Integrand size = 12, antiderivative size = 77 \[ \int (a+a \tanh (c+d x))^4 \, dx=8 a^4 x+\frac {8 a^4 \log (\cosh (c+d x))}{d}-\frac {4 a^4 \tanh (c+d x)}{d}-\frac {a (a+a \tanh (c+d x))^3}{3 d}-\frac {\left (a^2+a^2 \tanh (c+d x)\right )^2}{d} \]
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
Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.66 \[ \int (a+a \tanh (c+d x))^4 \, dx=-\frac {a^4 \left (4+24 \log (1-\tanh (c+d x))+21 \tanh (c+d x)+6 \tanh ^2(c+d x)+\tanh ^3(c+d x)\right )}{3 d} \]
-1/3*(a^4*(4 + 24*Log[1 - Tanh[c + d*x]] + 21*Tanh[c + d*x] + 6*Tanh[c + d *x]^2 + Tanh[c + d*x]^3))/d
Time = 0.44 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {3042, 3959, 3042, 3959, 3042, 3958, 26, 3042, 26, 3956}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (a \tanh (c+d x)+a)^4 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a-i a \tan (i c+i d x))^4dx\) |
\(\Big \downarrow \) 3959 |
\(\displaystyle 2 a \int (\tanh (c+d x) a+a)^3dx-\frac {a (a \tanh (c+d x)+a)^3}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a (a \tanh (c+d x)+a)^3}{3 d}+2 a \int (a-i a \tan (i c+i d x))^3dx\) |
\(\Big \downarrow \) 3959 |
\(\displaystyle 2 a \left (2 a \int (\tanh (c+d x) a+a)^2dx-\frac {a (a \tanh (c+d x)+a)^2}{2 d}\right )-\frac {a (a \tanh (c+d x)+a)^3}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a (a \tanh (c+d x)+a)^3}{3 d}+2 a \left (-\frac {a (a \tanh (c+d x)+a)^2}{2 d}+2 a \int (a-i a \tan (i c+i d x))^2dx\right )\) |
\(\Big \downarrow \) 3958 |
\(\displaystyle -\frac {a (a \tanh (c+d x)+a)^3}{3 d}+2 a \left (-\frac {a (a \tanh (c+d x)+a)^2}{2 d}+2 a \left (-2 i a^2 \int i \tanh (c+d x)dx-\frac {a^2 \tanh (c+d x)}{d}+2 a^2 x\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle 2 a \left (2 a \left (2 a^2 \int \tanh (c+d x)dx-\frac {a^2 \tanh (c+d x)}{d}+2 a^2 x\right )-\frac {a (a \tanh (c+d x)+a)^2}{2 d}\right )-\frac {a (a \tanh (c+d x)+a)^3}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a (a \tanh (c+d x)+a)^3}{3 d}+2 a \left (-\frac {a (a \tanh (c+d x)+a)^2}{2 d}+2 a \left (2 a^2 \int -i \tan (i c+i d x)dx-\frac {a^2 \tanh (c+d x)}{d}+2 a^2 x\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {a (a \tanh (c+d x)+a)^3}{3 d}+2 a \left (-\frac {a (a \tanh (c+d x)+a)^2}{2 d}+2 a \left (-2 i a^2 \int \tan (i c+i d x)dx-\frac {a^2 \tanh (c+d x)}{d}+2 a^2 x\right )\right )\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle 2 a \left (2 a \left (-\frac {a^2 \tanh (c+d x)}{d}+\frac {2 a^2 \log (\cosh (c+d x))}{d}+2 a^2 x\right )-\frac {a (a \tanh (c+d x)+a)^2}{2 d}\right )-\frac {a (a \tanh (c+d x)+a)^3}{3 d}\) |
-1/3*(a*(a + a*Tanh[c + d*x])^3)/d + 2*a*(-1/2*(a*(a + a*Tanh[c + d*x])^2) /d + 2*a*(2*a^2*x + (2*a^2*Log[Cosh[c + d*x]])/d - (a^2*Tanh[c + d*x])/d))
3.1.42.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(a^2 - b^2) *x, x] + (Simp[b^2*(Tan[c + d*x]/d), x] + Simp[2*a*b Int[Tan[c + d*x], x] , x]) /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[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] && GtQ[n , 1]
Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.62
method | result | size |
derivativedivides | \(\frac {a^{4} \left (-\frac {\tanh \left (d x +c \right )^{3}}{3}-2 \tanh \left (d x +c \right )^{2}-7 \tanh \left (d x +c \right )-8 \ln \left (\tanh \left (d x +c \right )-1\right )\right )}{d}\) | \(48\) |
default | \(\frac {a^{4} \left (-\frac {\tanh \left (d x +c \right )^{3}}{3}-2 \tanh \left (d x +c \right )^{2}-7 \tanh \left (d x +c \right )-8 \ln \left (\tanh \left (d x +c \right )-1\right )\right )}{d}\) | \(48\) |
