Integrand size = 10, antiderivative size = 109 \[ \int e^x \coth ^2(4 x) \, dx=e^x+\frac {e^x}{2 \left (1-e^{8 x}\right )}-\frac {\arctan \left (e^x\right )}{8}+\frac {\arctan \left (1-\sqrt {2} e^x\right )}{8 \sqrt {2}}-\frac {\arctan \left (1+\sqrt {2} e^x\right )}{8 \sqrt {2}}-\frac {\text {arctanh}\left (e^x\right )}{8}-\frac {\text {arctanh}\left (\frac {\sqrt {2} e^x}{1+e^{2 x}}\right )}{8 \sqrt {2}} \] Output:
exp(x)+exp(x)/(2-2*exp(8*x))-1/8*arctan(exp(x))-1/16*arctan(-1+2^(1/2)*exp (x))*2^(1/2)-1/16*arctan(1+2^(1/2)*exp(x))*2^(1/2)-1/8*arctanh(exp(x))-1/1 6*arctanh(2^(1/2)*exp(x)/(1+exp(2*x)))*2^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.23 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.04 \[ \int e^x \coth ^2(4 x) \, dx=\frac {e^{-15 x} \left (-44217-80225 e^{8 x}-15127 e^{16 x}+9361 e^{24 x}+9 \left (4913+8368 e^{8 x}+1486 e^{16 x}-1456 e^{24 x}+e^{32 x}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{8},1,\frac {9}{8},e^{8 x}\right )\right )}{9216}+\frac {64 e^{9 x} \left (1+e^{8 x}\right )^2 \, _4F_3\left (\frac {9}{8},2,2,2;1,1,\frac {33}{8};e^{8 x}\right )}{3825} \] Input:
Integrate[E^x*Coth[4*x]^2,x]
Output:
(-44217 - 80225*E^(8*x) - 15127*E^(16*x) + 9361*E^(24*x) + 9*(4913 + 8368* E^(8*x) + 1486*E^(16*x) - 1456*E^(24*x) + E^(32*x))*Hypergeometric2F1[1/8, 1, 9/8, E^(8*x)])/(9216*E^(15*x)) + (64*E^(9*x)*(1 + E^(8*x))^2*Hypergeom etricPFQ[{9/8, 2, 2, 2}, {1, 1, 33/8}, E^(8*x)])/3825
Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.23, 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.
\(\displaystyle \int e^x \coth ^2(4 x) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \int \frac {\left (e^{8 x}+1\right )^2}{\left (1-e^{8 x}\right )^2}de^x\) |
\(\Big \downarrow \) 915 |
\(\displaystyle \int \left (\frac {4 e^{8 x}}{\left (1-e^{8 x}\right )^2}+1\right )de^x\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{8} \arctan \left (e^x\right )+\frac {\arctan \left (1-\sqrt {2} e^x\right )}{8 \sqrt {2}}-\frac {\arctan \left (\sqrt {2} e^x+1\right )}{8 \sqrt {2}}-\frac {\text {arctanh}\left (e^x\right )}{8}+e^x+\frac {e^x}{2 \left (1-e^{8 x}\right )}+\frac {\log \left (-\sqrt {2} e^x+e^{2 x}+1\right )}{16 \sqrt {2}}-\frac {\log \left (\sqrt {2} e^x+e^{2 x}+1\right )}{16 \sqrt {2}}\) |
Input:
Int[E^x*Coth[4*x]^2,x]
Output:
E^x + E^x/(2*(1 - E^(8*x))) - ArcTan[E^x]/8 + ArcTan[1 - Sqrt[2]*E^x]/(8*S qrt[2]) - ArcTan[1 + Sqrt[2]*E^x]/(8*Sqrt[2]) - ArcTanh[E^x]/8 + Log[1 - S qrt[2]*E^x + E^(2*x)]/(16*Sqrt[2]) - Log[1 + Sqrt[2]*E^x + E^(2*x)]/(16*Sq rt[2])
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 higher order function than in optimal. Order 9 vs. order 3.
