3.42 \(\int (a+a \tanh (c+d x))^4 \, dx\)

Optimal. Leaf size=77 \[ -\frac {4 a^4 \tanh (c+d x)}{d}+\frac {8 a^4 \log (\cosh (c+d x))}{d}+8 a^4 x-\frac {\left (a^2 \tanh (c+d x)+a^2\right )^2}{d}-\frac {a (a \tanh (c+d x)+a)^3}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3478, 3477, 3475} \[ -\frac {4 a^4 \tanh (c+d x)}{d}-\frac {\left (a^2 \tanh (c+d x)+a^2\right )^2}{d}+\frac {8 a^4 \log (\cosh (c+d x))}{d}+8 a^4 x-\frac {a (a \tanh (c+d x)+a)^3}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Tanh[c + d*x])^4,x]

[Out]

8*a^4*x + (8*a^4*Log[Cosh[c + d*x]])/d - (4*a^4*Tanh[c + d*x])/d - (a*(a + a*Tanh[c + d*x])^3)/(3*d) - (a^2 +
a^2*Tanh[c + d*x])^2/d

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3477

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

Rule 3478

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n - 1))/(d*(n - 1)
), x] + Dist[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] && G
tQ[n, 1]

Rubi steps

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

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Mathematica [B]  time = 1.15, size = 178, normalized size = 2.31 \[ \frac {a^4 \text {sech}(c) \text {sech}^3(c+d x) (\sinh (4 d x)+\cosh (4 d x)) (12 \sinh (2 c+d x)-11 \sinh (2 c+3 d x)+6 d x \cosh (2 c+3 d x)+6 d x \cosh (4 c+3 d x)+6 \cosh (2 c+3 d x) \log (\cosh (c+d x))+6 \cosh (4 c+3 d x) \log (\cosh (c+d x))+6 \cosh (d x) (3 \log (\cosh (c+d x))+3 d x+1)+6 \cosh (2 c+d x) (3 \log (\cosh (c+d x))+3 d x+1)-21 \sinh (d x))}{6 d (\sinh (d x)+\cosh (d x))^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Tanh[c + d*x])^4,x]

[Out]

(a^4*Sech[c]*Sech[c + d*x]^3*(Cosh[4*d*x] + Sinh[4*d*x])*(6*d*x*Cosh[2*c + 3*d*x] + 6*d*x*Cosh[4*c + 3*d*x] +
6*Cosh[2*c + 3*d*x]*Log[Cosh[c + d*x]] + 6*Cosh[4*c + 3*d*x]*Log[Cosh[c + d*x]] + 6*Cosh[d*x]*(1 + 3*d*x + 3*L
og[Cosh[c + d*x]]) + 6*Cosh[2*c + d*x]*(1 + 3*d*x + 3*Log[Cosh[c + d*x]]) - 21*Sinh[d*x] + 12*Sinh[2*c + d*x]
- 11*Sinh[2*c + 3*d*x]))/(6*d*(Cosh[d*x] + Sinh[d*x])^4)

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fricas [B]  time = 0.48, size = 562, normalized size = 7.30 \[ \frac {4 \, {\left (18 \, a^{4} \cosh \left (d x + c\right )^{4} + 72 \, a^{4} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 18 \, a^{4} \sinh \left (d x + c\right )^{4} + 27 \, a^{4} \cosh \left (d x + c\right )^{2} + 11 \, a^{4} + 27 \, {\left (4 \, a^{4} \cosh \left (d x + c\right )^{2} + a^{4}\right )} \sinh \left (d x + c\right )^{2} + 6 \, {\left (a^{4} \cosh \left (d x + c\right )^{6} + 6 \, a^{4} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + a^{4} \sinh \left (d x + c\right )^{6} + 3 \, a^{4} \cosh \left (d x + c\right )^{4} + 3 \, a^{4} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, a^{4} \cosh \left (d x + c\right )^{2} + a^{4}\right )} \sinh \left (d x + c\right )^{4} + a^{4} + 4 \, {\left (5 \, a^{4} \cosh \left (d x + c\right )^{3} + 3 \, a^{4} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (5 \, a^{4} \cosh \left (d x + c\right )^{4} + 6 \, a^{4} \cosh \left (d x + c\right )^{2} + a^{4}\right )} \sinh \left (d x + c\right )^{2} + 6 \, {\left (a^{4} \cosh \left (d x + c\right )^{5} + 2 \, a^{4} \cosh \left (d x + c\right )^{3} + a^{4} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 18 \, {\left (4 \, a^{4} \cosh \left (d x + c\right )^{3} + 3 \, a^{4} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}}{3 \, {\left (d \cosh \left (d x + c\right )^{6} + 6 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + d \sinh \left (d x + c\right )^{6} + 3 \, d \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, d \cosh \left (d x + c\right )^{4} + 6 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{2} + 6 \, {\left (d \cosh \left (d x + c\right )^{5} + 2 \, d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(18*a^4*cosh(d*x + c)^4 + 72*a^4*cosh(d*x + c)*sinh(d*x + c)^3 + 18*a^4*sinh(d*x + c)^4 + 27*a^4*cosh(d*x
+ c)^2 + 11*a^4 + 27*(4*a^4*cosh(d*x + c)^2 + a^4)*sinh(d*x + c)^2 + 6*(a^4*cosh(d*x + c)^6 + 6*a^4*cosh(d*x +
c)*sinh(d*x + c)^5 + a^4*sinh(d*x + c)^6 + 3*a^4*cosh(d*x + c)^4 + 3*a^4*cosh(d*x + c)^2 + 3*(5*a^4*cosh(d*x
+ c)^2 + a^4)*sinh(d*x + c)^4 + a^4 + 4*(5*a^4*cosh(d*x + c)^3 + 3*a^4*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*a
^4*cosh(d*x + c)^4 + 6*a^4*cosh(d*x + c)^2 + a^4)*sinh(d*x + c)^2 + 6*(a^4*cosh(d*x + c)^5 + 2*a^4*cosh(d*x +
c)^3 + a^4*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 18*(4*a^4*cosh
(d*x + c)^3 + 3*a^4*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^6 + 6*d*cosh(d*x + c)*sinh(d*x + c)^5 + d*s
inh(d*x + c)^6 + 3*d*cosh(d*x + c)^4 + 3*(5*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^4 + 4*(5*d*cosh(d*x + c)^3 +
3*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*d*cosh(d*x + c)^2 + 3*(5*d*cosh(d*x + c)^4 + 6*d*cosh(d*x + c)^2 + d)*s
inh(d*x + c)^2 + 6*(d*cosh(d*x + c)^5 + 2*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c) + d)

