3.48 \(\int \frac {1}{(a+a \tanh (c+d x))^4} \, dx\)

Optimal. Leaf size=96 \[ -\frac {1}{16 d \left (a^4 \tanh (c+d x)+a^4\right )}+\frac {x}{16 a^4}-\frac {1}{16 d \left (a^2 \tanh (c+d x)+a^2\right )^2}-\frac {1}{12 a d (a \tanh (c+d x)+a)^3}-\frac {1}{8 d (a \tanh (c+d x)+a)^4} \]

[Out]

1/16*x/a^4-1/8/d/(a+a*tanh(d*x+c))^4-1/12/a/d/(a+a*tanh(d*x+c))^3-1/16/d/(a^2+a^2*tanh(d*x+c))^2-1/16/d/(a^4+a
^4*tanh(d*x+c))

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Rubi [A]  time = 0.06, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3479, 8} \[ -\frac {1}{16 d \left (a^4 \tanh (c+d x)+a^4\right )}-\frac {1}{16 d \left (a^2 \tanh (c+d x)+a^2\right )^2}+\frac {x}{16 a^4}-\frac {1}{12 a d (a \tanh (c+d x)+a)^3}-\frac {1}{8 d (a \tanh (c+d x)+a)^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Tanh[c + d*x])^(-4),x]

[Out]

x/(16*a^4) - 1/(8*d*(a + a*Tanh[c + d*x])^4) - 1/(12*a*d*(a + a*Tanh[c + d*x])^3) - 1/(16*d*(a^2 + a^2*Tanh[c
+ d*x])^2) - 1/(16*d*(a^4 + a^4*Tanh[c + d*x]))

Rule 8

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

Rubi steps

\begin {align*} \int \frac {1}{(a+a \tanh (c+d x))^4} \, dx &=-\frac {1}{8 d (a+a \tanh (c+d x))^4}+\frac {\int \frac {1}{(a+a \tanh (c+d x))^3} \, dx}{2 a}\\ &=-\frac {1}{8 d (a+a \tanh (c+d x))^4}-\frac {1}{12 a d (a+a \tanh (c+d x))^3}+\frac {\int \frac {1}{(a+a \tanh (c+d x))^2} \, dx}{4 a^2}\\ &=-\frac {1}{8 d (a+a \tanh (c+d x))^4}-\frac {1}{12 a d (a+a \tanh (c+d x))^3}-\frac {1}{16 d \left (a^2+a^2 \tanh (c+d x)\right )^2}+\frac {\int \frac {1}{a+a \tanh (c+d x)} \, dx}{8 a^3}\\ &=-\frac {1}{8 d (a+a \tanh (c+d x))^4}-\frac {1}{12 a d (a+a \tanh (c+d x))^3}-\frac {1}{16 d \left (a^2+a^2 \tanh (c+d x)\right )^2}-\frac {1}{16 d \left (a^4+a^4 \tanh (c+d x)\right )}+\frac {\int 1 \, dx}{16 a^4}\\ &=\frac {x}{16 a^4}-\frac {1}{8 d (a+a \tanh (c+d x))^4}-\frac {1}{12 a d (a+a \tanh (c+d x))^3}-\frac {1}{16 d \left (a^2+a^2 \tanh (c+d x)\right )^2}-\frac {1}{16 d \left (a^4+a^4 \tanh (c+d x)\right )}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 88, normalized size = 0.92 \[ \frac {\text {sech}^4(c+d x) (-32 \sinh (2 (c+d x))+24 d x \sinh (4 (c+d x))+3 \sinh (4 (c+d x))-64 \cosh (2 (c+d x))+3 (8 d x-1) \cosh (4 (c+d x))-36)}{384 a^4 d (\tanh (c+d x)+1)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Tanh[c + d*x])^(-4),x]

[Out]

