∫ ∘ ____________ a + b sinh2(e + fx) tanh5(e + fx) dx [457]

Optimal. Leaf size=187 \[ -\frac {\left (8 a^2-24 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{8 (a-b)^{3/2} f}+\frac {\left (8 a^2-24 a b+15 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)}}{8 (a-b)^2 f}+\frac {(8 a-7 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{8 (a-b)^2 f}-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 (a-b) f} \]

-1/8*(8*a^2-24*a*b+15*b^2)*arctanh((a+b*sinh(f*x+e)^2)^(1/2)/(a-b)^(1/2))/(a-b)^(3/2)/f+1/8*(8*a-7*b)*sech(f*x

+e)^2*(a+b*sinh(f*x+e)^2)^(3/2)/(a-b)^2/f-1/4*sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2)/(a-b)/f+1/8*(8*a^2-24*a*

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Rubi [A]
time = 0.17, antiderivative size = 187, 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 = {3273, 91, 79, 52, 65, 214} \begin {gather*} \frac {\left (8 a^2-24 a b+15 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)}}{8 f (a-b)^2}-\frac {\left (8 a^2-24 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{8 f (a-b)^{3/2}}-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 f (a-b)}+\frac {(8 a-7 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{8 f (a-b)^2} \end {gather*}

-1/8*((8*a^2 - 24*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/((a - b)^(3/2)*f) + ((8*a^2

- 24*a*b + 15*b^2)*Sqrt[a + b*Sinh[e + f*x]^2])/(8*(a - b)^2*f) + ((8*a - 7*b)*Sech[e + f*x]^2*(a + b*Sinh[e +

f*x]^2)^(3/2))/(8*(a - b)^2*f) - (Sech[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2))/(4*(a - b)*f)

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(

m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a

, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !IntegerQ[n] || (G

tQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub

st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &

& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f

))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1

) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,

f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || L

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d

)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In

t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*

(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ

[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))

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

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = Free

Factors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[x^((m - 1)/2)*((a + b*ff*x)^p/(1 - ff*x)^((m

+ 1)/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

\begin {align*} \int \sqrt {a+b \sinh ^2(e+f x)} \tanh ^5(e+f x) \, dx &=\frac {\text {Subst}\left (\int \frac {x^2 \sqrt {a+b x}}{(1+x)^3} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 (a-b) f}+\frac {\text {Subst}\left (\int \frac {\left (\frac {1}{2} (-4 a+3 b)+2 (a-b) x\right ) \sqrt {a+b x}}{(1+x)^2} \, dx,x,\sinh ^2(e+f x)\right )}{4 (a-b) f}\\ &=\frac {(8 a-7 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{8 (a-b)^2 f}-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 (a-b) f}+\frac {\left (8 a^2-24 a b+15 b^2\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{1+x} \, dx,x,\sinh ^2(e+f x)\right )}{16 (a-b)^2 f}\\ &=\frac {\left (8 a^2-24 a b+15 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)}}{8 (a-b)^2 f}+\frac {(8 a-7 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{8 (a-b)^2 f}-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 (a-b) f}+\frac {\left (8 a^2-24 a b+15 b^2\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{16 (a-b) f}\\ &=\frac {\left (8 a^2-24 a b+15 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)}}{8 (a-b)^2 f}+\frac {(8 a-7 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{8 (a-b)^2 f}-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 (a-b) f}+\frac {\left (8 a^2-24 a b+15 b^2\right ) \text {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^2(e+f x)}\right )}{8 (a-b) b f}\\ &=-\frac {\left (8 a^2-24 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{8 (a-b)^{3/2} f}+\frac {\left (8 a^2-24 a b+15 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)}}{8 (a-b)^2 f}+\frac {(8 a-7 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{8 (a-b)^2 f}-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 (a-b) f}\\ \end {align*}

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Mathematica [A]
time = 0.40, size = 151, normalized size = 0.81 \begin {gather*} -\frac {-\left ((8 a-7 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}\right )+2 (a-b) \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}+\left (8 a^2-24 a b+15 b^2\right ) \left (\sqrt {a-b} \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )-\sqrt {a+b \sinh ^2(e+f x)}\right )}{8 (a-b)^2 f} \end {gather*}

Integrate[Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]^5,x]

