3.37 \(\int (a \tanh ^4(x))^{3/2} \, dx\)

Optimal. Leaf size=69 \[ -\frac{1}{5} a \tanh ^3(x) \sqrt{a \tanh ^4(x)}-\frac{1}{3} a \tanh (x) \sqrt{a \tanh ^4(x)}+a x \coth ^2(x) \sqrt{a \tanh ^4(x)}-a \coth (x) \sqrt{a \tanh ^4(x)} \]

[Out]

-(a*Coth[x]*Sqrt[a*Tanh[x]^4]) + a*x*Coth[x]^2*Sqrt[a*Tanh[x]^4] - (a*Tanh[x]*Sqrt[a*Tanh[x]^4])/3 - (a*Tanh[x
]^3*Sqrt[a*Tanh[x]^4])/5

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Rubi [A]  time = 0.0262483, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3658, 3473, 8} \[ -\frac{1}{5} a \tanh ^3(x) \sqrt{a \tanh ^4(x)}-\frac{1}{3} a \tanh (x) \sqrt{a \tanh ^4(x)}+a x \coth ^2(x) \sqrt{a \tanh ^4(x)}-a \coth (x) \sqrt{a \tanh ^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Tanh[x]^4)^(3/2),x]

[Out]

-(a*Coth[x]*Sqrt[a*Tanh[x]^4]) + a*x*Coth[x]^2*Sqrt[a*Tanh[x]^4] - (a*Tanh[x]*Sqrt[a*Tanh[x]^4])/3 - (a*Tanh[x
]^3*Sqrt[a*Tanh[x]^4])/5

Rule 3658

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \left (a \tanh ^4(x)\right )^{3/2} \, dx &=\left (a \coth ^2(x) \sqrt{a \tanh ^4(x)}\right ) \int \tanh ^6(x) \, dx\\ &=-\frac{1}{5} a \tanh ^3(x) \sqrt{a \tanh ^4(x)}+\left (a \coth ^2(x) \sqrt{a \tanh ^4(x)}\right ) \int \tanh ^4(x) \, dx\\ &=-\frac{1}{3} a \tanh (x) \sqrt{a \tanh ^4(x)}-\frac{1}{5} a \tanh ^3(x) \sqrt{a \tanh ^4(x)}+\left (a \coth ^2(x) \sqrt{a \tanh ^4(x)}\right ) \int \tanh ^2(x) \, dx\\ &=-a \coth (x) \sqrt{a \tanh ^4(x)}-\frac{1}{3} a \tanh (x) \sqrt{a \tanh ^4(x)}-\frac{1}{5} a \tanh ^3(x) \sqrt{a \tanh ^4(x)}+\left (a \coth ^2(x) \sqrt{a \tanh ^4(x)}\right ) \int 1 \, dx\\ &=-a \coth (x) \sqrt{a \tanh ^4(x)}+a x \coth ^2(x) \sqrt{a \tanh ^4(x)}-\frac{1}{3} a \tanh (x) \sqrt{a \tanh ^4(x)}-\frac{1}{5} a \tanh ^3(x) \sqrt{a \tanh ^4(x)}\\ \end{align*}

Mathematica [A]  time = 0.147969, size = 56, normalized size = 0.81 \[ \frac{1}{240} \coth (x) \text{csch}^5(x) \left (a \tanh ^4(x)\right )^{3/2} (-50 \sinh (x)-25 \sinh (3 x)-23 \sinh (5 x)+150 x \cosh (x)+75 x \cosh (3 x)+15 x \cosh (5 x)) \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Tanh[x]^4)^(3/2),x]

[Out]

(Coth[x]*Csch[x]^5*(150*x*Cosh[x] + 75*x*Cosh[3*x] + 15*x*Cosh[5*x] - 50*Sinh[x] - 25*Sinh[3*x] - 23*Sinh[5*x]
)*(a*Tanh[x]^4)^(3/2))/240

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Maple [A]  time = 0.026, size = 46, normalized size = 0.7 \begin{align*} -{\frac{6\, \left ( \tanh \left ( x \right ) \right ) ^{5}+10\, \left ( \tanh \left ( x \right ) \right ) ^{3}+15\,\ln \left ( \tanh \left ( x \right ) -1 \right ) -15\,\ln \left ( 1+\tanh \left ( x \right ) \right ) +30\,\tanh \left ( x \right ) }{30\, \left ( \tanh \left ( x \right ) \right ) ^{6}} \left ( a \left ( \tanh \left ( x \right ) \right ) ^{4} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*tanh(x)^4)^(3/2),x)

