3.49 \(\int \frac{1}{(a+a \tanh (c+d x))^5} \, dx\)

Optimal. Leaf size=121 \[ -\frac{1}{32 d \left (a^5 \tanh (c+d x)+a^5\right )}-\frac{1}{32 a d \left (a^2 \tanh (c+d x)+a^2\right )^2}-\frac{1}{24 a^2 d (a \tanh (c+d x)+a)^3}+\frac{x}{32 a^5}-\frac{1}{16 a d (a \tanh (c+d x)+a)^4}-\frac{1}{10 d (a \tanh (c+d x)+a)^5} \]

[Out]

x/(32*a^5) - 1/(10*d*(a + a*Tanh[c + d*x])^5) - 1/(16*a*d*(a + a*Tanh[c + d*x])^4) - 1/(24*a^2*d*(a + a*Tanh[c
+ d*x])^3) - 1/(32*a*d*(a^2 + a^2*Tanh[c + d*x])^2) - 1/(32*d*(a^5 + a^5*Tanh[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.082235, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3479, 8} \[ -\frac{1}{32 d \left (a^5 \tanh (c+d x)+a^5\right )}-\frac{1}{32 a d \left (a^2 \tanh (c+d x)+a^2\right )^2}-\frac{1}{24 a^2 d (a \tanh (c+d x)+a)^3}+\frac{x}{32 a^5}-\frac{1}{16 a d (a \tanh (c+d x)+a)^4}-\frac{1}{10 d (a \tanh (c+d x)+a)^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Tanh[c + d*x])^(-5),x]

[Out]

x/(32*a^5) - 1/(10*d*(a + a*Tanh[c + d*x])^5) - 1/(16*a*d*(a + a*Tanh[c + d*x])^4) - 1/(24*a^2*d*(a + a*Tanh[c
+ d*x])^3) - 1/(32*a*d*(a^2 + a^2*Tanh[c + d*x])^2) - 1/(32*d*(a^5 + a^5*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))^5} \, dx &=-\frac{1}{10 d (a+a \tanh (c+d x))^5}+\frac{\int \frac{1}{(a+a \tanh (c+d x))^4} \, dx}{2 a}\\ &=-\frac{1}{10 d (a+a \tanh (c+d x))^5}-\frac{1}{16 a d (a+a \tanh (c+d x))^4}+\frac{\int \frac{1}{(a+a \tanh (c+d x))^3} \, dx}{4 a^2}\\ &=-\frac{1}{10 d (a+a \tanh (c+d x))^5}-\frac{1}{16 a d (a+a \tanh (c+d x))^4}-\frac{1}{24 a^2 d (a+a \tanh (c+d x))^3}+\frac{\int \frac{1}{(a+a \tanh (c+d x))^2} \, dx}{8 a^3}\\ &=-\frac{1}{10 d (a+a \tanh (c+d x))^5}-\frac{1}{16 a d (a+a \tanh (c+d x))^4}-\frac{1}{24 a^2 d (a+a \tanh (c+d x))^3}-\frac{1}{32 a^3 d (a+a \tanh (c+d x))^2}+\frac{\int \frac{1}{a+a \tanh (c+d x)} \, dx}{16 a^4}\\ &=-\frac{1}{10 d (a+a \tanh (c+d x))^5}-\frac{1}{16 a d (a+a \tanh (c+d x))^4}-\frac{1}{24 a^2 d (a+a \tanh (c+d x))^3}-\frac{1}{32 a^3 d (a+a \tanh (c+d x))^2}-\frac{1}{32 d \left (a^5+a^5 \tanh (c+d x)\right )}+\frac{\int 1 \, dx}{32 a^5}\\ &=\frac{x}{32 a^5}-\frac{1}{10 d (a+a \tanh (c+d x))^5}-\frac{1}{16 a d (a+a \tanh (c+d x))^4}-\frac{1}{24 a^2 d (a+a \tanh (c+d x))^3}-\frac{1}{32 a^3 d (a+a \tanh (c+d x))^2}-\frac{1}{32 d \left (a^5+a^5 \tanh (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.275754, size = 109, normalized size = 0.9 \[ \frac{\text{sech}^5(c+d x) (-100 \sinh (c+d x)-225 \sinh (3 (c+d x))+120 d x \sinh (5 (c+d x))+12 \sinh (5 (c+d x))-500 \cosh (c+d x)-375 \cosh (3 (c+d x))+120 d x \cosh (5 (c+d x))-12 \cosh (5 (c+d x)))}{3840 a^5 d (\tanh (c+d x)+1)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Tanh[c + d*x])^(-5),x]

[Out]

(Sech[c + d*x]^5*(-500*Cosh[c + d*x] - 375*Cosh[3*(c + d*x)] - 12*Cosh[5*(c + d*x)] + 120*d*x*Cosh[5*(c + d*x)
] - 100*Sinh[c + d*x] - 225*Sinh[3*(c + d*x)] + 12*Sinh[5*(c + d*x)] + 120*d*x*Sinh[5*(c + d*x)]))/(3840*a^5*d
*(1 + Tanh[c + d*x])^5)

