3.10.43 \(\int e^x \coth ^2(2 x) \text {csch}(2 x) \, dx\) [943]

Optimal. Leaf size=55 \[ -\frac {e^{3 x}}{\left (1-e^{4 x}\right )^2}+\frac {3 e^{3 x}}{4 \left (1-e^{4 x}\right )}+\frac {5 \text {ArcTan}\left (e^x\right )}{8}-\frac {5}{8} \tanh ^{-1}\left (e^x\right ) \]

[Out]

-exp(3*x)/(1-exp(4*x))^2+3/4*exp(3*x)/(1-exp(4*x))+5/8*arctan(exp(x))-5/8*arctanh(exp(x))

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Rubi [A]
time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2320, 12, 474, 468, 304, 209, 212} \begin {gather*} \frac {5 \text {ArcTan}\left (e^x\right )}{8}+\frac {3 e^{3 x}}{4 \left (1-e^{4 x}\right )}-\frac {e^{3 x}}{\left (1-e^{4 x}\right )^2}-\frac {5}{8} \tanh ^{-1}\left (e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x*Coth[2*x]^2*Csch[2*x],x]

[Out]

-(E^(3*x)/(1 - E^(4*x))^2) + (3*E^(3*x))/(4*(1 - E^(4*x))) + (5*ArcTan[E^x])/8 - (5*ArcTanh[E^x])/8

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 468

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] - Dist[(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] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(p + 1)]))

Rule 474

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] + Dist[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]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {align*} \int e^x \coth ^2(2 x) \text {csch}(2 x) \, dx &=\text {Subst}\left (\int \frac {2 x^2 \left (1+x^4\right )^2}{\left (-1+x^4\right )^3} \, dx,x,e^x\right )\\ &=2 \text {Subst}\left (\int \frac {x^2 \left (1+x^4\right )^2}{\left (-1+x^4\right )^3} \, dx,x,e^x\right )\\ &=-\frac {e^{3 x}}{\left (1-e^{4 x}\right )^2}+\frac {1}{4} \text {Subst}\left (\int \frac {x^2 \left (4+8 x^4\right )}{\left (-1+x^4\right )^2} \, dx,x,e^x\right )\\ &=-\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}{4} \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,e^x\right )\\ &=-\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}{8} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,e^x\right )+\frac {5}{8} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^x\right )\\ &=-\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}{8} \tan ^{-1}\left (e^x\right )-\frac {5}{8} \tanh ^{-1}\left (e^x\right )\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 2.33, size = 161, normalized size = 2.93 \begin {gather*} \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 ) \, _2F_1\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x*Coth[2*x]^2*Csch[2*x],x]

[Out]

(177023 + 244931*E^(4*x) + 43161*E^(8*x) - 26091*E^(12*x) - 7*(25289 + 24152*E^(4*x) - 10058*E^(8*x) - 9048*E^
(12*x) + 513*E^(16*x))*Hypergeometric2F1[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*Hyp
ergeometricPFQ[{7/4, 2, 2, 2, 2}, {1, 1, 1, 19/4}, E^(4*x)])/1155

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Maple [C] Result contains complex when optimal does not.
time = 1.51, size = 56, normalized size = 1.02

method	result	size
risch	\(-\frac {{\mathrm e}^{3 x} \left (3 \,{\mathrm e}^{4 x}+1\right )}{4 \left ({\mathrm e}^{4 x}-1\right )^{2}}-\frac {5 \ln \left ({\mathrm e}^{x}+1\right )}{16}+\frac {5 i \ln \left ({\mathrm e}^{x}+i\right )}{16}-\frac {5 i \ln \left ({\mathrm e}^{x}-i\right )}{16}+\frac {5 \ln \left ({\mathrm e}^{x}-1\right )}{16}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*coth(2*x)^2*csch(2*x),x,method=_RETURNVERBOSE)

[Out]

-1/4*exp(3*x)*(3*exp(4*x)+1)/(exp(4*x)-1)^2-5/16*ln(exp(x)+1)+5/16*I*ln(exp(x)+I)-5/16*I*ln(exp(x)-I)+5/16*ln(
exp(x)-1)

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Maxima [A]
time = 0.47, size = 47, normalized size = 0.85 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*coth(2*x)^2*csch(2*x),x, algorithm="maxima")

[Out]

-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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 557 vs. \(2 (39) = 78\).
time = 0.37, size = 557, normalized size = 10.13 \begin {gather*} -\frac {12 \, \cosh \left (x\right )^{7} + 420 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{4} + 252 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{5} + 84 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + 12 \, \sinh \left (x\right )^{7} + 4 \, {\left (105 \, \cosh \left (x\right )^{4} + 1\right )} \sinh \left (x\right )^{3} + 4 \, \cosh \left (x\right )^{3} + 12 \, {\left (21 \, \cosh \left (x\right )^{5} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 10 \, {\left (\cosh \left (x\right )^{8} + 56 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{5} + 28 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{6} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 1\right )} \sinh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 3 \, \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} - \cosh \left (x\right )^{3}\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 5 \, {\left (\cosh \left (x\right )^{8} + 56 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{5} + 28 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{6} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 1\right )} \sinh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 3 \, \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} - \cosh \left (x\right )^{3}\right )} \sinh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - 5 \, {\left (\cosh \left (x\right )^{8} + 56 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{5} + 28 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{6} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 1\right )} \sinh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 3 \, \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} - \cosh \left (x\right )^{3}\right )} \sinh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 12 \, {\left (7 \, \cosh \left (x\right )^{6} + \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )}{16 \, {\left (\cosh \left (x\right )^{8} + 56 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{5} + 28 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{6} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 1\right )} \sinh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 3 \, \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} - \cosh \left (x\right )^{3}\right )} \sinh \left (x\right ) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*coth(2*x)^2*csch(2*x),x, algorithm="fricas")

[Out]

-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*c
osh(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)*arctan(cosh(x) + sinh(x)) + 5*(cosh(x)^8 + 56*cosh(x)^3*sinh(x)^5 + 28*cosh(x)^2*sin
h(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 - cos
h(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))*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) + 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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{x} \coth ^{2}{\left (2 x \right )} \operatorname {csch}{\left (2 x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*coth(2*x)**2*csch(2*x),x)

[Out]

Integral(exp(x)*coth(2*x)**2*csch(2*x), x)

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Giac [A]
time = 0.40, size = 42, normalized size = 0.76 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*coth(2*x)^2*csch(2*x),x, algorithm="giac")

[Out]

-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))

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Mupad [B]
time = 1.87, size = 62, normalized size = 1.13 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((coth(2*x)^2*exp(x))/sinh(2*x),x)

[Out]

(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))

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