Integrand size = 14, antiderivative size = 34 \[ \int e^x \sec ^3\left (1-e^x\right ) \, dx=-\frac {1}{2} \text {arctanh}\left (\sin \left (1-e^x\right )\right )-\frac {1}{2} \sec \left (1-e^x\right ) \tan \left (1-e^x\right ) \] Output:
1/2*arctanh(sin(-1+exp(x)))+1/2*sec(-1+exp(x))*tan(-1+exp(x))
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int e^x \sec ^3\left (1-e^x\right ) \, dx=-\frac {1}{2} \text {arctanh}\left (\sin \left (1-e^x\right )\right )-\frac {1}{2} \sec \left (1-e^x\right ) \tan \left (1-e^x\right ) \] Input:
Integrate[E^x*Sec[1 - E^x]^3,x]
Output:
-1/2*ArcTanh[Sin[1 - E^x]] - (Sec[1 - E^x]*Tan[1 - E^x])/2
Time = 0.44 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2720, 3042, 4255, 3042, 4257}
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.
\(\displaystyle \int e^x \sec ^3\left (1-e^x\right ) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \int \sec ^3\left (1-e^x\right )de^x\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc \left (-e^x+1+\frac {\pi }{2}\right )^3de^x\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {1}{2} \int \sec \left (1-e^x\right )de^x-\frac {1}{2} \tan \left (1-e^x\right ) \sec \left (1-e^x\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \csc \left (1-e^x+\frac {\pi }{2}\right )de^x-\frac {1}{2} \tan \left (1-e^x\right ) \sec \left (1-e^x\right )\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -\frac {1}{2} \text {arctanh}\left (\sin \left (1-e^x\right )\right )-\frac {1}{2} \tan \left (1-e^x\right ) \sec \left (1-e^x\right )\) |
Input:
Int[E^x*Sec[1 - E^x]^3,x]
Output:
-1/2*ArcTanh[Sin[1 - E^x]] - (Sec[1 - E^x]*Tan[1 - E^x])/2
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]]]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.53 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\sec \left (-1+{\mathrm e}^{x}\right ) \tan \left (-1+{\mathrm e}^{x}\right )}{2}+\frac {\ln \left (\sec \left (-1+{\mathrm e}^{x}\right )+\tan \left (-1+{\mathrm e}^{x}\right )\right )}{2}\) | \(28\) |
default | \(\frac {\sec \left (-1+{\mathrm e}^{x}\right ) \tan \left (-1+{\mathrm e}^{x}\right )}{2}+\frac {\ln \left (\sec \left (-1+{\mathrm e}^{x}\right )+\tan \left (-1+{\mathrm e}^{x}\right )\right )}{2}\) | \(28\) |
norman | \(\frac {\tan \left (-\frac {1}{2}+\frac {{\mathrm e}^{x}}{2}\right )^{3}+\tan \left (-\frac {1}{2}+\frac {{\mathrm e}^{x}}{2}\right )}{{\left (\tan \left (-\frac {1}{2}+\frac {{\mathrm e}^{x}}{2}\right )^{2}-1\right )}^{2}}-\frac {\ln \left (\tan \left (-\frac {1}{2}+\frac {{\mathrm e}^{x}}{2}\right )-1\right )}{2}+\frac {\ln \left (\tan \left (-\frac {1}{2}+\frac {{\mathrm e}^{x}}{2}\right )+1\right )}{2}\) | \(57\) |
risch | \(-\frac {i \left ({\mathrm e}^{3 i \left (-1+{\mathrm e}^{x}\right )}-{\mathrm e}^{i \left (-1+{\mathrm e}^{x}\right )}\right )}{\left ({\mathrm e}^{2 i \left (-1+{\mathrm e}^{x}\right )}+1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (-1+{\mathrm e}^{x}\right )}+i\right )}{2}-\frac {\ln \left ({\mathrm e}^{i \left (-1+{\mathrm e}^{x}\right )}-i\right )}{2}\) | \(64\) |
parallelrisch | \(\frac {\left (-\cos \left (-2+2 \,{\mathrm e}^{x}\right )-1\right ) \ln \left (\tan \left (-\frac {1}{2}+\frac {{\mathrm e}^{x}}{2}\right )-1\right )+\left (\cos \left (-2+2 \,{\mathrm e}^{x}\right )+1\right ) \ln \left (\tan \left (-\frac {1}{2}+\frac {{\mathrm e}^{x}}{2}\right )+1\right )+2 \sin \left (-1+{\mathrm e}^{x}\right )}{2+2 \cos \left (-2+2 \,{\mathrm e}^{x}\right )}\) | \(65\) |
Input:
int(exp(x)*sec(-1+exp(x))^3,x,method=_RETURNVERBOSE)
Output:
1/2*sec(-1+exp(x))*tan(-1+exp(x))+1/2*ln(sec(-1+exp(x))+tan(-1+exp(x)))
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (21) = 42\).
