Integrand size = 10, antiderivative size = 74 \[ \int e^x \coth ^4(2 x) \, dx=e^x+\frac {4 e^x}{3 \left (1-e^{4 x}\right )^3}-\frac {13 e^x}{6 \left (1-e^{4 x}\right )^2}+\frac {53 e^x}{24 \left (1-e^{4 x}\right )}-\frac {11 \arctan \left (e^x\right )}{16}-\frac {11 \text {arctanh}\left (e^x\right )}{16} \] Output:
exp(x)+4/3*exp(x)/(1-exp(4*x))^3-13/6*exp(x)/(1-exp(4*x))^2+53*exp(x)/(24- 24*exp(4*x))-11/16*arctan(exp(x))-11/16*arctanh(exp(x))
Time = 11.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.03 \[ \int e^x \coth ^4(2 x) \, dx=\frac {1}{96} \left (96 e^x-\frac {128 e^x}{\left (-1+e^{4 x}\right )^3}-\frac {208 e^x}{\left (-1+e^{4 x}\right )^2}-\frac {212 e^x}{-1+e^{4 x}}-66 \arctan \left (e^x\right )+33 \log \left (1-e^x\right )-33 \log \left (1+e^x\right )\right ) \] Input:
Integrate[E^x*Coth[2*x]^4,x]
Output:
(96*E^x - (128*E^x)/(-1 + E^(4*x))^3 - (208*E^x)/(-1 + E^(4*x))^2 - (212*E ^x)/(-1 + E^(4*x)) - 66*ArcTan[E^x] + 33*Log[1 - E^x] - 33*Log[1 + E^x])/9 6
Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2720, 915, 2009}
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.
\(\displaystyle \int e^x \coth ^4(2 x) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \int \frac {\left (e^{4 x}+1\right )^4}{\left (1-e^{4 x}\right )^4}de^x\) |
\(\Big \downarrow \) 915 |
\(\displaystyle \int \left (\frac {8 e^{4 x} \left (e^{8 x}+1\right )}{\left (1-e^{4 x}\right )^4}+1\right )de^x\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {11}{16} \arctan \left (e^x\right )-\frac {11 \text {arctanh}\left (e^x\right )}{16}+e^x+\frac {53 e^x}{24 \left (1-e^{4 x}\right )}-\frac {13 e^x}{6 \left (1-e^{4 x}\right )^2}+\frac {4 e^x}{3 \left (1-e^{4 x}\right )^3}\) |
Input:
Int[E^x*Coth[2*x]^4,x]
Output:
E^x + (4*E^x)/(3*(1 - E^(4*x))^3) - (13*E^x)/(6*(1 - E^(4*x))^2) + (53*E^x )/(24*(1 - E^(4*x))) - (11*ArcTan[E^x])/16 - (11*ArcTanh[E^x])/16
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a , b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Result contains complex when optimal does not.
Time = 0.68 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.84
method | result | size |
risch | \({\mathrm e}^{x}-\frac {{\mathrm e}^{x} \left (53 \,{\mathrm e}^{8 x}-54 \,{\mathrm e}^{4 x}+33\right )}{24 \left ({\mathrm e}^{4 x}-1\right )^{3}}+\frac {11 i \ln \left ({\mathrm e}^{x}-i\right )}{32}-\frac {11 i \ln \left ({\mathrm e}^{x}+i\right )}{32}-\frac {11 \ln \left ({\mathrm e}^{x}+1\right )}{32}+\frac {11 \ln \left ({\mathrm e}^{x}-1\right )}{32}\) | \(62\) |
Input:
int(exp(x)*coth(2*x)^4,x,method=_RETURNVERBOSE)
Output:
exp(x)-1/24*exp(x)*(53*exp(8*x)-54*exp(4*x)+33)/(exp(4*x)-1)^3+11/32*I*ln( exp(x)-I)-11/32*I*ln(exp(x)+I)-11/32*ln(exp(x)+1)+11/32*ln(exp(x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 1097 vs. \(2 (49) = 98\).
