Integrand size = 14, antiderivative size = 55 \[ \int e^x \coth ^2(2 x) \text {csch}(2 x) \, dx=-\frac {e^{3 x}}{\left (1-e^{4 x}\right )^2}+\frac {3 e^{3 x}}{4 \left (1-e^{4 x}\right )}+\frac {5 \arctan \left (e^x\right )}{8}-\frac {5 \text {arctanh}\left (e^x\right )}{8} \] Output:
-exp(3*x)/(1-exp(4*x))^2+3*exp(3*x)/(4-4*exp(4*x))+5/8*arctan(exp(x))-5/8* arctanh(exp(x))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 2.32 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.93 \[ \int e^x \coth ^2(2 x) \text {csch}(2 x) \, dx=\frac {e^{-5 x} \left (177023+244931 e^{4 x}+43161 e^{8 x}-26091 e^{12 x}-7 \left (25289+24152 e^{4 x}-10058 e^{8 x}-9048 e^{12 x}+513 e^{16 x}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},e^{4 x}\right )\right )}{10752}-\frac {8 e^{7 x} \left (15+26 e^{4 x}+11 e^{8 x}\right ) \, _4F_3\left (\frac {7}{4},2,2,2;1,1,\frac {19}{4};e^{4 x}\right )}{1155}-\frac {16 e^{7 x} \left (1+e^{4 x}\right )^2 \, _5F_4\left (\frac {7}{4},2,2,2,2;1,1,1,\frac {19}{4};e^{4 x}\right )}{1155} \] Input:
Integrate[E^x*Coth[2*x]^2*Csch[2*x],x]
Output:
(177023 + 244931*E^(4*x) + 43161*E^(8*x) - 26091*E^(12*x) - 7*(25289 + 241 52*E^(4*x) - 10058*E^(8*x) - 9048*E^(12*x) + 513*E^(16*x))*Hypergeometric2 F1[3/4, 1, 7/4, E^(4*x)])/(10752*E^(5*x)) - (8*E^(7*x)*(15 + 26*E^(4*x) + 11*E^(8*x))*HypergeometricPFQ[{7/4, 2, 2, 2}, {1, 1, 19/4}, E^(4*x)])/1155 - (16*E^(7*x)*(1 + E^(4*x))^2*HypergeometricPFQ[{7/4, 2, 2, 2, 2}, {1, 1, 1, 19/4}, E^(4*x)])/1155
Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.25, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {2720, 27, 963, 27, 957, 827, 216, 219}
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.
\(\displaystyle \int e^x \coth ^2(2 x) \text {csch}(2 x) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \int -\frac {2 e^{2 x} \left (e^{4 x}+1\right )^2}{\left (1-e^{4 x}\right )^3}de^x\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -2 \int \frac {e^{2 x} \left (1+e^{4 x}\right )^2}{\left (1-e^{4 x}\right )^3}de^x\) |
\(\Big \downarrow \) 963 |
\(\displaystyle -2 \left (\frac {e^{3 x}}{2 \left (1-e^{4 x}\right )^2}-\frac {1}{8} \int \frac {4 e^{2 x} \left (1+2 e^{4 x}\right )}{\left (1-e^{4 x}\right )^2}de^x\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -2 \left (\frac {e^{3 x}}{2 \left (1-e^{4 x}\right )^2}-\frac {1}{2} \int \frac {e^{2 x} \left (1+2 e^{4 x}\right )}{\left (1-e^{4 x}\right )^2}de^x\right )\) |
\(\Big \downarrow \) 957 |
\(\displaystyle -2 \left (\frac {1}{2} \left (\frac {5}{4} \int \frac {e^{2 x}}{1-e^{4 x}}de^x-\frac {3 e^{3 x}}{4 \left (1-e^{4 x}\right )}\right )+\frac {e^{3 x}}{2 \left (1-e^{4 x}\right )^2}\right )\) |
\(\Big \downarrow \) 827 |
\(\displaystyle -2 \left (\frac {1}{2} \left (\frac {5}{4} \left (\frac {1}{2} \int \frac {1}{1-e^{2 x}}de^x-\frac {1}{2} \int \frac {1}{1+e^{2 x}}de^x\right )-\frac {3 e^{3 x}}{4 \left (1-e^{4 x}\right )}\right )+\frac {e^{3 x}}{2 \left (1-e^{4 x}\right )^2}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -2 \left (\frac {1}{2} \left (\frac {5}{4} \left (\frac {1}{2} \int \frac {1}{1-e^{2 x}}de^x-\frac {\arctan \left (e^x\right )}{2}\right )-\frac {3 e^{3 x}}{4 \left (1-e^{4 x}\right )}\right )+\frac {e^{3 x}}{2 \left (1-e^{4 x}\right )^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -2 \left (\frac {1}{2} \left (\frac {5}{4} \left (\frac {\text {arctanh}\left (e^x\right )}{2}-\frac {\arctan \left (e^x\right )}{2}\right )-\frac {3 e^{3 x}}{4 \left (1-e^{4 x}\right )}\right )+\frac {e^{3 x}}{2 \left (1-e^{4 x}\right )^2}\right )\) |