(Sech[c + d*x]^4*(-36 - 64*Cosh[2*(c + d*x)] + 3*(-1 + 8*d*x)*Cosh[4*(c + d*x)] - 32*Sinh[2*(c + d*x)] + 3*Sin
h[4*(c + d*x)] + 24*d*x*Sinh[4*(c + d*x)]))/(384*a^4*d*(1 + Tanh[c + d*x])^4)

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fricas [B]  time = 0.55, size = 220, normalized size = 2.29 \[ \frac {3 \, {\left (8 \, d x - 1\right )} \cosh \left (d x + c\right )^{4} + 12 \, {\left (8 \, d x + 1\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 3 \, {\left (8 \, d x - 1\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (9 \, {\left (8 \, d x - 1\right )} \cosh \left (d x + c\right )^{2} - 32\right )} \sinh \left (d x + c\right )^{2} - 64 \, \cosh \left (d x + c\right )^{2} + 4 \, {\left (3 \, {\left (8 \, d x + 1\right )} \cosh \left (d x + c\right )^{3} - 16 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 36}{384 \, {\left (a^{4} d \cosh \left (d x + c\right )^{4} + 4 \, a^{4} d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, a^{4} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, a^{4} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{4} d \sinh \left (d x + c\right )^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tanh(d*x+c))^4,x, algorithm="fricas")

[Out]

1/384*(3*(8*d*x - 1)*cosh(d*x + c)^4 + 12*(8*d*x + 1)*cosh(d*x + c)*sinh(d*x + c)^3 + 3*(8*d*x - 1)*sinh(d*x +
c)^4 + 2*(9*(8*d*x - 1)*cosh(d*x + c)^2 - 32)*sinh(d*x + c)^2 - 64*cosh(d*x + c)^2 + 4*(3*(8*d*x + 1)*cosh(d*
x + c)^3 - 16*cosh(d*x + c))*sinh(d*x + c) - 36)/(a^4*d*cosh(d*x + c)^4 + 4*a^4*d*cosh(d*x + c)^3*sinh(d*x + c
) + 6*a^4*d*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*a^4*d*cosh(d*x + c)*sinh(d*x + c)^3 + a^4*d*sinh(d*x + c)^4)

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giac [A]  time = 0.12, size = 64, normalized size = 0.67 \[ -\frac {\frac {{\left (48 \, e^{\left (6 \, d x + 6 \, c\right )} + 36 \, e^{\left (4 \, d x + 4 \, c\right )} + 16 \, e^{\left (2 \, d x + 2 \, c\right )} + 3\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{a^{4}} - \frac {24 \, {\left (d x + c\right )}}{a^{4}}}{384 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tanh(d*x+c))^4,x, algorithm="giac")

[Out]

-1/384*((48*e^(6*d*x + 6*c) + 36*e^(4*d*x + 4*c) + 16*e^(2*d*x + 2*c) + 3)*e^(-8*d*x - 8*c)/a^4 - 24*(d*x + c)
/a^4)/d

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maple [A]  time = 0.10, size = 108, normalized size = 1.12 \[ -\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{32 d \,a^{4}}-\frac {1}{8 d \,a^{4} \left (1+\tanh \left (d x +c \right )\right )^{4}}-\frac {1}{12 d \,a^{4} \left (1+\tanh \left (d x +c \right )\right )^{3}}-\frac {1}{16 d \,a^{4} \left (1+\tanh \left (d x +c \right )\right )^{2}}-\frac {1}{16 d \,a^{4} \left (1+\tanh \left (d x +c \right )\right )}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{32 d \,a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*tanh(d*x+c))^4,x)

[Out]

-1/32/d/a^4*ln(tanh(d*x+c)-1)-1/8/d/a^4/(1+tanh(d*x+c))^4-1/12/d/a^4/(1+tanh(d*x+c))^3-1/16/d/a^4/(1+tanh(d*x+
c))^2-1/16/d/a^4/(1+tanh(d*x+c))+1/32/d/a^4*ln(1+tanh(d*x+c))

