3.1.58 \(\int (a+b \tanh (c+d x))^4 \, dx\) [58]

Optimal. Leaf size=101 \[ \left (a^4+6 a^2 b^2+b^4\right ) x+\frac {4 a b \left (a^2+b^2\right ) \log (\cosh (c+d x))}{d}-\frac {b^2 \left (3 a^2+b^2\right ) \tanh (c+d x)}{d}-\frac {a b (a+b \tanh (c+d x))^2}{d}-\frac {b (a+b \tanh (c+d x))^3}{3 d} \]

[Out]

(a^4+6*a^2*b^2+b^4)*x+4*a*b*(a^2+b^2)*ln(cosh(d*x+c))/d-b^2*(3*a^2+b^2)*tanh(d*x+c)/d-a*b*(a+b*tanh(d*x+c))^2/
d-1/3*b*(a+b*tanh(d*x+c))^3/d

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Rubi [A]
time = 0.09, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3563, 3609, 3606, 3556} \begin {gather*} -\frac {b^2 \left (3 a^2+b^2\right ) \tanh (c+d x)}{d}+\frac {4 a b \left (a^2+b^2\right ) \log (\cosh (c+d x))}{d}+x \left (a^4+6 a^2 b^2+b^4\right )-\frac {b (a+b \tanh (c+d x))^3}{3 d}-\frac {a b (a+b \tanh (c+d x))^2}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Tanh[c + d*x])^4,x]

[Out]

(a^4 + 6*a^2*b^2 + b^4)*x + (4*a*b*(a^2 + b^2)*Log[Cosh[c + d*x]])/d - (b^2*(3*a^2 + b^2)*Tanh[c + d*x])/d - (
a*b*(a + b*Tanh[c + d*x])^2)/d - (b*(a + b*Tanh[c + d*x])^3)/(3*d)

Rule 3556

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

Rule 3563

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

Rule 3606

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c - b
*d)*x, x] + (Dist[b*c + a*d, Int[Tan[e + f*x], x], x] + Simp[b*d*(Tan[e + f*x]/f), x]) /; FreeQ[{a, b, c, d, e
, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]

Rule 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*
((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

Rubi steps

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

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Mathematica [A]
time = 0.24, size = 91, normalized size = 0.90 \begin {gather*} -\frac {3 (a+b)^4 \log (1-\tanh (c+d x))-3 (a-b)^4 \log (1+\tanh (c+d x))+6 b^2 \left (6 a^2+b^2\right ) \tanh (c+d x)+12 a b^3 \tanh ^2(c+d x)+2 b^4 \tanh ^3(c+d x)}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tanh[c + d*x])^4,x]

[Out]

-1/6*(3*(a + b)^4*Log[1 - Tanh[c + d*x]] - 3*(a - b)^4*Log[1 + Tanh[c + d*x]] + 6*b^2*(6*a^2 + b^2)*Tanh[c + d
*x] + 12*a*b^3*Tanh[c + d*x]^2 + 2*b^4*Tanh[c + d*x]^3)/d

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Maple [A]
time = 0.27, size = 134, normalized size = 1.33