Time = 0.62 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62
method | result | size |
risch | \({\mathrm e}^{x}-\frac {{\mathrm e}^{x}}{2 \left ({\mathrm e}^{8 x}-1\right )}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{16}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{16}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (65536 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{x}-16 \textit {\_R} \right )\right )+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{16}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{16}\) | \(68\) |
Input:
int(exp(x)*coth(4*x)^2,x,method=_RETURNVERBOSE)
Output:
exp(x)-1/2*exp(x)/(exp(8*x)-1)+1/16*I*ln(exp(x)-I)-1/16*I*ln(exp(x)+I)+sum (_R*ln(exp(x)-16*_R),_R=RootOf(65536*_Z^4+1))+1/16*ln(exp(x)-1)-1/16*ln(ex p(x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 943 vs. \(2 (73) = 146\).
Time = 0.12 (sec) , antiderivative size = 943, normalized size of antiderivative = 8.65 \[ \int e^x \coth ^2(4 x) \, dx=\text {Too large to display} \] Input:
integrate(exp(x)*coth(4*x)^2,x, algorithm="fricas")
Output:
1/32*(32*cosh(x)^9 + 1152*cosh(x)^7*sinh(x)^2 + 2688*cosh(x)^6*sinh(x)^3 + 4032*cosh(x)^5*sinh(x)^4 + 4032*cosh(x)^4*sinh(x)^5 + 2688*cosh(x)^3*sinh (x)^6 + 1152*cosh(x)^2*sinh(x)^7 + 288*cosh(x)*sinh(x)^8 + 32*sinh(x)^9 - 2*(sqrt(2)*cosh(x)^8 + 8*sqrt(2)*cosh(x)^7*sinh(x) + 28*sqrt(2)*cosh(x)^6* sinh(x)^2 + 56*sqrt(2)*cosh(x)^5*sinh(x)^3 + 70*sqrt(2)*cosh(x)^4*sinh(x)^ 4 + 56*sqrt(2)*cosh(x)^3*sinh(x)^5 + 28*sqrt(2)*cosh(x)^2*sinh(x)^6 + 8*sq rt(2)*cosh(x)*sinh(x)^7 + sqrt(2)*sinh(x)^8 - sqrt(2))*arctan(sqrt(2)*cosh (x) + sqrt(2)*sinh(x) + 1) - 2*(sqrt(2)*cosh(x)^8 + 8*sqrt(2)*cosh(x)^7*si nh(x) + 28*sqrt(2)*cosh(x)^6*sinh(x)^2 + 56*sqrt(2)*cosh(x)^5*sinh(x)^3 + 70*sqrt(2)*cosh(x)^4*sinh(x)^4 + 56*sqrt(2)*cosh(x)^3*sinh(x)^5 + 28*sqrt( 2)*cosh(x)^2*sinh(x)^6 + 8*sqrt(2)*cosh(x)*sinh(x)^7 + sqrt(2)*sinh(x)^8 - sqrt(2))*arctan(sqrt(2)*cosh(x) + sqrt(2)*sinh(x) - 1) - 4*(cosh(x)^8 + 8 *cosh(x)^7*sinh(x) + 28*cosh(x)^6*sinh(x)^2 + 56*cosh(x)^5*sinh(x)^3 + 70* cosh(x)^4*sinh(x)^4 + 56*cosh(x)^3*sinh(x)^5 + 28*cosh(x)^2*sinh(x)^6 + 8* cosh(x)*sinh(x)^7 + sinh(x)^8 - 1)*arctan(cosh(x) + sinh(x)) - (sqrt(2)*co sh(x)^8 + 8*sqrt(2)*cosh(x)^7*sinh(x) + 28*sqrt(2)*cosh(x)^6*sinh(x)^2 + 5 6*sqrt(2)*cosh(x)^5*sinh(x)^3 + 70*sqrt(2)*cosh(x)^4*sinh(x)^4 + 56*sqrt(2 )*cosh(x)^3*sinh(x)^5 + 28*sqrt(2)*cosh(x)^2*sinh(x)^6 + 8*sqrt(2)*cosh(x) *sinh(x)^7 + sqrt(2)*sinh(x)^8 - sqrt(2))*log((sqrt(2) + 2*cosh(x))/(cosh( x) - sinh(x))) + (sqrt(2)*cosh(x)^8 + 8*sqrt(2)*cosh(x)^7*sinh(x) + 28*...