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maxima [A]  time = 0.33, size = 67, normalized size = 0.70 \[ \frac {d x + c}{16 \, a^{4} d} - \frac {48 \, e^{\left (-2 \, d x - 2 \, c\right )} + 36 \, e^{\left (-4 \, d x - 4 \, c\right )} + 16 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{384 \, a^{4} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tanh(d*x+c))^4,x, algorithm="maxima")

[Out]

1/16*(d*x + c)/(a^4*d) - 1/384*(48*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 16*e^(-6*d*x - 6*c) + 3*e^(-8*d*x
- 8*c))/(a^4*d)

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mupad [B]  time = 1.14, size = 75, normalized size = 0.78 \[ \frac {x}{16\,a^4}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,a^4\,d}-\frac {3\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{32\,a^4\,d}-\frac {{\mathrm {e}}^{-6\,c-6\,d\,x}}{24\,a^4\,d}-\frac {{\mathrm {e}}^{-8\,c-8\,d\,x}}{128\,a^4\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + a*tanh(c + d*x))^4,x)

[Out]

x/(16*a^4) - exp(- 2*c - 2*d*x)/(8*a^4*d) - (3*exp(- 4*c - 4*d*x))/(32*a^4*d) - exp(- 6*c - 6*d*x)/(24*a^4*d)
- exp(- 8*c - 8*d*x)/(128*a^4*d)

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sympy [A]  time = 2.31, size = 694, normalized size = 7.23 \[ \begin {cases} \frac {3 d x \tanh ^{4}{\left (c + d x \right )}}{48 a^{4} d \tanh ^{4}{\left (c + d x \right )} + 192 a^{4} d \tanh ^{3}{\left (c + d x \right )} + 288 a^{4} d \tanh ^{2}{\left (c + d x \right )} + 192 a^{4} d \tanh {\left (c + d x \right )} + 48 a^{4} d} + \frac {12 d x \tanh ^{3}{\left (c + d x \right )}}{48 a^{4} d \tanh ^{4}{\left (c + d x \right )} + 192 a^{4} d \tanh ^{3}{\left (c + d x \right )} + 288 a^{4} d \tanh ^{2}{\left (c + d x \right )} + 192 a^{4} d \tanh {\left (c + d x \right )} + 48 a^{4} d} + \frac {18 d x \tanh ^{2}{\left (c + d x \right )}}{48 a^{4} d \tanh ^{4}{\left (c + d x \right )} + 192 a^{4} d \tanh ^{3}{\left (c + d x \right )} + 288 a^{4} d \tanh ^{2}{\left (c + d x \right )} + 192 a^{4} d \tanh {\left (c + d x \right )} + 48 a^{4} d} + \frac {12 d x \tanh {\left (c + d x \right )}}{48 a^{4} d \tanh ^{4}{\left (c + d x \right )} + 192 a^{4} d \tanh ^{3}{\left (c + d x \right )} + 288 a^{4} d \tanh ^{2}{\left (c + d x \right )} + 192 a^{4} d \tanh {\left (c + d x \right )} + 48 a^{4} d} + \frac {3 d x}{48 a^{4} d \tanh ^{4}{\left (c + d x \right )} + 192 a^{4} d \tanh ^{3}{\left (c + d x \right )} + 288 a^{4} d \tanh ^{2}{\left (c + d x \right )} + 192 a^{4} d \tanh {\left (c + d x \right )} + 48 a^{4} d} - \frac {3 \tanh ^{3}{\left (c + d x \right )}}{48 a^{4} d \tanh ^{4}{\left (c + d x \right )} + 192 a^{4} d \tanh ^{3}{\left (c + d x \right )} + 288 a^{4} d \tanh ^{2}{\left (c + d x \right )} + 192 a^{4} d \tanh {\left (c + d x \right )} + 48 a^{4} d} - \frac {12 \tanh ^{2}{\left (c + d x \right )}}{48 a^{4} d \tanh ^{4}{\left (c + d x \right )} + 192 a^{4} d \tanh ^{3}{\left (c + d x \right )} + 288 a^{4} d \tanh ^{2}{\left (c + d x \right )} + 192 a^{4} d \tanh {\left (c + d x \right )} + 48 a^{4} d} - \frac {19 \tanh {\left (c + d x \right )}}{48 a^{4} d \tanh ^{4}{\left (c + d x \right )} + 192 a^{4} d \tanh ^{3}{\left (c + d x \right )} + 288 a^{4} d \tanh ^{2}{\left (c + d x \right )} + 192 a^{4} d \tanh {\left (c + d x \right )} + 48 a^{4} d} - \frac {16}{48 a^{4} d \tanh ^{4}{\left (c + d x \right )} + 192 a^{4} d \tanh ^{3}{\left (c + d x \right )} + 288 a^{4} d \tanh ^{2}{\left (c + d x \right )} + 192 a^{4} d \tanh {\left (c + d x \right )} + 48 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x}{\left (a \tanh {\relax (c )} + a\right )^{4}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tanh(d*x+c))**4,x)

