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

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

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

[Out]

-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*

b+15*b^2)*(a+b*sinh(f*x+e)^2)^(1/2)/(a-b)^2/f

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Rubi [A] time = 0.22, 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 = {3194, 89, 78, 50, 63, 208} \[ \frac {\left (8 a^2-24 a b+15 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)}}{8 f (a-b)^2}-\frac {\left (8 a^2-24 a b+15 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{8 f (a-b)^{3/2}}-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 f (a-b)}+\frac {(8 a-7 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{8 f (a-b)^2} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 50

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]

Rule 63

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,

b, c, d, m, n, x]

Rule 78

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] || LtQ

[p, n]))))

Rule 89

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])))

Rule 208

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

b}, x] && NegQ[a/b]

Rule 3194

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]

Rubi steps

\begin {align*} \int \sqrt {a+b \sinh ^2(e+f x)} \tanh ^5(e+f x) \, dx &=\frac {\operatorname {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 {\operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.56, size = 151, normalized size = 0.81 \[ -\frac {\left (8 a^2-24 a b+15 b^2\right ) \left (\sqrt {a-b} \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )-\sqrt {a+b \sinh ^2(e+f x)}\right )+2 (a-b) \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}-(8 a-7 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{8 f (a-b)^2} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-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

t[a + b*Sinh[e + f*x]^2]))/((a - b)^2*f)

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fricas [B] time = 1.61, size = 4704, normalized size = 25.16 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-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^

2 - 24*a*b + 15*b^2)*sinh(f*x + e)^7 + 28*(3*(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)^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 + 15*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)*cosh(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 + 30*(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)*

arctan(-1/2*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*c

osh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2))/((a - b)*cosh(f*x + e) + (a - b)*sinh(f*x + e))) - 2*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)*cosh(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)/(cos

h(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

*cosh(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*co

sh(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))]

________________________________________________________________________________________

giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

1.95Unable to divide, perhaps due to rounding error%%%{%%{[262144,0]:[1,0,%%%{-1,[1]%%%}]%%},[10,13,13]%%%}+%%

%{%%{[%%%{-1572864,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[10,13,12]%%%}+%%%{%%{[%%%{3932160,[2]%%%},0]:[1,0,%%%{-

1,[1]%%%}]%%},[10,13,11]%%%}+%%%{%%{[%%%{-5242880,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[10,13,10]%%%}+%%%{%%{[%%

%{3932160,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[10,13,9]%%%}+%%%{%%{[%%%{-1572864,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}

]%%},[10,13,8]%%%}+%%%{%%{[%%%{262144,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[10,13,7]%%%}+%%%{%%%{-2621440,[1]%%%

},[9,13,13]%%%}+%%%{%%%{15728640,[2]%%%},[9,13,12]%%%}+%%%{%%%{-39321600,[3]%%%},[9,13,11]%%%}+%%%{%%%{5242880

0,[4]%%%},[9,13,10]%%%}+%%%{%%%{-39321600,[5]%%%},[9,13,9]%%%}+%%%{%%%{15728640,[6]%%%},[9,13,8]%%%}+%%%{%%%{-

2621440,[7]%%%},[9,13,7]%%%}+%%%{%%{[5242880,0]:[1,0,%%%{-1,[1]%%%}]%%},[8,13,14]%%%}+%%%{%%{[%%%{-24903680,[1

]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,13,13]%%%}+%%%{%%{[%%%{39321600,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,13,1

2]%%%}+%%%{%%{[%%%{-6553600,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,13,11]%%%}+%%%{%%{[%%%{-52428800,[4]%%%},0]:

[1,0,%%%{-1,[1]%%%}]%%},[8,13,10]%%%}+%%%{%%{[%%%{66846720,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,13,9]%%%}+%%%

{%%{[%%%{-34078720,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,13,8]%%%}+%%%{%%{[%%%{6553600,[7]%%%},0]:[1,0,%%%{-1,

[1]%%%}]%%},[8,13,7]%%%}+%%%{%%%{-41943040,[1]%%%},[7,13,14]%%%}+%%%{%%%{262144000,[2]%%%},[7,13,13]%%%}+%%%{%

%%{-692060160,[3]%%%},[7,13,12]%%%}+%%%{%%%{996147200,[4]%%%},[7,13,11]%%%}+%%%{%%%{-838860800,[5]%%%},[7,13,1

0]%%%}+%%%{%%%{408944640,[6]%%%},[7,13,9]%%%}+%%%{%%%{-104857600,[7]%%%},[7,13,8]%%%}+%%%{%%%{10485760,[8]%%%}

,[7,13,7]%%%}+%%%{%%{[41943040,0]:[1,0,%%%{-1,[1]%%%}]%%},[6,13,15]%%%}+%%%{%%{[%%%{-188743680,[1]%%%},0]:[1,0

,%%%{-1,[1]%%%}]%%},[6,13,14]%%%}+%%%{%%{[%%%{201850880,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,13,13]%%%}+%%%{%

