Integrand size = 15, antiderivative size = 203 \[ \int \frac {x^3}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {2}{5 c^4 \sqrt {\text {sech}(2 \log (c x))}}-\frac {2}{5 c^4 \left (c^2+\frac {1}{x^2}\right ) x^2 \sqrt {\text {sech}(2 \log (c x))}}+\frac {x^4}{5 \sqrt {\text {sech}(2 \log (c x))}}+\frac {2 \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) E\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{5 c^3 \left (c^4+\frac {1}{x^4}\right ) x \sqrt {\text {sech}(2 \log (c x))}}-\frac {\sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \operatorname {EllipticF}\left (2 \cot ^{-1}(c x),\frac {1}{2}\right )}{5 c^3 \left (c^4+\frac {1}{x^4}\right ) x \sqrt {\text {sech}(2 \log (c x))}} \] Output:
2/5/c^4/sech(2*ln(c*x))^(1/2)-2/5/c^4/(c^2+1/x^2)/x^2/sech(2*ln(c*x))^(1/2 )+1/5*x^4/sech(2*ln(c*x))^(1/2)+2/5*((c^4+1/x^4)/(c^2+1/x^2)^2)^(1/2)*(c^2 +1/x^2)*EllipticE(sin(2*arccot(c*x)),1/2*2^(1/2))/c^3/(c^4+1/x^4)/x/sech(2 *ln(c*x))^(1/2)-1/5*((c^4+1/x^4)/(c^2+1/x^2)^2)^(1/2)*(c^2+1/x^2)*InverseJ acobiAM(2*arccot(c*x),1/2*2^(1/2))/c^3/(c^4+1/x^4)/x/sech(2*ln(c*x))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.32 \[ \int \frac {x^3}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {\left (\frac {c^2 x^2}{1+c^4 x^4}\right )^{3/2} \left (1+c^4 x^4\right )^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-c^4 x^4\right )}{3 \sqrt {2} c^4} \] Input:
Integrate[x^3/Sqrt[Sech[2*Log[c*x]]],x]
Output:
(((c^2*x^2)/(1 + c^4*x^4))^(3/2)*(1 + c^4*x^4)^(3/2)*Hypergeometric2F1[-1/ 2, 3/4, 7/4, -(c^4*x^4)])/(3*Sqrt[2]*c^4)
Time = 0.49 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.19, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {6085, 6083, 858, 809, 847, 834, 761, 1510}
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.
\(\displaystyle \int \frac {x^3}{\sqrt {\text {sech}(2 \log (c x))}} \, dx\) |
\(\Big \downarrow \) 6085 |
\(\displaystyle \frac {\int \frac {c^3 x^3}{\sqrt {\text {sech}(2 \log (c x))}}d(c x)}{c^4}\) |
\(\Big \downarrow \) 6083 |
\(\displaystyle \frac {\int c^4 \sqrt {1+\frac {1}{c^4 x^4}} x^4d(c x)}{c^5 x \sqrt {\frac {1}{c^4 x^4}+1} \sqrt {\text {sech}(2 \log (c x))}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\frac {\int \frac {\sqrt {c^4 x^4+1}}{c^6 x^6}d\frac {1}{c x}}{c^5 x \sqrt {\frac {1}{c^4 x^4}+1} \sqrt {\text {sech}(2 \log (c x))}}\) |
\(\Big \downarrow \) 809 |
\(\displaystyle -\frac {\frac {2}{5} \int \frac {1}{c^2 x^2 \sqrt {c^4 x^4+1}}d\frac {1}{c x}-\frac {\sqrt {c^4 x^4+1}}{5 c^5 x^5}}{c^5 x \sqrt {\frac {1}{c^4 x^4}+1} \sqrt {\text {sech}(2 \log (c x))}}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle -\frac {\frac {2}{5} \left (\int \frac {c^2 x^2}{\sqrt {c^4 x^4+1}}d\frac {1}{c x}-\frac {\sqrt {c^4 x^4+1}}{c x}\right )-\frac {\sqrt {c^4 x^4+1}}{5 c^5 x^5}}{c^5 x \sqrt {\frac {1}{c^4 x^4}+1} \sqrt {\text {sech}(2 \log (c x))}}\) |
