*e^(12*e) + 210*a^8*b^4*e^(12*e) - 2
52*a^7*b^5*e^(12*e) + 210*a^6*b^6*e^(12*e) - 120*a^5*b^7*e^(12*e) + 45*a^4*b^8*e^(12*e) - 10*a^3*b^9*e^(12*e)
+ a^2*b^10*e^(12*e)))*e^(2*f*x) - (5*a^9*b^2*e^(6*e) - 42*a^8*b^3*e^(6*e) + 156*a^7*b^4*e^(6*e) - 336*a^6*b^5*
e^(6*e) + 462*a^5*b^6*e^(6*e) - 420*a^4*b^7*e^(6*e) + 252*a^3*b^8*e^(6*e) - 96*a^2*b^9*e^(6*e) + 21*a*b^10*e^(
6*e) - 2*b^11*e^(6*e))/(a^12*e^(12*e) - 10*a^11*b*e^(12*e) + 45*a^10*b^2*e^(12*e) - 120*a^9*b^3*e^(12*e) + 210
*a^8*b^4*e^(12*e) - 252*a^7*b^5*e^(12*e) + 210*a^6*b^6*e^(12*e) - 120*a^5*b^7*e^(12*e) + 45*a^4*b^8*e^(12*e) -
10*a^3*b^9*e^(12*e) + a^2*b^10*e^(12*e)))/(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) - 6*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*e^(6*e) - 2*a*b*e^(6*e) + b^2*e^(6*e))*sqrt(a - b)))*e^(6*e)/f

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\mathrm {cosh}\left (e+f\,x\right )\,{\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)*(a + b*sinh(e + f*x)^2)^(5/2)),x)

[Out]

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

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