[Out]

-1/30*(a*tanh(x)^4)^(3/2)*(6*tanh(x)^5+10*tanh(x)^3+15*ln(tanh(x)-1)-15*ln(1+tanh(x))+30*tanh(x))/tanh(x)^6

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Maxima [A]  time = 1.6101, size = 111, normalized size = 1.61 \begin{align*} a^{\frac{3}{2}} x - \frac{2 \,{\left (70 \, a^{\frac{3}{2}} e^{\left (-2 \, x\right )} + 140 \, a^{\frac{3}{2}} e^{\left (-4 \, x\right )} + 90 \, a^{\frac{3}{2}} e^{\left (-6 \, x\right )} + 45 \, a^{\frac{3}{2}} e^{\left (-8 \, x\right )} + 23 \, a^{\frac{3}{2}}\right )}}{15 \,{\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*tanh(x)^4)^(3/2),x, algorithm="maxima")

[Out]

a^(3/2)*x - 2/15*(70*a^(3/2)*e^(-2*x) + 140*a^(3/2)*e^(-4*x) + 90*a^(3/2)*e^(-6*x) + 45*a^(3/2)*e^(-8*x) + 23*
a^(3/2))/(5*e^(-2*x) + 10*e^(-4*x) + 10*e^(-6*x) + 5*e^(-8*x) + e^(-10*x) + 1)

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Fricas [B]  time = 2.52678, size = 6368, normalized size = 92.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*tanh(x)^4)^(3/2),x, algorithm="fricas")

[Out]