\(\Big \downarrow \) 834 |
\(\displaystyle -\frac {\frac {2}{5} \left (\int \frac {1}{\sqrt {c^4 x^4+1}}d\frac {1}{c x}-\int \frac {1-c^2 x^2}{\sqrt {c^4 x^4+1}}d\frac {1}{c x}-\frac {\sqrt {c^4 x^4+1}}{c x}\right )-\frac {\sqrt {c^4 x^4+1}}{5 c^5 x^5}}{c^5 x \sqrt {\frac {1}{c^4 x^4}+1} \sqrt {\text {sech}(2 \log (c x))}}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle -\frac {\frac {2}{5} \left (-\int \frac {1-c^2 x^2}{\sqrt {c^4 x^4+1}}d\frac {1}{c x}+\frac {\left (c^2 x^2+1\right ) \sqrt {\frac {c^4 x^4+1}{\left (c^2 x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {1}{c x}\right ),\frac {1}{2}\right )}{2 \sqrt {c^4 x^4+1}}-\frac {\sqrt {c^4 x^4+1}}{c x}\right )-\frac {\sqrt {c^4 x^4+1}}{5 c^5 x^5}}{c^5 x \sqrt {\frac {1}{c^4 x^4}+1} \sqrt {\text {sech}(2 \log (c x))}}\) |
\(\Big \downarrow \) 1510 |
\(\displaystyle -\frac {\frac {2}{5} \left (\frac {\left (c^2 x^2+1\right ) \sqrt {\frac {c^4 x^4+1}{\left (c^2 x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {1}{c x}\right ),\frac {1}{2}\right )}{2 \sqrt {c^4 x^4+1}}-\frac {\left (c^2 x^2+1\right ) \sqrt {\frac {c^4 x^4+1}{\left (c^2 x^2+1\right )^2}} E\left (2 \arctan \left (\frac {1}{c x}\right )|\frac {1}{2}\right )}{\sqrt {c^4 x^4+1}}-\frac {\sqrt {c^4 x^4+1}}{c x}+\frac {\sqrt {c^4 x^4+1}}{c x \left (c^2 x^2+1\right )}\right )-\frac {\sqrt {c^4 x^4+1}}{5 c^5 x^5}}{c^5 x \sqrt {\frac {1}{c^4 x^4}+1} \sqrt {\text {sech}(2 \log (c x))}}\) |
Input:
Int[x^3/Sqrt[Sech[2*Log[c*x]]],x]
Output:
-((-1/5*Sqrt[1 + c^4*x^4]/(c^5*x^5) + (2*(-(Sqrt[1 + c^4*x^4]/(c*x)) + Sqr t[1 + c^4*x^4]/(c*x*(1 + c^2*x^2)) - ((1 + c^2*x^2)*Sqrt[(1 + c^4*x^4)/(1 + c^2*x^2)^2]*EllipticE[2*ArcTan[1/(c*x)], 1/2])/Sqrt[1 + c^4*x^4] + ((1 + c^2*x^2)*Sqrt[(1 + c^4*x^4)/(1 + c^2*x^2)^2]*EllipticF[2*ArcTan[1/(c*x)], 1/2])/(2*Sqrt[1 + c^4*x^4])))/5)/(c^5*Sqrt[1 + 1/(c^4*x^4)]*x*Sqrt[Sech[2 *Log[c*x]]]))
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((e_.)*(x_))^(m_.)*Sech[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Simp[Sech[d*(a + b*Log[x])]^p*((1 + 1/(E^(2*a*d)*x^(2*b*d)))^p/x^((-b)* d*p)) Int[(e*x)^m*(1/(x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p)), x], x] /; FreeQ[{a, b, d, e, m, p}, x] && !IntegerQ[p]
Int[((e_.)*(x_))^(m_.)*Sech[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p _.), x_Symbol] :> Simp[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)) Subst[Int[ x^((m + 1)/n - 1)*Sech[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.66
method | result | size |
risch | \(\frac {\sqrt {2}\, x^{4}}{10 \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}+\frac {i \sqrt {-i c^{2} x^{2}+1}\, \sqrt {i c^{2} x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {i c^{2}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {i c^{2}}, i\right )\right ) \sqrt {2}\, x}{5 \sqrt {i c^{2}}\, \left (c^{4} x^{4}+1\right ) c^{2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}\) | \(134\) |