%{[%%%{403701760,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,13,12]%%%}+%%%{%%{[%%%{-1376256000,[4]%%%},0]:[1,0,%%%{

-1,[1]%%%}]%%},[6,13,11]%%%}+%%%{%%{[%%%{1688207360,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,13,10]%%%}+%%%{%%{[%

%%{-1082654720,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,13,9]%%%}+%%%{%%{[%%%{361758720,[7]%%%},0]:[1,0,%%%{-1,[1

]%%%}]%%},[6,13,8]%%%}+%%%{%%{[%%%{-49807360,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,13,7]%%%}+%%%{%%%{-25165824

0,[1]%%%},[5,13,15]%%%}+%%%{%%%{1719664640,[2]%%%},[5,13,14]%%%}+%%%{%%%{-5057282048,[3]%%%},[5,13,13]%%%}+%%%

{%%%{8323596288,[4]%%%},[5,13,12]%%%}+%%%{%%%{-8330936320,[5]%%%},[5,13,11]%%%}+%%%{%%%{5138022400,[6]%%%},[5,

13,10]%%%}+%%%{%%%{-1871708160,[7]%%%},[5,13,9]%%%}+%%%{%%%{354418688,[8]%%%},[5,13,8]%%%}+%%%{%%%{-24117248,[

9]%%%},[5,13,7]%%%}+%%%{%%{[167772160,0]:[1,0,%%%{-1,[1]%%%}]%%},[4,13,16]%%%}+%%%{%%{[%%%{-880803840,[1]%%%},

0]:[1,0,%%%{-1,[1]%%%}]%%},[4,13,15]%%%}+%%%{%%{[%%%{1373634560,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,13,14]%%

%}+%%%{%%{[%%%{1009254400,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,13,13]%%%}+%%%{%%{[%%%{-6716129280,[4]%%%},0]:

[1,0,%%%{-1,[1]%%%}]%%},[4,13,12]%%%}+%%%{%%{[%%%{10881597440,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,13,11]%%%}

+%%%{%%{[%%%{-9395240960,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,13,10]%%%}+%%%{%%{[%%%{4694999040,[7]%%%},0]:[1

,0,%%%{-1,[1]%%%}]%%},[4,13,9]%%%}+%%%{%%{[%%%{-1284505600,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,13,8]%%%}+%%%

{%%{[%%%{149422080,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,13,7]%%%}+%%%{%%%{-671088640,[1]%%%},[3,13,16]%%%}+%%

%{%%%{5200936960,[2]%%%},[3,13,15]%%%}+%%%{%%%{-17741905920,[3]%%%},[3,13,14]%%%}+%%%{%%%{34907095040,[4]%%%},

[3,13,13]%%%}+%%%{%%%{-43557847040,[5]%%%},[3,13,12]%%%}+%%%{%%%{35641098240,[6]%%%},[3,13,11]%%%}+%%%{%%%{-19

042140160,[7]%%%},[3,13,10]%%%}+%%%{%%%{6364856320,[8]%%%},[3,13,9]%%%}+%%%{%%%{-1195376640,[9]%%%},[3,13,8]%%

%}+%%%{%%%{94371840,[10]%%%},[3,13,7]%%%}+%%%{%%{[335544320,0]:[1,0,%%%{-1,[1]%%%}]%%},[2,13,17]%%%}+%%%{%%{[%

%%{-2348810240,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,13,16]%%%}+%%%{%%{[%%%{6668943360,[2]%%%},0]:[1,0,%%%{-1,

[1]%%%}]%%},[2,13,15]%%%}+%%%{%%{[%%%{-8912896000,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,13,14]%%%}+%%%{%%{[%%%

{2507407360,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,13,13]%%%}+%%%{%%{[%%%{10058465280,[5]%%%},0]:[1,0,%%%{-1,[1

]%%%}]%%},[2,13,12]%%%}+%%%{%%{[%%%{-17292328960,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,13,11]%%%}+%%%{%%{[%%%{

13961789440,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,13,10]%%%}+%%%{%%{[%%%{-6429081600,[8]%%%},0]:[1,0,%%%{-1,[1

]%%%}]%%},[2,13,9]%%%}+%%%{%%{[%%%{1627914240,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,13,8]%%%}+%%%{%%{[%%%{-176

947200,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,13,7]%%%}+%%%{%%%{-671088640,[1]%%%},[1,13,17]%%%}+%%%{%%%{60397

97760,[2]%%%},[1,13,16]%%%}+%%%{%%%{-24410849280,[3]%%%},[1,13,15]%%%}+%%%{%%%{58342768640,[4]%%%},[1,13,14]%%

%}+%%%{%%%{-91312619520,[5]%%%},[1,13,13]%%%}+%%%{%%%{97784954880,[6]%%%},[1,13,12]%%%}+%%%{%%%{-72558837760,[