1/15*(15*a*x*cosh(x)^10 + 15*(a*x*e^(4*x) + 2*a*x*e^(2*x) + a*x)*sinh(x)^10 + 150*(a*x*cosh(x)*e^(4*x) + 2*a*x
*cosh(x)*e^(2*x) + a*x*cosh(x))*sinh(x)^9 + 15*(5*a*x + 6*a)*cosh(x)^8 + 15*(45*a*x*cosh(x)^2 + 5*a*x + (45*a*
x*cosh(x)^2 + 5*a*x + 6*a)*e^(4*x) + 2*(45*a*x*cosh(x)^2 + 5*a*x + 6*a)*e^(2*x) + 6*a)*sinh(x)^8 + 120*(15*a*x
*cosh(x)^3 + (5*a*x + 6*a)*cosh(x) + (15*a*x*cosh(x)^3 + (5*a*x + 6*a)*cosh(x))*e^(4*x) + 2*(15*a*x*cosh(x)^3
+ (5*a*x + 6*a)*cosh(x))*e^(2*x))*sinh(x)^7 + 30*(5*a*x + 6*a)*cosh(x)^6 + 30*(105*a*x*cosh(x)^4 + 14*(5*a*x +
6*a)*cosh(x)^2 + 5*a*x + (105*a*x*cosh(x)^4 + 14*(5*a*x + 6*a)*cosh(x)^2 + 5*a*x + 6*a)*e^(4*x) + 2*(105*a*x*
cosh(x)^4 + 14*(5*a*x + 6*a)*cosh(x)^2 + 5*a*x + 6*a)*e^(2*x) + 6*a)*sinh(x)^6 + 60*(63*a*x*cosh(x)^5 + 14*(5*
a*x + 6*a)*cosh(x)^3 + 3*(5*a*x + 6*a)*cosh(x) + (63*a*x*cosh(x)^5 + 14*(5*a*x + 6*a)*cosh(x)^3 + 3*(5*a*x + 6
*a)*cosh(x))*e^(4*x) + 2*(63*a*x*cosh(x)^5 + 14*(5*a*x + 6*a)*cosh(x)^3 + 3*(5*a*x + 6*a)*cosh(x))*e^(2*x))*si
nh(x)^5 + 10*(15*a*x + 28*a)*cosh(x)^4 + 10*(315*a*x*cosh(x)^6 + 105*(5*a*x + 6*a)*cosh(x)^4 + 45*(5*a*x + 6*a
)*cosh(x)^2 + 15*a*x + (315*a*x*cosh(x)^6 + 105*(5*a*x + 6*a)*cosh(x)^4 + 45*(5*a*x + 6*a)*cosh(x)^2 + 15*a*x
+ 28*a)*e^(4*x) + 2*(315*a*x*cosh(x)^6 + 105*(5*a*x + 6*a)*cosh(x)^4 + 45*(5*a*x + 6*a)*cosh(x)^2 + 15*a*x + 2
8*a)*e^(2*x) + 28*a)*sinh(x)^4 + 40*(45*a*x*cosh(x)^7 + 21*(5*a*x + 6*a)*cosh(x)^5 + 15*(5*a*x + 6*a)*cosh(x)^
3 + (15*a*x + 28*a)*cosh(x) + (45*a*x*cosh(x)^7 + 21*(5*a*x + 6*a)*cosh(x)^5 + 15*(5*a*x + 6*a)*cosh(x)^3 + (1
5*a*x + 28*a)*cosh(x))*e^(4*x) + 2*(45*a*x*cosh(x)^7 + 21*(5*a*x + 6*a)*cosh(x)^5 + 15*(5*a*x + 6*a)*cosh(x)^3
+ (15*a*x + 28*a)*cosh(x))*e^(2*x))*sinh(x)^3 + 5*(15*a*x + 28*a)*cosh(x)^2 + 5*(135*a*x*cosh(x)^8 + 84*(5*a*
x + 6*a)*cosh(x)^6 + 90*(5*a*x + 6*a)*cosh(x)^4 + 12*(15*a*x + 28*a)*cosh(x)^2 + 15*a*x + (135*a*x*cosh(x)^8 +
84*(5*a*x + 6*a)*cosh(x)^6 + 90*(5*a*x + 6*a)*cosh(x)^4 + 12*(15*a*x + 28*a)*cosh(x)^2 + 15*a*x + 28*a)*e^(4*
x) + 2*(135*a*x*cosh(x)^8 + 84*(5*a*x + 6*a)*cosh(x)^6 + 90*(5*a*x + 6*a)*cosh(x)^4 + 12*(15*a*x + 28*a)*cosh(
x)^2 + 15*a*x + 28*a)*e^(2*x) + 28*a)*sinh(x)^2 + 15*a*x + (15*a*x*cosh(x)^10 + 15*(5*a*x + 6*a)*cosh(x)^8 + 3
0*(5*a*x + 6*a)*cosh(x)^6 + 10*(15*a*x + 28*a)*cosh(x)^4 + 5*(15*a*x + 28*a)*cosh(x)^2 + 15*a*x + 46*a)*e^(4*x
) + 2*(15*a*x*cosh(x)^10 + 15*(5*a*x + 6*a)*cosh(x)^8 + 30*(5*a*x + 6*a)*cosh(x)^6 + 10*(15*a*x + 28*a)*cosh(x
)^4 + 5*(15*a*x + 28*a)*cosh(x)^2 + 15*a*x + 46*a)*e^(2*x) + 10*(15*a*x*cosh(x)^9 + 12*(5*a*x + 6*a)*cosh(x)^7
+ 18*(5*a*x + 6*a)*cosh(x)^5 + 4*(15*a*x + 28*a)*cosh(x)^3 + (15*a*x + 28*a)*cosh(x) + (15*a*x*cosh(x)^9 + 12
*(5*a*x + 6*a)*cosh(x)^7 + 18*(5*a*x + 6*a)*cosh(x)^5 + 4*(15*a*x + 28*a)*cosh(x)^3 + (15*a*x + 28*a)*cosh(x))