Time = 0.11 (sec) , antiderivative size = 1097, normalized size of antiderivative = 14.82 \[ \int e^x \coth ^4(2 x) \, dx=\text {Too large to display} \] Input:
integrate(exp(x)*coth(2*x)^4,x, algorithm="fricas")
Output:
1/96*(96*cosh(x)^13 + 27456*cosh(x)^3*sinh(x)^10 + 7488*cosh(x)^2*sinh(x)^ 11 + 1248*cosh(x)*sinh(x)^12 + 96*sinh(x)^13 + 20*(3432*cosh(x)^4 - 25)*si nh(x)^9 - 500*cosh(x)^9 + 36*(3432*cosh(x)^5 - 125*cosh(x))*sinh(x)^8 + 14 4*(1144*cosh(x)^6 - 125*cosh(x)^2)*sinh(x)^7 + 48*(3432*cosh(x)^7 - 875*co sh(x)^3)*sinh(x)^6 + 72*(1716*cosh(x)^8 - 875*cosh(x)^4 + 7)*sinh(x)^5 + 5 04*cosh(x)^5 + 120*(572*cosh(x)^9 - 525*cosh(x)^5 + 21*cosh(x))*sinh(x)^4 + 48*(572*cosh(x)^10 - 875*cosh(x)^6 + 105*cosh(x)^2)*sinh(x)^3 + 144*(52* cosh(x)^11 - 125*cosh(x)^7 + 35*cosh(x)^3)*sinh(x)^2 - 66*(cosh(x)^12 + 22 0*cosh(x)^3*sinh(x)^9 + 66*cosh(x)^2*sinh(x)^10 + 12*cosh(x)*sinh(x)^11 + sinh(x)^12 + 3*(165*cosh(x)^4 - 1)*sinh(x)^8 - 3*cosh(x)^8 + 24*(33*cosh(x )^5 - cosh(x))*sinh(x)^7 + 84*(11*cosh(x)^6 - cosh(x)^2)*sinh(x)^6 + 24*(3 3*cosh(x)^7 - 7*cosh(x)^3)*sinh(x)^5 + 3*(165*cosh(x)^8 - 70*cosh(x)^4 + 1 )*sinh(x)^4 + 3*cosh(x)^4 + 4*(55*cosh(x)^9 - 42*cosh(x)^5 + 3*cosh(x))*si nh(x)^3 + 6*(11*cosh(x)^10 - 14*cosh(x)^6 + 3*cosh(x)^2)*sinh(x)^2 + 12*(c osh(x)^11 - 2*cosh(x)^7 + cosh(x)^3)*sinh(x) - 1)*arctan(cosh(x) + sinh(x) ) - 33*(cosh(x)^12 + 220*cosh(x)^3*sinh(x)^9 + 66*cosh(x)^2*sinh(x)^10 + 1 2*cosh(x)*sinh(x)^11 + sinh(x)^12 + 3*(165*cosh(x)^4 - 1)*sinh(x)^8 - 3*co sh(x)^8 + 24*(33*cosh(x)^5 - cosh(x))*sinh(x)^7 + 84*(11*cosh(x)^6 - cosh( x)^2)*sinh(x)^6 + 24*(33*cosh(x)^7 - 7*cosh(x)^3)*sinh(x)^5 + 3*(165*cosh( x)^8 - 70*cosh(x)^4 + 1)*sinh(x)^4 + 3*cosh(x)^4 + 4*(55*cosh(x)^9 - 42...
\[ \int e^x \coth ^4(2 x) \, dx=\int e^{x} \coth ^{4}{\left (2 x \right )}\, dx \] Input:
integrate(exp(x)*coth(2*x)**4,x)
Output:
Integral(exp(x)*coth(2*x)**4, x)
Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.82 \[ \int e^x \coth ^4(2 x) \, dx=-\frac {53 \, e^{\left (9 \, x\right )} - 54 \, e^{\left (5 \, x\right )} + 33 \, e^{x}}{24 \, {\left (e^{\left (12 \, x\right )} - 3 \, e^{\left (8 \, x\right )} + 3 \, e^{\left (4 \, x\right )} - 1\right )}} - \frac {11}{16} \, \arctan \left (e^{x}\right ) + e^{x} - \frac {11}{32} \, \log \left (e^{x} + 1\right ) + \frac {11}{32} \, \log \left (e^{x} - 1\right ) \] Input:
integrate(exp(x)*coth(2*x)^4,x, algorithm="maxima")
Output:
-1/24*(53*e^(9*x) - 54*e^(5*x) + 33*e^x)/(e^(12*x) - 3*e^(8*x) + 3*e^(4*x) - 1) - 11/16*arctan(e^x) + e^x - 11/32*log(e^x + 1) + 11/32*log(e^x - 1)
Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.68 \[ \int e^x \coth ^4(2 x) \, dx=-\frac {53 \, e^{\left (9 \, x\right )} - 54 \, e^{\left (5 \, x\right )} + 33 \, e^{x}}{24 \, {\left (e^{\left (4 \, x\right )} - 1\right )}^{3}} - \frac {11}{16} \, \arctan \left (e^{x}\right ) + e^{x} - \frac {11}{32} \, \log \left (e^{x} + 1\right ) + \frac {11}{32} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \] Input:
integrate(exp(x)*coth(2*x)^4,x, algorithm="giac")
Output:
-1/24*(53*e^(9*x) - 54*e^(5*x) + 33*e^x)/(e^(4*x) - 1)^3 - 11/16*arctan(e^ x) + e^x - 11/32*log(e^x + 1) + 11/32*log(abs(e^x - 1))
Time = 2.57 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.49 \[ \int e^x \coth ^4(2 x) \, dx=\frac {11\,\ln \left (\frac {11}{8}-\frac {11\,{\mathrm {e}}^x}{8}\right )}{32}-\frac {11\,\ln \left (-\frac {11\,{\mathrm {e}}^x}{8}-\frac {11}{8}\right )}{32}+{\mathrm {e}}^x-\frac {37\,{\mathrm {e}}^x}{24\,\left ({\mathrm {e}}^{4\,x}-1\right )}-\frac {\frac {2\,{\mathrm {e}}^{9\,x}}{3}+\frac {2\,{\mathrm {e}}^x}{3}}{3\,{\mathrm {e}}^{4\,x}-3\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{12\,x}-1}-\frac {5\,{\mathrm {e}}^x}{6\,\left ({\mathrm {e}}^{8\,x}-2\,{\mathrm {e}}^{4\,x}+1\right )}-\frac {\ln \left (-\frac {11\,{\mathrm {e}}^x}{8}-\frac {11}{8}{}\mathrm {i}\right )\,11{}\mathrm {i}}{32}+\frac {\ln \left (-\frac {11\,{\mathrm {e}}^x}{8}+\frac {11}{8}{}\mathrm {i}\right )\,11{}\mathrm {i}}{32} \] Input:
int(coth(2*x)^4*exp(x),x)
Output:
(11*log(11/8 - (11*exp(x))/8))/32 - (11*log(- (11*exp(x))/8 - 11/8))/32 - (log(- (11*exp(x))/8 - 11i/8)*11i)/32 + (log(11i/8 - (11*exp(x))/8)*11i)/3 2 + exp(x) - (37*exp(x))/(24*(exp(4*x) - 1)) - ((2*exp(9*x))/3 + (2*exp(x) )/3)/(3*exp(4*x) - 3*exp(8*x) + exp(12*x) - 1) - (5*exp(x))/(6*(exp(8*x) - 2*exp(4*x) + 1))
Time = 0.25 (sec) , antiderivative size = 186, normalized size of antiderivative = 2.51 \[ \int e^x \coth ^4(2 x) \, dx=\frac {-66 e^{12 x} \mathit {atan} \left (e^{x}\right )+198 e^{8 x} \mathit {atan} \left (e^{x}\right )-198 e^{4 x} \mathit {atan} \left (e^{x}\right )+66 \mathit {atan} \left (e^{x}\right )+96 e^{13 x}+33 e^{12 x} \mathrm {log}\left (e^{x}-1\right )-33 e^{12 x} \mathrm {log}\left (e^{x}+1\right )-500 e^{9 x}-99 e^{8 x} \mathrm {log}\left (e^{x}-1\right )+99 e^{8 x} \mathrm {log}\left (e^{x}+1\right )+504 e^{5 x}+99 e^{4 x} \mathrm {log}\left (e^{x}-1\right )-99 e^{4 x} \mathrm {log}\left (e^{x}+1\right )-228 e^{x}-33 \,\mathrm {log}\left (e^{x}-1\right )+33 \,\mathrm {log}\left (e^{x}+1\right )}{96 e^{12 x}-288 e^{8 x}+288 e^{4 x}-96} \] Input:
int(exp(x)*coth(2*x)^4,x)
Output:
( - 66*e**(12*x)*atan(e**x) + 198*e**(8*x)*atan(e**x) - 198*e**(4*x)*atan( e**x) + 66*atan(e**x) + 96*e**(13*x) + 33*e**(12*x)*log(e**x - 1) - 33*e** (12*x)*log(e**x + 1) - 500*e**(9*x) - 99*e**(8*x)*log(e**x - 1) + 99*e**(8 *x)*log(e**x + 1) + 504*e**(5*x) + 99*e**(4*x)*log(e**x - 1) - 99*e**(4*x) *log(e**x + 1) - 228*e**x - 33*log(e**x - 1) + 33*log(e**x + 1))/(96*(e**( 12*x) - 3*e**(8*x) + 3*e**(4*x) - 1))