7]%%%},[1,13,11]%%%}+%%%{%%%{36836474880,[8]%%%},[1,13,10]%%%}+%%%{%%%{-12244746240,[9]%%%},[1,13,9]%%%}+%%%{%

%%{2406481920,[10]%%%},[1,13,8]%%%}+%%%{%%%{-212336640,[11]%%%},[1,13,7]%%%}+%%%{%%{[268435456,0]:[1,0,%%%{-1,

[1]%%%}]%%},[0,13,18]%%%}+%%%{%%{[%%%{-2617245696,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,13,17]%%%}+%%%{%%{[%%%

{11576279040,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,13,16]%%%}+%%%{%%{[%%%{-30660362240,[3]%%%},0]:[1,0,%%%{-1,

[1]%%%}]%%},[0,13,15]%%%}+%%%{%%{[%%%{54027878400,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,13,14]%%%}+%%%{%%{[%%%

{-66507767808,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,13,13]%%%}+%%%{%%{[%%%{58359021568,[6]%%%},0]:[1,0,%%%{-1,

[1]%%%}]%%},[0,13,12]%%%}+%%%{%%{[%%%{-36502241280,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,13,11]%%%}+%%%{%%{[%%

%{15948840960,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,13,10]%%%}+%%%{%%{[%%%{-4636016640,[9]%%%},0]:[1,0,%%%{-1,

[1]%%%}]%%},[0,13,9]%%%}+%%%{%%{[%%%{806879232,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,13,8]%%%}+%%%{%%{[%%%{-6

3700992,[11]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,13,7]%%%} / %%%{%%{poly1[%%%{1,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%

%},[10,0,0]%%%}+%%%{%%%{-10,[3]%%%},[9,0,0]%%%}+%%%{%%{[%%%{20,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,0,1]%%%}+

%%%{%%{poly1[%%%{25,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[8,0,0]%%%}+%%%{%%%{-160,[3]%%%},[7,0,1]%%%}+%%%{%%%{40

,[4]%%%},[7,0,0]%%%}+%%%{%%{poly1[%%%{160,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,2]%%%}+%%%{%%{[%%%{240,[3]%%

%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,1]%%%}+%%%{%%{poly1[%%%{-190,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[6,0,0]%%%}

+%%%{%%%{-960,[3]%%%},[5,0,2]%%%}+%%%{%%%{800,[4]%%%},[5,0,1]%%%}+%%%{%%%{-92,[5]%%%},[5,0,0]%%%}+%%%{%%{[%%%{

640,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,3]%%%}+%%%{%%{poly1[%%%{480,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,

0,2]%%%}+%%%{%%{[%%%{-1480,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[4,0,1]%%%}+%%%{%%{poly1[%%%{570,[5]%%%},0]:[1,0

,%%%{-1,[1]%%%}]%%},[4,0,0]%%%}+%%%{%%%{-2560,[3]%%%},[3,0,3]%%%}+%%%{%%%{4480,[4]%%%},[3,0,2]%%%}+%%%{%%%{-24

00,[5]%%%},[3,0,1]%%%}+%%%{%%%{360,[6]%%%},[3,0,0]%%%}+%%%{%%{[%%%{1280,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,

0,4]%%%}+%%%{%%{[%%%{-1280,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,3]%%%}+%%%{%%{poly1[%%%{-1440,[4]%%%},0]:[1

,0,%%%{-1,[1]%%%}]%%},[2,0,2]%%%}+%%%{%%{[%%%{2160,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,1]%%%}+%%%{%%{poly1

[%%%{-675,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[2,0,0]%%%}+%%%{%%%{-2560,[3]%%%},[1,0,4]%%%}+%%%{%%%{7680,[4]%%%

},[1,0,3]%%%}+%%%{%%%{-8640,[5]%%%},[1,0,2]%%%}+%%%{%%%{4320,[6]%%%},[1,0,1]%%%}+%%%{%%%{-810,[7]%%%},[1,0,0]%

%%}+%%%{%%{[%%%{1024,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,5]%%%}+%%%{%%{[%%%{-3840,[3]%%%},0]:[1,0,%%%{-1,[

1]%%%}]%%},[0,0,4]%%%}+%%%{%%{[%%%{5760,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,3]%%%}+%%%{%%{poly1[%%%{-4320,

[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,2]%%%}+%%%{%%{[%%%{1620,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,1]%%%}

+%%%{%%{poly1[%%%{-243,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[0,0,0]%%%} Error: Bad Argument Value

________________________________________________________________________________________

maple [C] time = 0.25, size = 43, normalized size = 0.23 \[ \frac {\mathit {`\,int/indef0`\,}\left (\frac {\sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, \left (\sinh ^{5}\left (f x +e \right )\right )}{\cosh \left (f x +e \right )^{6}}, \sinh \left (f x +e \right )\right )}{f} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

`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

________________________________________________________________________________________

maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sinh \left (f x + e\right )^{2} + a} \tanh \left (f x + e\right )^{5}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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