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.53 \[ \int e^x \sec ^3\left (1-e^x\right ) \, dx=\frac {\cos \left (e^{x} - 1\right )^{2} \log \left (\sin \left (e^{x} - 1\right ) + 1\right ) - \cos \left (e^{x} - 1\right )^{2} \log \left (-\sin \left (e^{x} - 1\right ) + 1\right ) + 2 \, \sin \left (e^{x} - 1\right )}{4 \, \cos \left (e^{x} - 1\right )^{2}} \] Input:
integrate(exp(x)*sec(-1+exp(x))^3,x, algorithm="fricas")
Output:
1/4*(cos(e^x - 1)^2*log(sin(e^x - 1) + 1) - cos(e^x - 1)^2*log(-sin(e^x - 1) + 1) + 2*sin(e^x - 1))/cos(e^x - 1)^2
\[ \int e^x \sec ^3\left (1-e^x\right ) \, dx=\int e^{x} \sec ^{3}{\left (e^{x} - 1 \right )}\, dx \] Input:
integrate(exp(x)*sec(-1+exp(x))**3,x)
Output:
Integral(exp(x)*sec(exp(x) - 1)**3, x)
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15 \[ \int e^x \sec ^3\left (1-e^x\right ) \, dx=-\frac {\sin \left (e^{x} - 1\right )}{2 \, {\left (\sin \left (e^{x} - 1\right )^{2} - 1\right )}} + \frac {1}{4} \, \log \left (\sin \left (e^{x} - 1\right ) + 1\right ) - \frac {1}{4} \, \log \left (\sin \left (e^{x} - 1\right ) - 1\right ) \] Input:
integrate(exp(x)*sec(-1+exp(x))^3,x, algorithm="maxima")
Output:
-1/2*sin(e^x - 1)/(sin(e^x - 1)^2 - 1) + 1/4*log(sin(e^x - 1) + 1) - 1/4*l og(sin(e^x - 1) - 1)
Time = 0.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.21 \[ \int e^x \sec ^3\left (1-e^x\right ) \, dx=-\frac {\sin \left (e^{x} - 1\right )}{2 \, {\left (\sin \left (e^{x} - 1\right )^{2} - 1\right )}} + \frac {1}{4} \, \log \left (\sin \left (e^{x} - 1\right ) + 1\right ) - \frac {1}{4} \, \log \left (-\sin \left (e^{x} - 1\right ) + 1\right ) \] Input:
integrate(exp(x)*sec(-1+exp(x))^3,x, algorithm="giac")
Output:
-1/2*sin(e^x - 1)/(sin(e^x - 1)^2 - 1) + 1/4*log(sin(e^x - 1) + 1) - 1/4*l og(-sin(e^x - 1) + 1)
Time = 1.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.29 \[ \int e^x \sec ^3\left (1-e^x\right ) \, dx=-\mathrm {atan}\left ({\mathrm {e}}^{-\mathrm {i}}\,{\mathrm {e}}^{{\mathrm {e}}^x\,1{}\mathrm {i}}\right )\,1{}\mathrm {i}-\frac {{\mathrm {e}}^{-\mathrm {i}}\,{\mathrm {e}}^{{\mathrm {e}}^x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{{\mathrm {e}}^{-2{}\mathrm {i}}\,{\mathrm {e}}^{{\mathrm {e}}^x\,2{}\mathrm {i}}+1}+\frac {{\mathrm {e}}^{-\mathrm {i}}\,{\mathrm {e}}^{{\mathrm {e}}^x\,1{}\mathrm {i}}\,2{}\mathrm {i}}{2\,{\mathrm {e}}^{-2{}\mathrm {i}}\,{\mathrm {e}}^{{\mathrm {e}}^x\,2{}\mathrm {i}}+{\mathrm {e}}^{-4{}\mathrm {i}}\,{\mathrm {e}}^{{\mathrm {e}}^x\,4{}\mathrm {i}}+1} \] Input:
int(exp(x)/cos(exp(x) - 1)^3,x)
Output:
(exp(-1i)*exp(exp(x)*1i)*2i)/(2*exp(-2i)*exp(exp(x)*2i) + exp(-4i)*exp(exp (x)*4i) + 1) - (exp(-1i)*exp(exp(x)*1i)*1i)/(exp(-2i)*exp(exp(x)*2i) + 1) - atan(exp(-1i)*exp(exp(x)*1i))*1i
Time = 0.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.62 \[ \int e^x \sec ^3\left (1-e^x\right ) \, dx=\frac {-\mathrm {log}\left (\tan \left (\frac {e^{x}}{2}-\frac {1}{2}\right )-1\right ) \sin \left (e^{x}-1\right )^{2}+\mathrm {log}\left (\tan \left (\frac {e^{x}}{2}-\frac {1}{2}\right )-1\right )+\mathrm {log}\left (\tan \left (\frac {e^{x}}{2}-\frac {1}{2}\right )+1\right ) \sin \left (e^{x}-1\right )^{2}-\mathrm {log}\left (\tan \left (\frac {e^{x}}{2}-\frac {1}{2}\right )+1\right )-\sin \left (e^{x}-1\right )}{2 \sin \left (e^{x}-1\right )^{2}-2} \] Input:
int(exp(x)*sec(-1+exp(x))^3,x)
Output:
( - log(tan((e**x - 1)/2) - 1)*sin(e**x - 1)**2 + log(tan((e**x - 1)/2) - 1) + log(tan((e**x - 1)/2) + 1)*sin(e**x - 1)**2 - log(tan((e**x - 1)/2) + 1) - sin(e**x - 1))/(2*(sin(e**x - 1)**2 - 1))