∫ tanh5 (e+fx)______∘ a + b sinh2(e + fx) dx [479]

Optimal. Leaf size=142 \[ -\frac {\left (8 a^2-8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{8 (a-b)^{5/2} f}+\frac {(8 a-5 b) \text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{8 (a-b)^2 f}-\frac {\text {sech}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{4 (a-b) f} \]

-1/8*(8*a^2-8*a*b+3*b^2)*arctanh((a+b*sinh(f*x+e)^2)^(1/2)/(a-b)^(1/2))/(a-b)^(5/2)/f+1/8*(8*a-5*b)*sech(f*x+e

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

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Rubi [A]
time = 0.13, antiderivative size = 142, 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 = {3273, 91, 79, 65, 214} \begin {gather*} -\frac {\left (8 a^2-8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{8 f (a-b)^{5/2}}-\frac {\text {sech}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{4 f (a-b)}+\frac {(8 a-5 b) \text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{8 f (a-b)^2} \end {gather*}

-1/8*((8*a^2 - 8*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/((a - b)^(5/2)*f) + ((8*a - 5*

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

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

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Mathematica [A]
time = 0.33, size = 116, normalized size = 0.82 \begin {gather*} \frac {\left (-8 a^2+8 a b-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )+\sqrt {a-b} \text {sech}^2(e+f x) \left (8 a-5 b-2 (a-b) \text {sech}^2(e+f x)\right ) \sqrt {a+b \sinh ^2(e+f x)}}{8 (a-b)^{5/2} f} \end {gather*}

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

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

- 5*b - 2*(a - b)*Sech[e + f*x]^2)*Sqrt[a + b*Sinh[e + f*x]^2])/(8*(a - b)^(5/2)*f)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.45, size = 43, normalized size = 0.30


method	result	size

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

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

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

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

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1952 vs. \(2 (126) = 252\).
time = 0.64, size = 4100, normalized size = 28.87 \begin {gather*} \text {Too large to display} \end {gather*}

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

[1/16*(((8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^8 + 8*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + (8

*a^2 - 8*a*b + 3*b^2)*sinh(f*x + e)^8 + 4*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^6 + 4*(7*(8*a^2 - 8*a*b + 3*b^

2)*cosh(f*x + e)^2 + 8*a^2 - 8*a*b + 3*b^2)*sinh(f*x + e)^6 + 8*(7*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^3 + 3

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

*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^4 + 30*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^2 + 24*a^2 - 24*a*b + 9*b^2)*

sinh(f*x + e)^4 + 8*(7*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^5 + 10*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^3 +

3*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^2 + 4*(7*(8

*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^6 + 15*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^4 + 9*(8*a^2 - 8*a*b + 3*b^2)

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

*cosh(f*x + e)^7 + 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^5 + 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^3 + (8*

a^2 - 8*a*b + 3*b^2)*cosh(f*x + e))*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*co

sh(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)*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)) + 4*sqrt(2)*((8*a^2 - 13*a*b + 5*b^2)*cosh(f*x + e)^5 + 5*(8*a^2 - 13*a*b + 5*b^2)*cosh(f*x + e)*sin

h(f*x + e)^4 + (8*a^2 - 13*a*b + 5*b^2)*sinh(f*x + e)^5 + 2*(4*a^2 - 5*a*b + b^2)*cosh(f*x + e)^3 + 2*(5*(8*a^

2 - 13*a*b + 5*b^2)*cosh(f*x + e)^2 + 4*a^2 - 5*a*b + b^2)*sinh(f*x + e)^3 + 2*(5*(8*a^2 - 13*a*b + 5*b^2)*cos

h(f*x + e)^3 + 3*(4*a^2 - 5*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^2 + (8*a^2 - 13*a*b + 5*b^2)*cosh(f*x + e)

+ (5*(8*a^2 - 13*a*b + 5*b^2)*cosh(f*x + e)^4 + 6*(4*a^2 - 5*a*b + b^2)*cosh(f*x + e)^2 + 8*a^2 - 13*a*b + 5*

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^3 - 3*a^2*b + 3*a*b^2 - b^3)*f*cosh(f*x + e)^8 + 8*(a^3 - 3*a^2*b + 3*a

*b^2 - b^3)*f*cosh(f*x + e)*sinh(f*x + e)^7 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f*sinh(f*x + e)^8 + 4*(a^3 - 3*a

