∫ tanh3 (e+fx)___ (a+b sinh2(e+fx))3∕2 dx [491]

3.5.91 \(\int \frac {\tanh ^3(e+f x)}{(a+b \sinh ^2(e+f x))^{3/2}} \, dx\) [491]

Optimal. Leaf size=122 \[ -\frac {(2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{2 (a-b)^{5/2} f}+\frac {2 a+b}{2 (a-b)^2 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\text {sech}^2(e+f x)}{2 (a-b) f \sqrt {a+b \sinh ^2(e+f x)}} \]

[Out]

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

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

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Rubi [A]
time = 0.10, antiderivative size = 122, 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, 79, 53, 65, 214} \begin {gather*} \frac {2 a+b}{2 f (a-b)^2 \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{2 f (a-b)^{5/2}}+\frac {\text {sech}^2(e+f x)}{2 f (a-b) \sqrt {a+b \sinh ^2(e+f x)}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[Tanh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

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

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

Rule 53

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

)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,

x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[

c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

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 79

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

tQ[p, n]))))

Rule 214

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

x] && NegQ[a/b]

Rule 3273

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 \frac {\tanh ^3(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {x}{(1+x)^2 (a+b x)^{3/2}} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=\frac {\text {sech}^2(e+f x)}{2 (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(2 a+b) \text {Subst}\left (\int \frac {1}{(1+x) (a+b x)^{3/2}} \, dx,x,\sinh ^2(e+f x)\right )}{4 (a-b) f}\\ &=\frac {2 a+b}{2 (a-b)^2 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\text {sech}^2(e+f x)}{2 (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(2 a+b) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{4 (a-b)^2 f}\\ &=\frac {2 a+b}{2 (a-b)^2 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\text {sech}^2(e+f x)}{2 (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(2 a+b) \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 )}{2 (a-b)^2 b f}\\ &=-\frac {(2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{2 (a-b)^{5/2} f}+\frac {2 a+b}{2 (a-b)^2 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\text {sech}^2(e+f x)}{2 (a-b) 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 = 0.09, size = 79, normalized size = 0.65 \begin {gather*} \frac {(2 a+b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \sinh ^2(e+f x)}{a-b}\right )+(a-b) \text {sech}^2(e+f x)}{2 (a-b)^2 f \sqrt {a+b \sinh ^2(e+f x)}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[Tanh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

((2*a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sinh[e + f*x]^2)/(a - b)] + (a - b)*Sech[e + f*x]^2)/(2*(a -

b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.07, size = 103, normalized size = 0.84


method	result	size

default	\(\frac {\mathit {`\,int/indef0`\,}\left (-\frac {\left (\sinh ^{3}\left (f x +e \right )\right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, \left (\cosh ^{2}\left (f x +e \right )\right )}{-b^{2} \left (\cosh ^{10}\left (f x +e \right )\right )+\left (-2 a b +2 b^{2}\right ) \left (\cosh ^{8}\left (f x +e \right )\right )+\left (-a^{2}+2 a b -b^{2}\right ) \left (\cosh ^{6}\left (f x +e \right )\right )}, \sinh \left (f x +e \right )\right )}{f}\)	\(103\)
risch	\(\text {Expression too large to display}\)	\(289421\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

`int/indef0`(-sinh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(1/2)*cosh(f*x+e)^2/(-b^2*cosh(f*x+e)^10+(-2*a*b+2*b^2)*cosh(f

*x+e)^8+(-a^2+2*a*b-b^2)*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*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(tanh(f*x + e)^3/(b*sinh(f*x + e)^2 + a)^(3/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1927 vs. \(2 (106) = 212\).
time = 0.56, size = 4050, normalized size = 33.20 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(((2*a*b + b^2)*cosh(f*x + e)^8 + 8*(2*a*b + b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + (2*a*b + b^2)*sinh(f*x

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

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

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

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

b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(2*a^2 + a*b)*cosh(f*x + e)^2 + 4*(7*(2*a*b + b^2)*cosh(f*x + e)^6 + 1

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

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

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

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

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

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

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

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

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

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

a*b^3)*f)*sinh(f*x + e)^6 + 2*(4*a^4 - 13*a^3*b + 15*a^2*b^2 - 7*a*b^3 + b^4)*f*cosh(f*x + e)^4 + 8*(7*(a^3*b

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

f*x + e)^5 + 2*(35*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f*cosh(f*x + e)^4 + 30*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b

^3)*f*cosh(f*x + e)^2 + (4*a^4 - 13*a^3*b + 15*a^2*b^2 - 7*a*b^3 + b^4)*f)*sinh(f*x + e)^4 + 4*(a^4 - 3*a^3*b

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

- 3*a^3*b + 3*a^2*b^2 - a*b^3)*f*cosh(f*x + e)^3 + (4*a^4 - 13*a^3*b + 15*a^2*b^2 - 7*a*b^3 + b^4)*f*cosh(f*x

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

*b^2 - a*b^3)*f*cosh(f*x + e)^4 + 3*(4*a^4 - 13*a^3*b + 15*a^2*b^2 - 7*a*b^3 + b^4)*f*cosh(f*x + e)^2 + (a^4 -

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

^2 + 3*a*b^3 - b^4)*f*cosh(f*x + e)^7 + 3*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*f*cosh(f*x + e)^5 + (4*a^4 - 13*

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

inh(f*x + e)), -1/2*(((2*a*b + b^2)*cosh(f*x + e)^8 + 8*(2*a*b + b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + (2*a*b +

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

nh(f*x + e)^6 + 8*(7*(2*a*b + b^2)*cosh(f*x + e)^3 + 3*(2*a^2 + a*b)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(8*a^2

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

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

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

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

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

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

+ e)^5 + 5*(2*a^2 - a*b - b^2)*cosh(f*x + e)*si...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{3}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(f*x+e)**3/(a+b*sinh(f*x+e)**2)**(3/2),x)

[Out]

Integral(tanh(e + f*x)**3/(a + b*sinh(e + f*x)**2)**(3/2), 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*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

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

0.91Error: Bad Argument Type

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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 )}^3}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(e + f*x)^3/(a + b*sinh(e + f*x)^2)^(3/2),x)

[Out]

int(tanh(e + f*x)^3/(a + b*sinh(e + f*x)^2)^(3/2), x)

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