*e^(4*x) + 2*(15*a*x*cosh(x)^9 + 12*(5*a*x + 6*a)*cosh(x)^7 + 18*(5*a*x + 6*a)*cosh(x)^5 + 4*(15*a*x + 28*a)*c
osh(x)^3 + (15*a*x + 28*a)*cosh(x))*e^(2*x))*sinh(x) + 46*a)*sqrt((a*e^(8*x) - 4*a*e^(6*x) + 6*a*e^(4*x) - 4*a
*e^(2*x) + a)/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))/((e^(4*x) - 2*e^(2*x) + 1)*sinh(x)^10 + cosh(
x)^10 + 10*(cosh(x)*e^(4*x) - 2*cosh(x)*e^(2*x) + cosh(x))*sinh(x)^9 + 5*(9*cosh(x)^2 + (9*cosh(x)^2 + 1)*e^(4
*x) - 2*(9*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^8 + 5*cosh(x)^8 + 40*(3*cosh(x)^3 + (3*cosh(x)^3 + cosh(x))*e^(
4*x) - 2*(3*cosh(x)^3 + cosh(x))*e^(2*x) + cosh(x))*sinh(x)^7 + 10*(21*cosh(x)^4 + 14*cosh(x)^2 + (21*cosh(x)^
4 + 14*cosh(x)^2 + 1)*e^(4*x) - 2*(21*cosh(x)^4 + 14*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^6 + 10*cosh(x)^6 + 4*
(63*cosh(x)^5 + 70*cosh(x)^3 + (63*cosh(x)^5 + 70*cosh(x)^3 + 15*cosh(x))*e^(4*x) - 2*(63*cosh(x)^5 + 70*cosh(
x)^3 + 15*cosh(x))*e^(2*x) + 15*cosh(x))*sinh(x)^5 + 10*(21*cosh(x)^6 + 35*cosh(x)^4 + 15*cosh(x)^2 + (21*cosh
(x)^6 + 35*cosh(x)^4 + 15*cosh(x)^2 + 1)*e^(4*x) - 2*(21*cosh(x)^6 + 35*cosh(x)^4 + 15*cosh(x)^2 + 1)*e^(2*x)
+ 1)*sinh(x)^4 + 10*cosh(x)^4 + 40*(3*cosh(x)^7 + 7*cosh(x)^5 + 5*cosh(x)^3 + (3*cosh(x)^7 + 7*cosh(x)^5 + 5*c
osh(x)^3 + cosh(x))*e^(4*x) - 2*(3*cosh(x)^7 + 7*cosh(x)^5 + 5*cosh(x)^3 + cosh(x))*e^(2*x) + cosh(x))*sinh(x)
^3 + 5*(9*cosh(x)^8 + 28*cosh(x)^6 + 30*cosh(x)^4 + 12*cosh(x)^2 + (9*cosh(x)^8 + 28*cosh(x)^6 + 30*cosh(x)^4
+ 12*cosh(x)^2 + 1)*e^(4*x) - 2*(9*cosh(x)^8 + 28*cosh(x)^6 + 30*cosh(x)^4 + 12*cosh(x)^2 + 1)*e^(2*x) + 1)*si
nh(x)^2 + 5*cosh(x)^2 + (cosh(x)^10 + 5*cosh(x)^8 + 10*cosh(x)^6 + 10*cosh(x)^4 + 5*cosh(x)^2 + 1)*e^(4*x) - 2
*(cosh(x)^10 + 5*cosh(x)^8 + 10*cosh(x)^6 + 10*cosh(x)^4 + 5*cosh(x)^2 + 1)*e^(2*x) + 10*(cosh(x)^9 + 4*cosh(x
)^7 + 6*cosh(x)^5 + 4*cosh(x)^3 + (cosh(x)^9 + 4*cosh(x)^7 + 6*cosh(x)^5 + 4*cosh(x)^3 + cosh(x))*e^(4*x) - 2*
(cosh(x)^9 + 4*cosh(x)^7 + 6*cosh(x)^5 + 4*cosh(x)^3 + cosh(x))*e^(2*x) + cosh(x))*sinh(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \tanh ^{4}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*tanh(x)**4)**(3/2),x)

[Out]

Integral((a*tanh(x)**4)**(3/2), x)

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Giac [A]  time = 1.18484, size = 61, normalized size = 0.88 \begin{align*} \frac{1}{15} \, a^{\frac{3}{2}}{\left (15 \, x + \frac{2 \,{\left (45 \, e^{\left (8 \, x\right )} + 90 \, e^{\left (6 \, x\right )} + 140 \, e^{\left (4 \, x\right )} + 70 \, e^{\left (2 \, x\right )} + 23\right )}}{{\left (e^{\left (2 \, x\right )} + 1\right )}^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*tanh(x)^4)^(3/2),x, algorithm="giac")

[Out]

1/15*a^(3/2)*(15*x + 2*(45*e^(8*x) + 90*e^(6*x) + 140*e^(4*x) + 70*e^(2*x) + 23)/(e^(2*x) + 1)^5)