3.47 \(\int \frac{1}{(a+a \tanh (c+d x))^3} \, dx\)

Optimal. Leaf size=73 \[ -\frac{1}{8 d \left (a^3 \tanh (c+d x)+a^3\right )}+\frac{x}{8 a^3}-\frac{1}{8 a d (a \tanh (c+d x)+a)^2}-\frac{1}{6 d (a \tanh (c+d x)+a)^3} \]

[Out]

x/(8*a^3) - 1/(6*d*(a + a*Tanh[c + d*x])^3) - 1/(8*a*d*(a + a*Tanh[c + d*x])^2) - 1/(8*d*(a^3 + a^3*Tanh[c + d
*x]))

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Rubi [A]  time = 0.0441102, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3479, 8} \[ -\frac{1}{8 d \left (a^3 \tanh (c+d x)+a^3\right )}+\frac{x}{8 a^3}-\frac{1}{8 a d (a \tanh (c+d x)+a)^2}-\frac{1}{6 d (a \tanh (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Tanh[c + d*x])^(-3),x]

[Out]

x/(8*a^3) - 1/(6*d*(a + a*Tanh[c + d*x])^3) - 1/(8*a*d*(a + a*Tanh[c + d*x])^2) - 1/(8*d*(a^3 + a^3*Tanh[c + d
*x]))

Rule 3479

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(a + b*Tan[c + d*x])^n)/(2*b*d*n), x] +
Dist[1/(2*a), Int[(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{1}{(a+a \tanh (c+d x))^3} \, dx &=-\frac{1}{6 d (a+a \tanh (c+d x))^3}+\frac{\int \frac{1}{(a+a \tanh (c+d x))^2} \, dx}{2 a}\\ &=-\frac{1}{6 d (a+a \tanh (c+d x))^3}-\frac{1}{8 a d (a+a \tanh (c+d x))^2}+\frac{\int \frac{1}{a+a \tanh (c+d x)} \, dx}{4 a^2}\\ &=-\frac{1}{6 d (a+a \tanh (c+d x))^3}-\frac{1}{8 a d (a+a \tanh (c+d x))^2}-\frac{1}{8 d \left (a^3+a^3 \tanh (c+d x)\right )}+\frac{\int 1 \, dx}{8 a^3}\\ &=\frac{x}{8 a^3}-\frac{1}{6 d (a+a \tanh (c+d x))^3}-\frac{1}{8 a d (a+a \tanh (c+d x))^2}-\frac{1}{8 d \left (a^3+a^3 \tanh (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.24618, size = 83, normalized size = 1.14 \[ \frac{\text{sech}^3(c+d x) (-9 \sinh (c+d x)+12 d x \sinh (3 (c+d x))+2 \sinh (3 (c+d x))-27 \cosh (c+d x)+2 (6 d x-1) \cosh (3 (c+d x)))}{96 a^3 d (\tanh (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Tanh[c + d*x])^(-3),x]

[Out]

(Sech[c + d*x]^3*(-27*Cosh[c + d*x] + 2*(-1 + 6*d*x)*Cosh[3*(c + d*x)] - 9*Sinh[c + d*x] + 2*Sinh[3*(c + d*x)]
+ 12*d*x*Sinh[3*(c + d*x)]))/(96*a^3*d*(1 + Tanh[c + d*x])^3)

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Maple [A]  time = 0.019, size = 90, normalized size = 1.2 \begin{align*} -{\frac{1}{6\,{a}^{3}d \left ( \tanh \left ( dx+c \right ) +1 \right ) ^{3}}}-{\frac{1}{8\,{a}^{3}d \left ( \tanh \left ( dx+c \right ) +1 \right ) ^{2}}}-{\frac{1}{8\,{a}^{3}d \left ( \tanh \left ( dx+c \right ) +1 \right ) }}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) }{16\,{a}^{3}d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{16\,{a}^{3}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*tanh(d*x+c))^3,x)

[Out]

-1/6/a^3/d/(tanh(d*x+c)+1)^3-1/8/a^3/d/(tanh(d*x+c)+1)^2-1/8/a^3/d/(tanh(d*x+c)+1)+1/16/a^3/d*ln(tanh(d*x+c)+1
)-1/16/a^3/d*ln(tanh(d*x+c)-1)

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Maxima [A]  time = 1.13624, size = 76, normalized size = 1.04 \begin{align*} \frac{d x + c}{8 \, a^{3} d} - \frac{18 \, e^{\left (-2 \, d x - 2 \, c\right )} + 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, e^{\left (-6 \, d x - 6 \, c\right )}}{96 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tanh(d*x+c))^3,x, algorithm="maxima")

[Out]

1/8*(d*x + c)/(a^3*d) - 1/96*(18*e^(-2*d*x - 2*c) + 9*e^(-4*d*x - 4*c) + 2*e^(-6*d*x - 6*c))/(a^3*d)

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Fricas [B]  time = 2.11768, size = 428, normalized size = 5.86 \begin{align*} \frac{2 \,{\left (6 \, d x - 1\right )} \cosh \left (d x + c\right )^{3} + 6 \,{\left (6 \, d x - 1\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 2 \,{\left (6 \, d x + 1\right )} \sinh \left (d x + c\right )^{3} + 3 \,{\left (2 \,{\left (6 \, d x + 1\right )} \cosh \left (d x + c\right )^{2} - 3\right )} \sinh \left (d x + c\right ) - 27 \, \cosh \left (d x + c\right )}{96 \,{\left (a^{3} d \cosh \left (d x + c\right )^{3} + 3 \, a^{3} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{3} d \sinh \left (d x + c\right )^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tanh(d*x+c))^3,x, algorithm="fricas")