^2*b + 3*a*b^2 - b^3)*f*cosh(f*x + e)^6 + 4*(7*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f*cosh(f*x + e)^2 + (a^3 - 3*a^

2*b + 3*a*b^2 - b^3)*f)*sinh(f*x + e)^6 + 6*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f*cosh(f*x + e)^4 + 8*(7*(a^3 - 3*

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

+ 2*(35*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f*cosh(f*x + e)^4 + 30*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f*cosh(f*x + e)

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

2 + 8*(7*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f*cosh(f*x + e)^5 + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f*cosh(f*x + e

)^3 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(7*(a^3 - 3*a^2*b + 3*a*b^2 - b^3

)*f*cosh(f*x + e)^6 + 15*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f*cosh(f*x + e)^4 + 9*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)

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

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

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

+ e)), -1/8*(((8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^8 + 8*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)*sinh(f*x + e)

^7 + (8*a^2 - 8*a*b + 3*b^2)*sinh(f*x + e)^8 + 4*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^6 + 4*(7*(8*a^2 - 8*a*b

+ 3*b^2)*cosh(f*x + e)^2 + 8*a^2 - 8*a*b + 3*b^2)*sinh(f*x + e)^6 + 8*(7*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e

)^3 + 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 6*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^4 + 2

*(35*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^4 + 30*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^2 + 24*a^2 - 24*a*b +

9*b^2)*sinh(f*x + e)^4 + 8*(7*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^5 + 10*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x +

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

4*(7*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^6 + 15*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^4 + 9*(8*a^2 - 8*a*b +

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

3*b^2)*cosh(f*x + e)^7 + 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^5 + 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(f*x + e)^

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{5}{\left (e + f x \right )}}{\sqrt {a + b \sinh ^{2}{\left (e + f x \right )}}}\, dx \end {gather*}

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

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2265 vs. \(2 (126) = 252\).
time = 9.72, size = 2265, normalized size = 15.95 \begin {gather*} \text {Too large to display} \end {gather*}

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

-1/12*(24*arctan(-(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))*e^e/sqrt(-b) - 3*(15*a^2*e^e - 20*a*b*e^e + 8*b^2*e^e)*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 - 2*a*

b + b^2)*sqrt(a - b)) + 2*(21*(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))^7*a^2*e^e - 12*(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))^7*a*b*e^e + 243*(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))^6*a^2*sqrt(b)*e^e - 276*(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))^6*a*b^(3/2)*e^e + 96*(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))^6*b^(5/2)*e^e + 436*(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))^5*a^3*e^e - 363*(sqrt(b)*e^(2*f*x + 2*e) - s

qrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*a^2*b*e^e + 180*(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))^5*a*b^2*e^e - 64*(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))^5*b^3*e^e + 1796*(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))^4*a^3*sqrt(b)*e^

e - 2421*(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))^4

*a^2*b^(3/2)*e^e + 1164*(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))^4*a*b^(5/2)*e^e - 224*(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))^4*b^(7/2)*e^e + 1840*(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))^3*a^4*e^e - 1176*(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))^3*a^3*b*e^e - 1497*(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))^3*a^2*b^2*e^e + 1532*(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))^3*a*b^3*e^e - 384*(sqrt(b)*e^(2*f*x + 2*e) - s

qrt(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*b^4*e^e + 7056*(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))^2*a^4*sqrt(b)*e^e - 18136*(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))^2*a^3*b^(3/2)*e

^e + 18561*(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))

^2*a^2*b^(5/2)*e^e - 8988*(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))^2*a*b^(7/2)*e^e + 1696*(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))^2*b^(9/2)*e^e + 4800*(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))*a^5*e^e - 17328*(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))*a^4*b*e^e + 27364*(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))*a^3*b^2*e^e - 22737*(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))*a^2*b^3*e^e + 9564*(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))*a*b^4*e^e - 1600*(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))*b^5*e^e - 1344*a^5*sqrt(b)*e^e

+ 6128*a^4*b^(3/2)*e^e - 10284*a^3*b^(5/2)*e^e + 8193*a^2*b^(7/2)*e^e - 3164*a*b^(9/2)*e^e + 480*b^(11/2)*e^e

)/(((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))^2 + 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) +

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

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

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

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3.5.79 \(\int \frac {\tanh ^5(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx\) [479]