method	result	size
derivativedivides	\(\frac {-\frac {b^{4} \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-2 a \,b^{3} \left (\tanh ^{2}\left (d x +c \right )\right )-6 a^{2} b^{2} \tanh \left (d x +c \right )-b^{4} \tanh \left (d x +c \right )-\frac {\left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\left (a^{4}-4 a^{3} b +6 a^{2} b^{2}-4 a \,b^{3}+b^{4}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )}{2}}{d}\)	\(134\)
default	\(\frac {-\frac {b^{4} \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-2 a \,b^{3} \left (\tanh ^{2}\left (d x +c \right )\right )-6 a^{2} b^{2} \tanh \left (d x +c \right )-b^{4} \tanh \left (d x +c \right )-\frac {\left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\left (a^{4}-4 a^{3} b +6 a^{2} b^{2}-4 a \,b^{3}+b^{4}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )}{2}}{d}\)	\(134\)
risch	\(x \,a^{4}-4 a^{3} b x +6 a^{2} b^{2} x -4 b^{3} a x +b^{4} x -\frac {8 b \,a^{3} c}{d}-\frac {8 b^{3} a c}{d}+\frac {4 b^{2} \left (9 a^{2} {\mathrm e}^{4 d x +4 c}+6 a b \,{\mathrm e}^{4 d x +4 c}+3 b^{2} {\mathrm e}^{4 d x +4 c}+18 a^{2} {\mathrm e}^{2 d x +2 c}+6 a b \,{\mathrm e}^{2 d x +2 c}+3 b^{2} {\mathrm e}^{2 d x +2 c}+9 a^{2}+2 b^{2}\right )}{3 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}+\frac {4 b \,a^{3} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d}+\frac {4 b^{3} a \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d}\)	\(211\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tanh(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/3*b^4*tanh(d*x+c)^3-2*a*b^3*tanh(d*x+c)^2-6*a^2*b^2*tanh(d*x+c)-b^4*tanh(d*x+c)-1/2*(a^4+4*a^3*b+6*a^2
*b^2+4*a*b^3+b^4)*ln(tanh(d*x+c)-1)+1/2*(a^4-4*a^3*b+6*a^2*b^2-4*a*b^3+b^4)*ln(tanh(d*x+c)+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (99) = 198\).
time = 0.47, size = 201, normalized size = 1.99 \begin {gather*} \frac {1}{3} \, b^{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 b^{3} {\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^{2} b^{2} {\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^{3} b \log \left (\cosh \left (d x + c\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^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*b^3*(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^2*b^2*(x + c/d - 2/(d*(e^(-2*d*x - 2*c) + 1))) + a^4*x + 4*a^3*b
*log(cosh(d*x + c))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1389 vs. \(2 (99) = 198\).
time = 0.37, size = 1389, normalized size = 13.75 \begin {gather*} \frac {3 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x \cosh \left (d x + c\right )^{6} + 18 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 3 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x \sinh \left (d x + c\right )^{6} + 3 \, {\left (12 \, a^{2} b^{2} + 8 \, a b^{3} + 4 \, b^{4} + 3 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (15 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x \cosh \left (d x + c\right )^{2} + 12 \, a^{2} b^{2} + 8 \, a b^{3} + 4 \, b^{4} + 3 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x\right )} \sinh \left (d x + c\right )^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4} + 12 \, {\left (5 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x \cosh \left (d x + c\right )^{3} + {\left (12 \, a^{2} b^{2} + 8 \, a b^{3} + 4 \, b^{4} + 3 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x + 3 \, {\left (24 \, a^{2} b^{2} + 8 \, a b^{3} + 4 \, b^{4} + 3 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (15 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x \cosh \left (d x + c\right )^{4} + 24 \, a^{2} b^{2} + 8 \, a b^{3} + 4 \, b^{4} + 3 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x + 6 \, {\left (12 \, a^{2} b^{2} + 8 \, a b^{3} + 4 \, b^{4} + 3 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 12 \, {\left ({\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{6} + 6 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + {\left (a^{3} b + a b^{3}\right )} \sinh \left (d x + c\right )^{6} + 3 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (a^{3} b + a b^{3} + 5 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} + a^{3} b + a b^{3} + 4 \, {\left (5 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{4} + a^{3} b + a b^{3} + 6 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 6 \, {\left ({\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{5} + 2 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{3} b + a b^{3}\right )} \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 ) + 6 \, {\left (3 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x \cosh \left (d x + c\right )^{5} + 2 \, {\left (12 \, a^{2} b^{2} + 8 \, a b^{3} + 4 \, b^{4} + 3 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x\right )} \cosh \left (d x + c\right )^{3} + {\left (24 \, a^{2} b^{2} + 8 \, a b^{3} + 4 \, b^{4} + 3 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} d x\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*d*x*cosh(d*x + c)^6 + 18*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b