-1/8*(-((8*a - 7*b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2)) + 2*(a - b)*Sech[e + f*x]^4*(a + b*Sinh[e +

f*x]^2)^(3/2) + (8*a^2 - 24*a*b + 15*b^2)*(Sqrt[a - b]*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]] - Sqr

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.43, size = 43, normalized size = 0.23


method	result	size

default	\(\frac {\mathit {`\,int/indef0`\,}\left (\frac {\sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, \left (\sinh ^{5}\left (f x +e \right )\right )}{\cosh \left (f x +e \right )^{6}}, \sinh \left (f x +e \right )\right )}{f}\)	\(43\)

int((a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)^5,x,method=_RETURNVERBOSE)

`int/indef0`((a+b*sinh(f*x+e)^2)^(1/2)*sinh(f*x+e)^5/cosh(f*x+e)^6,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*}

integrate((a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)^5,x, algorithm="maxima")

integrate(sqrt(b*sinh(f*x + e)^2 + a)*tanh(f*x + e)^5, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2254 vs. \(2 (167) = 334\).
time = 1.13, size = 4704, normalized size = 25.16 \begin {gather*} \text {Too large to display} \end {gather*}

integrate((a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)^5,x, algorithm="fricas")

[-1/16*(((8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)^9 + 9*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)*sinh(f*x + e)^8

+ (8*a^2 - 24*a*b + 15*b^2)*sinh(f*x + e)^9 + 4*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)^7 + 4*(9*(8*a^2 - 24*

a*b + 15*b^2)*cosh(f*x + e)^2 + 8*a^2 - 24*a*b + 15*b^2)*sinh(f*x + e)^7 + 28*(3*(8*a^2 - 24*a*b + 15*b^2)*cos

h(f*x + e)^3 + (8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e))*sinh(f*x + e)^6 + 6*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x

+ e)^5 + 6*(21*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)^4 + 14*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)^2 + 8*a

^2 - 24*a*b + 15*b^2)*sinh(f*x + e)^5 + 2*(63*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)^5 + 70*(8*a^2 - 24*a*b +

15*b^2)*cosh(f*x + e)^3 + 15*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e))*sinh(f*x + e)^4 + 4*(8*a^2 - 24*a*b + 1

5*b^2)*cosh(f*x + e)^3 + 4*(21*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)^6 + 35*(8*a^2 - 24*a*b + 15*b^2)*cosh(f

*x + e)^4 + 15*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)^2 + 8*a^2 - 24*a*b + 15*b^2)*sinh(f*x + e)^3 + 12*(3*(8

*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)^7 + 7*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)^5 + 5*(8*a^2 - 24*a*b + 15

*b^2)*cosh(f*x + e)^3 + (8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e))*sinh(f*x + e)^2 + (8*a^2 - 24*a*b + 15*b^2)*c

osh(f*x + e) + (9*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)^8 + 28*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)^6 + 3

0*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)^4 + 12*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)^2 + 8*a^2 - 24*a*b +

15*b^2)*sinh(f*x + e))*sqrt(a - b)*log((b*cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*sinh(f*x + e)^3 + b*sinh(f*x + e

)^4 + 2*(4*a - 3*b)*cosh(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2 + 4*a - 3*b)*sinh(f*x + e)^2 + 4*sqrt(2)*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))*(cosh(f*x + e) + sinh(f*x + e)) + 4*(b*cosh(f*x + e)^3 + (4*a - 3*b)*cosh(f*x + e))*sinh(f*x

+ e) + 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)*si

nh(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)) - 4*sqrt(2)*(2*(a^

2 - 2*a*b + b^2)*cosh(f*x + e)^8 + 16*(a^2 - 2*a*b + b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + 2*(a^2 - 2*a*b + b^2

)*sinh(f*x + e)^8 + (16*a^2 - 33*a*b + 17*b^2)*cosh(f*x + e)^6 + (56*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 + 16*

a^2 - 33*a*b + 17*b^2)*sinh(f*x + e)^6 + 2*(56*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^3 + 3*(16*a^2 - 33*a*b + 17*b

^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(10*a^2 - 21*a*b + 11*b^2)*cosh(f*x + e)^4 + (140*(a^2 - 2*a*b + b^2)*c

osh(f*x + e)^4 + 15*(16*a^2 - 33*a*b + 17*b^2)*cosh(f*x + e)^2 + 20*a^2 - 42*a*b + 22*b^2)*sinh(f*x + e)^4 + 4