Input:
int(x^3/sech(2*ln(x*c))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/10*2^(1/2)*x^4/(c^2*x^2/(c^4*x^4+1))^(1/2)+1/5*I/(I*c^2)^(1/2)*(1-I*c^2* x^2)^(1/2)*(1+I*c^2*x^2)^(1/2)/(c^4*x^4+1)/c^2*(EllipticF(x*(I*c^2)^(1/2), I)-EllipticE(x*(I*c^2)^(1/2),I))*2^(1/2)*x/(c^2*x^2/(c^4*x^4+1))^(1/2)
Time = 0.11 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.59 \[ \int \frac {x^3}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {4 \, \sqrt {\frac {1}{2}} \sqrt {c^{4}} c x^{2} \left (-\frac {1}{c^{4}}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {1}{c^{4}}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 4 \, \sqrt {\frac {1}{2}} \sqrt {c^{4}} c x^{2} \left (-\frac {1}{c^{4}}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {1}{c^{4}}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + \sqrt {2} {\left (c^{8} x^{8} + 3 \, c^{4} x^{4} + 2\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{10 \, c^{6} x^{2}} \] Input:
integrate(x^3/sech(2*log(c*x))^(1/2),x, algorithm="fricas")
Output:
1/10*(4*sqrt(1/2)*sqrt(c^4)*c*x^2*(-1/c^4)^(3/4)*elliptic_e(arcsin((-1/c^4 )^(1/4)/x), -1) - 4*sqrt(1/2)*sqrt(c^4)*c*x^2*(-1/c^4)^(3/4)*elliptic_f(ar csin((-1/c^4)^(1/4)/x), -1) + sqrt(2)*(c^8*x^8 + 3*c^4*x^4 + 2)*sqrt(c^2*x ^2/(c^4*x^4 + 1)))/(c^6*x^2)
\[ \int \frac {x^3}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\int \frac {x^{3}}{\sqrt {\operatorname {sech}{\left (2 \log {\left (c x \right )} \right )}}}\, dx \] Input:
integrate(x**3/sech(2*ln(c*x))**(1/2),x)
Output:
Integral(x**3/sqrt(sech(2*log(c*x))), x)
\[ \int \frac {x^3}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\int { \frac {x^{3}}{\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}} \,d x } \] Input:
integrate(x^3/sech(2*log(c*x))^(1/2),x, algorithm="maxima")
Output:
integrate(x^3/sqrt(sech(2*log(c*x))), x)
Exception generated. \[ \int \frac {x^3}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3/sech(2*log(c*x))^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly exception caught Unable to convert to real %%{poly1[1.0000000000000000000000000000000,0.000000000000 000000000
Timed out. \[ \int \frac {x^3}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\int \frac {x^3}{\sqrt {\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}}} \,d x \] Input:
int(x^3/(1/cosh(2*log(c*x)))^(1/2),x)
Output:
int(x^3/(1/cosh(2*log(c*x)))^(1/2), x)
\[ \int \frac {x^3}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\int \frac {\sqrt {\mathrm {sech}\left (2 \,\mathrm {log}\left (c x \right )\right )}\, x^{3}}{\mathrm {sech}\left (2 \,\mathrm {log}\left (c x \right )\right )}d x \] Input:
int(x^3/sech(2*log(c*x))^(1/2),x)
Output:
int((sqrt(sech(2*log(c*x)))*x**3)/sech(2*log(c*x)),x)