Input:
Int[E^x*Coth[2*x]^2*Csch[2*x],x]
Output:
-2*(E^(3*x)/(2*(1 - E^(4*x))^2) + ((-3*E^(3*x))/(4*(1 - E^(4*x))) + (5*(-1 /2*ArcTan[E^x] + ArcTanh[E^x]/2))/4)/2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a *b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* (p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N eQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 , m, (-n)*(p + 1)]))
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^2, x_Symbol] :> Simp[(-(b*c - a*d)^2)*(e*x)^(m + 1)*((a + b*x^n)^(p + 1) /(a*b^2*e*n*(p + 1))), x] + Simp[1/(a*b^2*n*(p + 1)) Int[(e*x)^m*(a + b*x ^n)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1]
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 = 0.39 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02
method | result | size |
risch | \(-\frac {{\mathrm e}^{3 x} \left (3 \,{\mathrm e}^{4 x}+1\right )}{4 \left ({\mathrm e}^{4 x}-1\right )^{2}}+\frac {5 i \ln \left ({\mathrm e}^{x}+i\right )}{16}-\frac {5 i \ln \left ({\mathrm e}^{x}-i\right )}{16}+\frac {5 \ln \left ({\mathrm e}^{x}-1\right )}{16}-\frac {5 \ln \left (1+{\mathrm e}^{x}\right )}{16}\) | \(56\) |
Input:
int(exp(x)*coth(2*x)^2*csch(2*x),x,method=_RETURNVERBOSE)
Output:
-1/4*exp(x)^3*(3*exp(x)^4+1)/(exp(x)^4-1)^2+5/16*I*ln(exp(x)+I)-5/16*I*ln( exp(x)-I)+5/16*ln(exp(x)-1)-5/16*ln(1+exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 557 vs. \(2 (39) = 78\).
Time = 0.09 (sec) , antiderivative size = 557, normalized size of antiderivative = 10.13 \[ \int e^x \coth ^2(2 x) \text {csch}(2 x) \, dx=\text {Too large to display} \] Input:
integrate(exp(x)*coth(2*x)^2*csch(2*x),x, algorithm="fricas")
Output:
-1/16*(12*cosh(x)^7 + 420*cosh(x)^3*sinh(x)^4 + 252*cosh(x)^2*sinh(x)^5 + 84*cosh(x)*sinh(x)^6 + 12*sinh(x)^7 + 4*(105*cosh(x)^4 + 1)*sinh(x)^3 + 4* cosh(x)^3 + 12*(21*cosh(x)^5 + cosh(x))*sinh(x)^2 - 10*(cosh(x)^8 + 56*cos h(x)^3*sinh(x)^5 + 28*cosh(x)^2*sinh(x)^6 + 8*cosh(x)*sinh(x)^7 + sinh(x)^ 8 + 2*(35*cosh(x)^4 - 1)*sinh(x)^4 - 2*cosh(x)^4 + 8*(7*cosh(x)^5 - cosh(x ))*sinh(x)^3 + 4*(7*cosh(x)^6 - 3*cosh(x)^2)*sinh(x)^2 + 8*(cosh(x)^7 - co sh(x)^3)*sinh(x) + 1)*arctan(cosh(x) + sinh(x)) + 5*(cosh(x)^8 + 56*cosh(x )^3*sinh(x)^5 + 28*cosh(x)^2*sinh(x)^6 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 2*(35*cosh(x)^4 - 1)*sinh(x)^4 - 2*cosh(x)^4 + 8*(7*cosh(x)^5 - cosh(x))* sinh(x)^3 + 4*(7*cosh(x)^6 - 3*cosh(x)^2)*sinh(x)^2 + 8*(cosh(x)^7 - cosh( x)^3)*sinh(x) + 1)*log(cosh(x) + sinh(x) + 1) - 5*(cosh(x)^8 + 56*cosh(x)^ 3*sinh(x)^5 + 28*cosh(x)^2*sinh(x)^6 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 2 *(35*cosh(x)^4 - 1)*sinh(x)^4 - 2*cosh(x)^4 + 8*(7*cosh(x)^5 - cosh(x))*si nh(x)^3 + 4*(7*cosh(x)^6 - 3*cosh(x)^2)*sinh(x)^2 + 8*(cosh(x)^7 - cosh(x) ^3)*sinh(x) + 1)*log(cosh(x) + sinh(x) - 1) + 12*(7*cosh(x)^6 + cosh(x)^2) *sinh(x))/(cosh(x)^8 + 56*cosh(x)^3*sinh(x)^5 + 28*cosh(x)^2*sinh(x)^6 + 8 *cosh(x)*sinh(x)^7 + sinh(x)^8 + 2*(35*cosh(x)^4 - 1)*sinh(x)^4 - 2*cosh(x )^4 + 8*(7*cosh(x)^5 - cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 - 3*cosh(x)^2)* sinh(x)^2 + 8*(cosh(x)^7 - cosh(x)^3)*sinh(x) + 1)