parallelrisch | \(-\frac {\tanh \left (d x +c \right )^{3} a^{4}+6 \tanh \left (d x +c \right )^{2} a^{4}+24 \ln \left (1-\tanh \left (d x +c \right )\right ) a^{4}+21 a^{4} \tanh \left (d x +c \right )}{3 d}\) | \(59\) |
risch | \(-\frac {16 a^{4} c}{d}+\frac {4 a^{4} \left (18 \,{\mathrm e}^{4 d x +4 c}+27 \,{\mathrm e}^{2 d x +2 c}+11\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3}}+\frac {8 a^{4} \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d}\) | \(76\) |
parts | \(x \,a^{4}+\frac {a^{4} \left (-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {4 a^{4} \ln \left (\cosh \left (d x +c \right )\right )}{d}+\frac {6 a^{4} \left (-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {4 a^{4} \left (-\frac {\tanh \left (d x +c \right )^{2}}{2}-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}-\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}\) | \(150\) |
Leaf count of result is larger than twice the leaf count of optimal. 562 vs. \(2 (75) = 150\).
Time = 0.25 (sec) , antiderivative size = 562, normalized size of antiderivative = 7.30 \[ \int (a+a \tanh (c+d x))^4 \, dx=\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 )}} \]
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*co sh(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*c osh(d*x + c)^4 + 6*a^4*cosh(d*x + c)^2 + a^4)*sinh(d*x + c)^2 + 6*(a^4*cos h(d*x + c)^5 + 2*a^4*cosh(d*x + c)^3 + a^4*cosh(d*x + c))*sinh(d*x + c))*l og(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*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*co sh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c) + d)
Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int (a+a \tanh (c+d x))^4 \, dx=\begin {cases} 16 a^{4} x - \frac {8 a^{4} \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} - \frac {a^{4} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {2 a^{4} \tanh ^{2}{\left (c + d x \right )}}{d} - \frac {7 a^{4} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \tanh {\left (c \right )} + a\right )^{4} & \text {otherwise} \end {cases} \]
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))
Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (75) = 150\).
Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.55 \[ \int (a+a \tanh (c+d x))^4 \, dx=\frac {1}{3} \, a^{4} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + 4 \, a^{4} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + 6 \, a^{4} {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + a^{4} x + \frac {4 \, a^{4} \log \left (\cosh \left (d x + c\right )\right )}{d} \]
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(cosh(d*x + c))/d
Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int (a+a \tanh (c+d x))^4 \, dx=\frac {4 \, {\left (6 \, a^{4} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + \frac {18 \, a^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{4} e^{\left (2 \, d x + 2 \, c\right )} + 11 \, a^{4}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}\right )}}{3 \, d} \]
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
Time = 1.70 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.69 \[ \int (a+a \tanh (c+d x))^4 \, dx=16\,a^4\,x-\frac {a^4\,\left (24\,\ln \left (\mathrm {tanh}\left (c+d\,x\right )+1\right )+21\,\mathrm {tanh}\left (c+d\,x\right )+6\,{\mathrm {tanh}\left (c+d\,x\right )}^2+{\mathrm {tanh}\left (c+d\,x\right )}^3\right )}{3\,d} \]