________________________________________________________________________________________

Maple [A]  time = 0.027, size = 126, normalized size = 1. \begin{align*} -{\frac{1}{10\,d{a}^{5} \left ( \tanh \left ( dx+c \right ) +1 \right ) ^{5}}}-{\frac{1}{16\,d{a}^{5} \left ( \tanh \left ( dx+c \right ) +1 \right ) ^{4}}}-{\frac{1}{24\,d{a}^{5} \left ( \tanh \left ( dx+c \right ) +1 \right ) ^{3}}}-{\frac{1}{32\,d{a}^{5} \left ( \tanh \left ( dx+c \right ) +1 \right ) ^{2}}}-{\frac{1}{32\,d{a}^{5} \left ( \tanh \left ( dx+c \right ) +1 \right ) }}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) }{64\,d{a}^{5}}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{64\,d{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*tanh(d*x+c))^5,x)

[Out]

-1/10/d/a^5/(tanh(d*x+c)+1)^5-1/16/d/a^5/(tanh(d*x+c)+1)^4-1/24/d/a^5/(tanh(d*x+c)+1)^3-1/32/d/a^5/(tanh(d*x+c
)+1)^2-1/32/d/a^5/(tanh(d*x+c)+1)+1/64/d/a^5*ln(tanh(d*x+c)+1)-1/64/d/a^5*ln(tanh(d*x+c)-1)

________________________________________________________________________________________

Maxima [A]  time = 1.09581, size = 105, normalized size = 0.87 \begin{align*} \frac{d x + c}{32 \, a^{5} d} - \frac{300 \, e^{\left (-2 \, d x - 2 \, c\right )} + 300 \, e^{\left (-4 \, d x - 4 \, c\right )} + 200 \, e^{\left (-6 \, d x - 6 \, c\right )} + 75 \, e^{\left (-8 \, d x - 8 \, c\right )} + 12 \, e^{\left (-10 \, d x - 10 \, c\right )}}{3840 \, a^{5} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tanh(d*x+c))^5,x, algorithm="maxima")

[Out]

1/32*(d*x + c)/(a^5*d) - 1/3840*(300*e^(-2*d*x - 2*c) + 300*e^(-4*d*x - 4*c) + 200*e^(-6*d*x - 6*c) + 75*e^(-8
*d*x - 8*c) + 12*e^(-10*d*x - 10*c))/(a^5*d)

________________________________________________________________________________________

Fricas [B]  time = 2.19692, size = 798, normalized size = 6.6 \begin{align*} \frac{12 \,{\left (10 \, d x - 1\right )} \cosh \left (d x + c\right )^{5} + 60 \,{\left (10 \, d x - 1\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 12 \,{\left (10 \, d x + 1\right )} \sinh \left (d x + c\right )^{5} + 15 \,{\left (8 \,{\left (10 \, d x + 1\right )} \cosh \left (d x + c\right )^{2} - 15\right )} \sinh \left (d x + c\right )^{3} - 375 \, \cosh \left (d x + c\right )^{3} + 15 \,{\left (8 \,{\left (10 \, d x - 1\right )} \cosh \left (d x + c\right )^{3} - 75 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 5 \,{\left (12 \,{\left (10 \, d x + 1\right )} \cosh \left (d x + c\right )^{4} - 135 \, \cosh \left (d x + c\right )^{2} - 20\right )} \sinh \left (d x + c\right ) - 500 \, \cosh \left (d x + c\right )}{3840 \,{\left (a^{5} d \cosh \left (d x + c\right )^{5} + 5 \, a^{5} d \cosh \left (d x + c\right )^{4} \sinh \left (d x + c\right ) + 10 \, a^{5} d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{2} + 10 \, a^{5} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{3} + 5 \, a^{5} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + a^{5} d \sinh \left (d x + c\right )^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tanh(d*x+c))^5,x, algorithm="fricas")

[Out]

1/3840*(12*(10*d*x - 1)*cosh(d*x + c)^5 + 60*(10*d*x - 1)*cosh(d*x + c)*sinh(d*x + c)^4 + 12*(10*d*x + 1)*sinh
(d*x + c)^5 + 15*(8*(10*d*x + 1)*cosh(d*x + c)^2 - 15)*sinh(d*x + c)^3 - 375*cosh(d*x + c)^3 + 15*(8*(10*d*x -
1)*cosh(d*x + c)^3 - 75*cosh(d*x + c))*sinh(d*x + c)^2 + 5*(12*(10*d*x + 1)*cosh(d*x + c)^4 - 135*cosh(d*x +
c)^2 - 20)*sinh(d*x + c) - 500*cosh(d*x + c))/(a^5*d*cosh(d*x + c)^5 + 5*a^5*d*cosh(d*x + c)^4*sinh(d*x + c) +
10*a^5*d*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*a^5*d*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*a^5*d*cosh(d*x + c)*s
inh(d*x + c)^4 + a^5*d*sinh(d*x + c)^5)

________________________________________________________________________________________

Sympy [A]  time = 4.17607, size = 1018, normalized size = 8.41 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tanh(d*x+c))**5,x)