\[ \int e^x \coth ^2(2 x) \text {csch}(2 x) \, dx=\int e^{x} \coth ^{2}{\left (2 x \right )} \operatorname {csch}{\left (2 x \right )}\, dx \] Input:
integrate(exp(x)*coth(2*x)**2*csch(2*x),x)
Output:
Integral(exp(x)*coth(2*x)**2*csch(2*x), x)
Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int e^x \coth ^2(2 x) \text {csch}(2 x) \, dx=-\frac {3 \, e^{\left (7 \, x\right )} + e^{\left (3 \, x\right )}}{4 \, {\left (e^{\left (8 \, x\right )} - 2 \, e^{\left (4 \, x\right )} + 1\right )}} + \frac {5}{8} \, \arctan \left (e^{x}\right ) - \frac {5}{16} \, \log \left (e^{x} + 1\right ) + \frac {5}{16} \, \log \left (e^{x} - 1\right ) \] Input:
integrate(exp(x)*coth(2*x)^2*csch(2*x),x, algorithm="maxima")
Output:
-1/4*(3*e^(7*x) + e^(3*x))/(e^(8*x) - 2*e^(4*x) + 1) + 5/8*arctan(e^x) - 5 /16*log(e^x + 1) + 5/16*log(e^x - 1)
Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.76 \[ \int e^x \coth ^2(2 x) \text {csch}(2 x) \, dx=-\frac {3 \, e^{\left (7 \, x\right )} + e^{\left (3 \, x\right )}}{4 \, {\left (e^{\left (4 \, x\right )} - 1\right )}^{2}} + \frac {5}{8} \, \arctan \left (e^{x}\right ) - \frac {5}{16} \, \log \left (e^{x} + 1\right ) + \frac {5}{16} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \] Input:
integrate(exp(x)*coth(2*x)^2*csch(2*x),x, algorithm="giac")
Output:
-1/4*(3*e^(7*x) + e^(3*x))/(e^(4*x) - 1)^2 + 5/8*arctan(e^x) - 5/16*log(e^ x + 1) + 5/16*log(abs(e^x - 1))
Time = 2.76 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.13 \[ \int e^x \coth ^2(2 x) \text {csch}(2 x) \, dx=\frac {5\,\ln \left (\frac {25\,{\mathrm {e}}^x}{16}-\frac {25}{16}\right )}{16}-\frac {5\,\ln \left (\frac {25\,{\mathrm {e}}^x}{16}+\frac {25}{16}\right )}{16}-\frac {5\,\mathrm {atan}\left ({\mathrm {e}}^{-x}\right )}{8}-\frac {{\mathrm {e}}^{3\,x}}{{\mathrm {e}}^{8\,x}-2\,{\mathrm {e}}^{4\,x}+1}-\frac {3\,{\mathrm {e}}^{3\,x}}{4\,\left ({\mathrm {e}}^{4\,x}-1\right )} \] Input:
int((coth(2*x)^2*exp(x))/sinh(2*x),x)
Output:
(5*log((25*exp(x))/16 - 25/16))/16 - (5*log((25*exp(x))/16 + 25/16))/16 - (5*atan(exp(-x)))/8 - exp(3*x)/(exp(8*x) - 2*exp(4*x) + 1) - (3*exp(3*x))/ (4*(exp(4*x) - 1))
Time = 0.22 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.36 \[ \int e^x \coth ^2(2 x) \text {csch}(2 x) \, dx=\frac {10 e^{8 x} \mathit {atan} \left (e^{x}\right )-20 e^{4 x} \mathit {atan} \left (e^{x}\right )+10 \mathit {atan} \left (e^{x}\right )+5 e^{8 x} \mathrm {log}\left (e^{x}-1\right )-5 e^{8 x} \mathrm {log}\left (e^{x}+1\right )-12 e^{7 x}-10 e^{4 x} \mathrm {log}\left (e^{x}-1\right )+10 e^{4 x} \mathrm {log}\left (e^{x}+1\right )-4 e^{3 x}+5 \,\mathrm {log}\left (e^{x}-1\right )-5 \,\mathrm {log}\left (e^{x}+1\right )}{16 e^{8 x}-32 e^{4 x}+16} \] Input:
int(exp(x)*coth(2*x)^2*csch(2*x),x)
Output:
(10*e**(8*x)*atan(e**x) - 20*e**(4*x)*atan(e**x) + 10*atan(e**x) + 5*e**(8 *x)*log(e**x - 1) - 5*e**(8*x)*log(e**x + 1) - 12*e**(7*x) - 10*e**(4*x)*l og(e**x - 1) + 10*e**(4*x)*log(e**x + 1) - 4*e**(3*x) + 5*log(e**x - 1) - 5*log(e**x + 1))/(16*(e**(8*x) - 2*e**(4*x) + 1))