\[ \int e^x \coth ^2(4 x) \, dx=\int e^{x} \coth ^{2}{\left (4 x \right )}\, dx \] Input:
integrate(exp(x)*coth(4*x)**2,x)
Output:
Integral(exp(x)*coth(4*x)**2, x)
Time = 0.12 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int e^x \coth ^2(4 x) \, dx=-\frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) - \frac {1}{16} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) - \frac {1}{32} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{32} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {e^{x}}{2 \, {\left (e^{\left (8 \, x\right )} - 1\right )}} - \frac {1}{8} \, \arctan \left (e^{x}\right ) + e^{x} - \frac {1}{16} \, \log \left (e^{x} + 1\right ) + \frac {1}{16} \, \log \left (e^{x} - 1\right ) \] Input:
integrate(exp(x)*coth(4*x)^2,x, algorithm="maxima")
Output:
-1/16*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^x)) - 1/16*sqrt(2)*arctan( -1/2*sqrt(2)*(sqrt(2) - 2*e^x)) - 1/32*sqrt(2)*log(sqrt(2)*e^x + e^(2*x) + 1) + 1/32*sqrt(2)*log(-sqrt(2)*e^x + e^(2*x) + 1) - 1/2*e^x/(e^(8*x) - 1) - 1/8*arctan(e^x) + e^x - 1/16*log(e^x + 1) + 1/16*log(e^x - 1)
Time = 0.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.01 \[ \int e^x \coth ^2(4 x) \, dx=-\frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) - \frac {1}{16} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) - \frac {1}{32} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{32} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {e^{x}}{2 \, {\left (e^{\left (8 \, x\right )} - 1\right )}} - \frac {1}{8} \, \arctan \left (e^{x}\right ) + e^{x} - \frac {1}{16} \, \log \left (e^{x} + 1\right ) + \frac {1}{16} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \] Input:
integrate(exp(x)*coth(4*x)^2,x, algorithm="giac")
Output:
-1/16*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^x)) - 1/16*sqrt(2)*arctan( -1/2*sqrt(2)*(sqrt(2) - 2*e^x)) - 1/32*sqrt(2)*log(sqrt(2)*e^x + e^(2*x) + 1) + 1/32*sqrt(2)*log(-sqrt(2)*e^x + e^(2*x) + 1) - 1/2*e^x/(e^(8*x) - 1) - 1/8*arctan(e^x) + e^x - 1/16*log(e^x + 1) + 1/16*log(abs(e^x - 1))
Time = 2.52 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.12 \[ \int e^x \coth ^2(4 x) \, dx=\frac {\ln \left (\frac {1}{2}-\frac {{\mathrm {e}}^x}{2}\right )}{16}-\frac {\ln \left (-\frac {{\mathrm {e}}^x}{2}-\frac {1}{2}\right )}{16}-\frac {\mathrm {atan}\left ({\mathrm {e}}^x\right )}{8}+{\mathrm {e}}^x-\frac {{\mathrm {e}}^x}{2\,\left ({\mathrm {e}}^{8\,x}-1\right )}-\frac {\sqrt {2}\,\mathrm {atan}\left (2\,\sqrt {2}\,\left (\frac {{\mathrm {e}}^x}{2}-\frac {\sqrt {2}}{4}\right )\right )}{16}-\frac {\sqrt {2}\,\mathrm {atan}\left (2\,\sqrt {2}\,\left (\frac {{\mathrm {e}}^x}{2}+\frac {\sqrt {2}}{4}\right )\right )}{16}+\frac {\sqrt {2}\,\ln \left ({\left (\frac {{\mathrm {e}}^x}{2}-\frac {\sqrt {2}}{4}\right )}^2+\frac {1}{8}\right )}{32}-\frac {\sqrt {2}\,\ln \left ({\left (\frac {{\mathrm {e}}^x}{2}+\frac {\sqrt {2}}{4}\right )}^2+\frac {1}{8}\right )}{32} \] Input:
int(coth(4*x)^2*exp(x),x)
Output:
log(1/2 - exp(x)/2)/16 - log(- exp(x)/2 - 1/2)/16 - atan(exp(x))/8 + exp(x ) - exp(x)/(2*(exp(8*x) - 1)) - (2^(1/2)*atan(2*2^(1/2)*(exp(x)/2 - 2^(1/2 )/4)))/16 - (2^(1/2)*atan(2*2^(1/2)*(exp(x)/2 + 2^(1/2)/4)))/16 + (2^(1/2) *log((exp(x)/2 - 2^(1/2)/4)^2 + 1/8))/32 - (2^(1/2)*log((exp(x)/2 + 2^(1/2 )/4)^2 + 1/8))/32
Time = 0.26 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.31 \[ \int e^x \coth ^2(4 x) \, dx=\frac {-4 e^{8 x} \mathit {atan} \left (e^{x}\right )+4 \mathit {atan} \left (e^{x}\right )-2 e^{8 x} \sqrt {2}\, \mathit {atan} \left (\frac {2 e^{x}-\sqrt {2}}{\sqrt {2}}\right )+2 \sqrt {2}\, \mathit {atan} \left (\frac {2 e^{x}-\sqrt {2}}{\sqrt {2}}\right )-2 e^{8 x} \sqrt {2}\, \mathit {atan} \left (\frac {2 e^{x}+\sqrt {2}}{\sqrt {2}}\right )+2 \sqrt {2}\, \mathit {atan} \left (\frac {2 e^{x}+\sqrt {2}}{\sqrt {2}}\right )+32 e^{9 x}+e^{8 x} \sqrt {2}\, \mathrm {log}\left (e^{2 x}-e^{x} \sqrt {2}+1\right )-e^{8 x} \sqrt {2}\, \mathrm {log}\left (e^{2 x}+e^{x} \sqrt {2}+1\right )+2 e^{8 x} \mathrm {log}\left (e^{x}-1\right )-2 e^{8 x} \mathrm {log}\left (e^{x}+1\right )-48 e^{x}-\sqrt {2}\, \mathrm {log}\left (e^{2 x}-e^{x} \sqrt {2}+1\right )+\sqrt {2}\, \mathrm {log}\left (e^{2 x}+e^{x} \sqrt {2}+1\right )-2 \,\mathrm {log}\left (e^{x}-1\right )+2 \,\mathrm {log}\left (e^{x}+1\right )}{32 e^{8 x}-32} \] Input:
int(exp(x)*coth(4*x)^2,x)
Output:
( - 4*e**(8*x)*atan(e**x) + 4*atan(e**x) - 2*e**(8*x)*sqrt(2)*atan((2*e**x - sqrt(2))/sqrt(2)) + 2*sqrt(2)*atan((2*e**x - sqrt(2))/sqrt(2)) - 2*e**( 8*x)*sqrt(2)*atan((2*e**x + sqrt(2))/sqrt(2)) + 2*sqrt(2)*atan((2*e**x + s qrt(2))/sqrt(2)) + 32*e**(9*x) + e**(8*x)*sqrt(2)*log(e**(2*x) - e**x*sqrt (2) + 1) - e**(8*x)*sqrt(2)*log(e**(2*x) + e**x*sqrt(2) + 1) + 2*e**(8*x)* log(e**x - 1) - 2*e**(8*x)*log(e**x + 1) - 48*e**x - sqrt(2)*log(e**(2*x) - e**x*sqrt(2) + 1) + sqrt(2)*log(e**(2*x) + e**x*sqrt(2) + 1) - 2*log(e** x - 1) + 2*log(e**x + 1))/(32*(e**(8*x) - 1))