[Out]

Piecewise((15*d*x*tanh(c + d*x)**5/(480*a**5*d*tanh(c + d*x)**5 + 2400*a**5*d*tanh(c + d*x)**4 + 4800*a**5*d*t
anh(c + d*x)**3 + 4800*a**5*d*tanh(c + d*x)**2 + 2400*a**5*d*tanh(c + d*x) + 480*a**5*d) + 75*d*x*tanh(c + d*x
)**4/(480*a**5*d*tanh(c + d*x)**5 + 2400*a**5*d*tanh(c + d*x)**4 + 4800*a**5*d*tanh(c + d*x)**3 + 4800*a**5*d*
tanh(c + d*x)**2 + 2400*a**5*d*tanh(c + d*x) + 480*a**5*d) + 150*d*x*tanh(c + d*x)**3/(480*a**5*d*tanh(c + d*x
)**5 + 2400*a**5*d*tanh(c + d*x)**4 + 4800*a**5*d*tanh(c + d*x)**3 + 4800*a**5*d*tanh(c + d*x)**2 + 2400*a**5*
d*tanh(c + d*x) + 480*a**5*d) + 150*d*x*tanh(c + d*x)**2/(480*a**5*d*tanh(c + d*x)**5 + 2400*a**5*d*tanh(c + d
*x)**4 + 4800*a**5*d*tanh(c + d*x)**3 + 4800*a**5*d*tanh(c + d*x)**2 + 2400*a**5*d*tanh(c + d*x) + 480*a**5*d)
+ 75*d*x*tanh(c + d*x)/(480*a**5*d*tanh(c + d*x)**5 + 2400*a**5*d*tanh(c + d*x)**4 + 4800*a**5*d*tanh(c + d*x
)**3 + 4800*a**5*d*tanh(c + d*x)**2 + 2400*a**5*d*tanh(c + d*x) + 480*a**5*d) + 15*d*x/(480*a**5*d*tanh(c + d*
x)**5 + 2400*a**5*d*tanh(c + d*x)**4 + 4800*a**5*d*tanh(c + d*x)**3 + 4800*a**5*d*tanh(c + d*x)**2 + 2400*a**5
*d*tanh(c + d*x) + 480*a**5*d) - 15*tanh(c + d*x)**4/(480*a**5*d*tanh(c + d*x)**5 + 2400*a**5*d*tanh(c + d*x)*
*4 + 4800*a**5*d*tanh(c + d*x)**3 + 4800*a**5*d*tanh(c + d*x)**2 + 2400*a**5*d*tanh(c + d*x) + 480*a**5*d) - 7
5*tanh(c + d*x)**3/(480*a**5*d*tanh(c + d*x)**5 + 2400*a**5*d*tanh(c + d*x)**4 + 4800*a**5*d*tanh(c + d*x)**3
+ 4800*a**5*d*tanh(c + d*x)**2 + 2400*a**5*d*tanh(c + d*x) + 480*a**5*d) - 155*tanh(c + d*x)**2/(480*a**5*d*ta
nh(c + d*x)**5 + 2400*a**5*d*tanh(c + d*x)**4 + 4800*a**5*d*tanh(c + d*x)**3 + 4800*a**5*d*tanh(c + d*x)**2 +
2400*a**5*d*tanh(c + d*x) + 480*a**5*d) - 175*tanh(c + d*x)/(480*a**5*d*tanh(c + d*x)**5 + 2400*a**5*d*tanh(c
+ d*x)**4 + 4800*a**5*d*tanh(c + d*x)**3 + 4800*a**5*d*tanh(c + d*x)**2 + 2400*a**5*d*tanh(c + d*x) + 480*a**5
*d) - 128/(480*a**5*d*tanh(c + d*x)**5 + 2400*a**5*d*tanh(c + d*x)**4 + 4800*a**5*d*tanh(c + d*x)**3 + 4800*a*
*5*d*tanh(c + d*x)**2 + 2400*a**5*d*tanh(c + d*x) + 480*a**5*d), Ne(d, 0)), (x/(a*tanh(c) + a)**5, True))

________________________________________________________________________________________

Giac [A]  time = 1.20414, size = 99, normalized size = 0.82 \begin{align*} \frac{120 \, d x -{\left (300 \, e^{\left (8 \, d x + 8 \, c\right )} + 300 \, e^{\left (6 \, d x + 6 \, c\right )} + 200 \, e^{\left (4 \, d x + 4 \, c\right )} + 75 \, e^{\left (2 \, d x + 2 \, c\right )} + 12\right )} e^{\left (-10 \, d x - 10 \, c\right )} + 120 \, c}{3840 \, a^{5} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tanh(d*x+c))^5,x, algorithm="giac")

[Out]

1/3840*(120*d*x - (300*e^(8*d*x + 8*c) + 300*e^(6*d*x + 6*c) + 200*e^(4*d*x + 4*c) + 75*e^(2*d*x + 2*c) + 12)*
e^(-10*d*x - 10*c) + 120*c)/(a^5*d)