*(28*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^5 + 5*(16*a^2 - 33*a*b + 17*b^2)*cosh(f*x + e)^3 + 2*(10*a^2 - 21*a*b +

11*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + (16*a^2 - 33*a*b + 17*b^2)*cosh(f*x + e)^2 + (56*(a^2 - 2*a*b + b^2)

*cosh(f*x + e)^6 + 15*(16*a^2 - 33*a*b + 17*b^2)*cosh(f*x + e)^4 + 12*(10*a^2 - 21*a*b + 11*b^2)*cosh(f*x + e)

^2 + 16*a^2 - 33*a*b + 17*b^2)*sinh(f*x + e)^2 + 2*a^2 - 4*a*b + 2*b^2 + 2*(8*(a^2 - 2*a*b + b^2)*cosh(f*x + e

)^7 + 3*(16*a^2 - 33*a*b + 17*b^2)*cosh(f*x + e)^5 + 4*(10*a^2 - 21*a*b + 11*b^2)*cosh(f*x + e)^3 + (16*a^2 -

33*a*b + 17*b^2)*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^2 - 2*a*b + b^2)*f*cosh(f*x + e)^9 + 9*(a^2

- 2*a*b + b^2)*f*cosh(f*x + e)*sinh(f*x + e)^8 + (a^2 - 2*a*b + b^2)*f*sinh(f*x + e)^9 + 4*(a^2 - 2*a*b + b^2)

*f*cosh(f*x + e)^7 + 4*(9*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^2 + (a^2 - 2*a*b + b^2)*f)*sinh(f*x + e)^7 + 6*(

a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^5 + 28*(3*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^3 + (a^2 - 2*a*b + b^2)*f*cos

h(f*x + e))*sinh(f*x + e)^6 + 6*(21*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^4 + 14*(a^2 - 2*a*b + b^2)*f*cosh(f*x

+ e)^2 + (a^2 - 2*a*b + b^2)*f)*sinh(f*x + e)^5 + 4*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^3 + 2*(63*(a^2 - 2*a*b

+ b^2)*f*cosh(f*x + e)^5 + 70*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^3 + 15*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e))

*sinh(f*x + e)^4 + 4*(21*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^6 + 35*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^4 + 15

*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^2 + (a^2 - 2*a*b + b^2)*f)*sinh(f*x + e)^3 + (a^2 - 2*a*b + b^2)*f*cosh(f

*x + e) + 12*(3*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^7 + 7*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^5 + 5*(a^2 - 2*a

*b + b^2)*f*cosh(f*x + e)^3 + (a^2 - 2*a*b + b^2)*f*cosh(f*x + e))*sinh(f*x + e)^2 + (9*(a^2 - 2*a*b + b^2)*f*

cosh(f*x + e)^8 + 28*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^6 + 30*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^4 + 12*(a^

2 - 2*a*b + b^2)*f*cosh(f*x + e)^2 + (a^2 - 2*a*b + b^2)*f)*sinh(f*x + e)), -1/8*(((8*a^2 - 24*a*b + 15*b^2)*c

osh(f*x + e)^9 + 9*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)*sinh(f*x + e)^8 + (8*a^2 - 24*a*b + 15*b^2)*sinh(f*

x + e)^9 + 4*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)^7 + 4*(9*(8*a^2 - 24*a*b + 15*b^2)*cosh(f*x + e)^2 + 8*a^

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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 )}} \tanh ^{5}{\left (e + f x \right )}\, dx \end {gather*}

integrate((a+b*sinh(f*x+e)**2)**(1/2)*tanh(f*x+e)**5,x)

Integral(sqrt(a + b*sinh(e + f*x)**2)*tanh(e + f*x)**5, 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*}

integrate((a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)^5,x, algorithm="giac")

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Evaluation time:

0.93Unable to divide, perhaps due to rounding error%%%{%%{[262144,0]:[1,0,%%%{-1,[1]%%%}]%%},[10,13,13]%%%}+%%

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

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3.5.57 \(\int \sqrt {a+b \sinh ^2(e+f x)} \tanh ^5(e+f x) \, dx\) [457]