[Out]

Piecewise((3*d*x*tanh(c + d*x)**4/(48*a**4*d*tanh(c + d*x)**4 + 192*a**4*d*tanh(c + d*x)**3 + 288*a**4*d*tanh(
c + d*x)**2 + 192*a**4*d*tanh(c + d*x) + 48*a**4*d) + 12*d*x*tanh(c + d*x)**3/(48*a**4*d*tanh(c + d*x)**4 + 19
2*a**4*d*tanh(c + d*x)**3 + 288*a**4*d*tanh(c + d*x)**2 + 192*a**4*d*tanh(c + d*x) + 48*a**4*d) + 18*d*x*tanh(
c + d*x)**2/(48*a**4*d*tanh(c + d*x)**4 + 192*a**4*d*tanh(c + d*x)**3 + 288*a**4*d*tanh(c + d*x)**2 + 192*a**4
*d*tanh(c + d*x) + 48*a**4*d) + 12*d*x*tanh(c + d*x)/(48*a**4*d*tanh(c + d*x)**4 + 192*a**4*d*tanh(c + d*x)**3
+ 288*a**4*d*tanh(c + d*x)**2 + 192*a**4*d*tanh(c + d*x) + 48*a**4*d) + 3*d*x/(48*a**4*d*tanh(c + d*x)**4 + 1
92*a**4*d*tanh(c + d*x)**3 + 288*a**4*d*tanh(c + d*x)**2 + 192*a**4*d*tanh(c + d*x) + 48*a**4*d) - 3*tanh(c +
d*x)**3/(48*a**4*d*tanh(c + d*x)**4 + 192*a**4*d*tanh(c + d*x)**3 + 288*a**4*d*tanh(c + d*x)**2 + 192*a**4*d*t
anh(c + d*x) + 48*a**4*d) - 12*tanh(c + d*x)**2/(48*a**4*d*tanh(c + d*x)**4 + 192*a**4*d*tanh(c + d*x)**3 + 28
8*a**4*d*tanh(c + d*x)**2 + 192*a**4*d*tanh(c + d*x) + 48*a**4*d) - 19*tanh(c + d*x)/(48*a**4*d*tanh(c + d*x)*
*4 + 192*a**4*d*tanh(c + d*x)**3 + 288*a**4*d*tanh(c + d*x)**2 + 192*a**4*d*tanh(c + d*x) + 48*a**4*d) - 16/(4
8*a**4*d*tanh(c + d*x)**4 + 192*a**4*d*tanh(c + d*x)**3 + 288*a**4*d*tanh(c + d*x)**2 + 192*a**4*d*tanh(c + d*
x) + 48*a**4*d), Ne(d, 0)), (x/(a*tanh(c) + a)**4, True))

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