^3 + b^4)*d*x*cosh(d*x + c)*sinh(d*x + c)^5 + 3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*d*x*sinh(d*x + c)^
6 + 3*(12*a^2*b^2 + 8*a*b^3 + 4*b^4 + 3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*d*x)*cosh(d*x + c)^4 + 3*(
15*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*d*x*cosh(d*x + c)^2 + 12*a^2*b^2 + 8*a*b^3 + 4*b^4 + 3*(a^4 - 4
*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*d*x)*sinh(d*x + c)^4 + 36*a^2*b^2 + 8*b^4 + 12*(5*(a^4 - 4*a^3*b + 6*a^2*b
^2 - 4*a*b^3 + b^4)*d*x*cosh(d*x + c)^3 + (12*a^2*b^2 + 8*a*b^3 + 4*b^4 + 3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b
^3 + b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*d*x + 3*(24*a^2*
b^2 + 8*a*b^3 + 4*b^4 + 3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*d*x)*cosh(d*x + c)^2 + 3*(15*(a^4 - 4*a^
3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*d*x*cosh(d*x + c)^4 + 24*a^2*b^2 + 8*a*b^3 + 4*b^4 + 3*(a^4 - 4*a^3*b + 6*a^2
*b^2 - 4*a*b^3 + b^4)*d*x + 6*(12*a^2*b^2 + 8*a*b^3 + 4*b^4 + 3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*d*
x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 12*((a^3*b + a*b^3)*cosh(d*x + c)^6 + 6*(a^3*b + a*b^3)*cosh(d*x + c)*si
nh(d*x + c)^5 + (a^3*b + a*b^3)*sinh(d*x + c)^6 + 3*(a^3*b + a*b^3)*cosh(d*x + c)^4 + 3*(a^3*b + a*b^3 + 5*(a^
3*b + a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^3*b + a*b^3 + 4*(5*(a^3*b + a*b^3)*cosh(d*x + c)^3 + 3*(a^3*
b + a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(a^3*b + a*b^3)*cosh(d*x + c)^2 + 3*(5*(a^3*b + a*b^3)*cosh(d*x
+ c)^4 + a^3*b + a*b^3 + 6*(a^3*b + a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 6*((a^3*b + a*b^3)*cosh(d*x + c)
^5 + 2*(a^3*b + a*b^3)*cosh(d*x + c)^3 + (a^3*b + a*b^3)*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(co
sh(d*x + c) - sinh(d*x + c))) + 6*(3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*d*x*cosh(d*x + c)^5 + 2*(12*a
^2*b^2 + 8*a*b^3 + 4*b^4 + 3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*d*x)*cosh(d*x + c)^3 + (24*a^2*b^2 +
8*a*b^3 + 4*b^4 + 3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*d*x)*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*sinh(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)*sinh(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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Sympy [A]
time = 0.13, size = 144, normalized size = 1.43 \begin {gather*} \begin {cases} a^{4} x + 4 a^{3} b x - \frac {4 a^{3} b \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} + 6 a^{2} b^{2} x - \frac {6 a^{2} b^{2} \tanh {\left (c + d x \right )}}{d} + 4 a b^{3} x - \frac {4 a b^{3} \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} - \frac {2 a b^{3} \tanh ^{2}{\left (c + d x \right )}}{d} + b^{4} x - \frac {b^{4} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {b^{4} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tanh {\left (c \right )}\right )^{4} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**4*x + 4*a**3*b*x - 4*a**3*b*log(tanh(c + d*x) + 1)/d + 6*a**2*b**2*x - 6*a**2*b**2*tanh(c + d*x)
/d + 4*a*b**3*x - 4*a*b**3*log(tanh(c + d*x) + 1)/d - 2*a*b**3*tanh(c + d*x)**2/d + b**4*x - b**4*tanh(c + d*x
)**3/(3*d) - b**4*tanh(c + d*x)/d, Ne(d, 0)), (x*(a + b*tanh(c))**4, True))

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Giac [A]
time = 0.43, size = 152, normalized size = 1.50 \begin {gather*} \frac {3 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} {\left (d x + c\right )} + 12 \, {\left (a^{3} b + a b^{3}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + \frac {4 \, {\left (9 \, a^{2} b^{2} + 2 \, b^{4} + 3 \, {\left (3 \, a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, {\left (6 \, a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.09, size = 113, normalized size = 1.12 \begin {gather*} x\,\left (a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4\right )-\frac {b^4\,{\mathrm {tanh}\left (c+d\,x\right )}^3}{3\,d}-\frac {\ln \left (\mathrm {tanh}\left (c+d\,x\right )+1\right )\,\left (4\,a^3\,b+4\,a\,b^3\right )}{d}-\frac {2\,a\,b^3\,{\mathrm {tanh}\left (c+d\,x\right )}^2}{d}-\frac {b^2\,\mathrm {tanh}\left (c+d\,x\right )\,\left (6\,a^2+b^2\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tanh(c + d*x))^4,x)

[Out]

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

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