[Out]

1/96*(2*(6*d*x - 1)*cosh(d*x + c)^3 + 6*(6*d*x - 1)*cosh(d*x + c)*sinh(d*x + c)^2 + 2*(6*d*x + 1)*sinh(d*x + c
)^3 + 3*(2*(6*d*x + 1)*cosh(d*x + c)^2 - 3)*sinh(d*x + c) - 27*cosh(d*x + c))/(a^3*d*cosh(d*x + c)^3 + 3*a^3*d
*cosh(d*x + c)^2*sinh(d*x + c) + 3*a^3*d*cosh(d*x + c)*sinh(d*x + c)^2 + a^3*d*sinh(d*x + c)^3)

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Sympy [A]  time = 2.01707, size = 430, normalized size = 5.89 \begin{align*} \begin{cases} \frac{3 d x \tanh ^{3}{\left (c + d x \right )}}{24 a^{3} d \tanh ^{3}{\left (c + d x \right )} + 72 a^{3} d \tanh ^{2}{\left (c + d x \right )} + 72 a^{3} d \tanh{\left (c + d x \right )} + 24 a^{3} d} + \frac{9 d x \tanh ^{2}{\left (c + d x \right )}}{24 a^{3} d \tanh ^{3}{\left (c + d x \right )} + 72 a^{3} d \tanh ^{2}{\left (c + d x \right )} + 72 a^{3} d \tanh{\left (c + d x \right )} + 24 a^{3} d} + \frac{9 d x \tanh{\left (c + d x \right )}}{24 a^{3} d \tanh ^{3}{\left (c + d x \right )} + 72 a^{3} d \tanh ^{2}{\left (c + d x \right )} + 72 a^{3} d \tanh{\left (c + d x \right )} + 24 a^{3} d} + \frac{3 d x}{24 a^{3} d \tanh ^{3}{\left (c + d x \right )} + 72 a^{3} d \tanh ^{2}{\left (c + d x \right )} + 72 a^{3} d \tanh{\left (c + d x \right )} + 24 a^{3} d} - \frac{3 \tanh ^{2}{\left (c + d x \right )}}{24 a^{3} d \tanh ^{3}{\left (c + d x \right )} + 72 a^{3} d \tanh ^{2}{\left (c + d x \right )} + 72 a^{3} d \tanh{\left (c + d x \right )} + 24 a^{3} d} - \frac{9 \tanh{\left (c + d x \right )}}{24 a^{3} d \tanh ^{3}{\left (c + d x \right )} + 72 a^{3} d \tanh ^{2}{\left (c + d x \right )} + 72 a^{3} d \tanh{\left (c + d x \right )} + 24 a^{3} d} - \frac{10}{24 a^{3} d \tanh ^{3}{\left (c + d x \right )} + 72 a^{3} d \tanh ^{2}{\left (c + d x \right )} + 72 a^{3} d \tanh{\left (c + d x \right )} + 24 a^{3} d} & \text{for}\: d \neq 0 \\\frac{x}{\left (a \tanh{\left (c \right )} + a\right )^{3}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tanh(d*x+c))**3,x)

[Out]

Piecewise((3*d*x*tanh(c + d*x)**3/(24*a**3*d*tanh(c + d*x)**3 + 72*a**3*d*tanh(c + d*x)**2 + 72*a**3*d*tanh(c
+ d*x) + 24*a**3*d) + 9*d*x*tanh(c + d*x)**2/(24*a**3*d*tanh(c + d*x)**3 + 72*a**3*d*tanh(c + d*x)**2 + 72*a**
3*d*tanh(c + d*x) + 24*a**3*d) + 9*d*x*tanh(c + d*x)/(24*a**3*d*tanh(c + d*x)**3 + 72*a**3*d*tanh(c + d*x)**2
+ 72*a**3*d*tanh(c + d*x) + 24*a**3*d) + 3*d*x/(24*a**3*d*tanh(c + d*x)**3 + 72*a**3*d*tanh(c + d*x)**2 + 72*a
**3*d*tanh(c + d*x) + 24*a**3*d) - 3*tanh(c + d*x)**2/(24*a**3*d*tanh(c + d*x)**3 + 72*a**3*d*tanh(c + d*x)**2
+ 72*a**3*d*tanh(c + d*x) + 24*a**3*d) - 9*tanh(c + d*x)/(24*a**3*d*tanh(c + d*x)**3 + 72*a**3*d*tanh(c + d*x
)**2 + 72*a**3*d*tanh(c + d*x) + 24*a**3*d) - 10/(24*a**3*d*tanh(c + d*x)**3 + 72*a**3*d*tanh(c + d*x)**2 + 72
*a**3*d*tanh(c + d*x) + 24*a**3*d), Ne(d, 0)), (x/(a*tanh(c) + a)**3, True))

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Giac [A]  time = 1.19553, size = 69, normalized size = 0.95 \begin{align*} \frac{12 \, d x -{\left (18 \, e^{\left (4 \, d x + 4 \, c\right )} + 9 \, e^{\left (2 \, d x + 2 \, c\right )} + 2\right )} e^{\left (-6 \, d x - 6 \, c\right )} + 12 \, c}{96 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tanh(d*x+c))^3,x, algorithm="giac")

[Out]

1/96*(12*d*x - (18*e^(4*d*x + 4*c) + 9*e^(2*d*x + 2*c) + 2)*e^(-6*d*x - 6*c) + 12*c)/(a^3*d)