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giac [A]  time = 0.13, size = 71, normalized size = 0.92 \[ \frac {4 \, {\left (6 \, a^{4} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + \frac {18 \, a^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{4} e^{\left (2 \, d x + 2 \, c\right )} + 11 \, a^{4}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}\right )}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(6*a^4*log(e^(2*d*x + 2*c) + 1) + (18*a^4*e^(4*d*x + 4*c) + 27*a^4*e^(2*d*x + 2*c) + 11*a^4)/(e^(2*d*x + 2
*c) + 1)^3)/d

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maple [A]  time = 0.01, size = 65, normalized size = 0.84 \[ -\frac {a^{4} \left (\tanh ^{3}\left (d x +c \right )\right )}{3 d}-\frac {2 a^{4} \left (\tanh ^{2}\left (d x +c \right )\right )}{d}-\frac {7 a^{4} \tanh \left (d x +c \right )}{d}-\frac {8 a^{4} \ln \left (\tanh \left (d x +c \right )-1\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/d*a^4*tanh(d*x+c)^3-2/d*a^4*tanh(d*x+c)^2-7*a^4*tanh(d*x+c)/d-8/d*a^4*ln(tanh(d*x+c)-1)

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maxima [B]  time = 0.43, size = 196, normalized size = 2.55 \[ \frac {1}{3} \, a^{4} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + 4 \, a^{4} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + 6 \, a^{4} {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + a^{4} x + \frac {4 \, a^{4} \log \left (\cosh \left (d x + c\right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^4*(3*x + 3*c/d - 4*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + 2)/(d*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x -
4*c) + e^(-6*d*x - 6*c) + 1))) + 4*a^4*(x + c/d + log(e^(-2*d*x - 2*c) + 1)/d + 2*e^(-2*d*x - 2*c)/(d*(2*e^(-2
*d*x - 2*c) + e^(-4*d*x - 4*c) + 1))) + 6*a^4*(x + c/d - 2/(d*(e^(-2*d*x - 2*c) + 1))) + a^4*x + 4*a^4*log(cos
h(d*x + c))/d

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mupad [B]  time = 0.11, size = 53, normalized size = 0.69 \[ 16\,a^4\,x-\frac {a^4\,\left (24\,\ln \left (\mathrm {tanh}\left (c+d\,x\right )+1\right )+21\,\mathrm {tanh}\left (c+d\,x\right )+6\,{\mathrm {tanh}\left (c+d\,x\right )}^2+{\mathrm {tanh}\left (c+d\,x\right )}^3\right )}{3\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*tanh(c + d*x))^4,x)

[Out]

16*a^4*x - (a^4*(24*log(tanh(c + d*x) + 1) + 21*tanh(c + d*x) + 6*tanh(c + d*x)^2 + tanh(c + d*x)^3))/(3*d)

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sympy [A]  time = 0.34, size = 76, normalized size = 0.99 \[ \begin {cases} 16 a^{4} x - \frac {8 a^{4} \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} - \frac {a^{4} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {2 a^{4} \tanh ^{2}{\left (c + d x \right )}}{d} - \frac {7 a^{4} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \tanh {\relax (c )} + a\right )^{4} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((16*a**4*x - 8*a**4*log(tanh(c + d*x) + 1)/d - a**4*tanh(c + d*x)**3/(3*d) - 2*a**4*tanh(c + d*x)**2
/d - 7*a**4*tanh(c + d*x)/d, Ne(d, 0)), (x*(a*tanh(c) + a)**4, True))

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