Integrand size = 18, antiderivative size = 383 \[ \int \frac {x^3}{\sqrt {a+a \cosh (c+d x)}} \, dx=\frac {4 x^3 \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cosh (c+d x)}}-\frac {12 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {12 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {48 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^3 \sqrt {a+a \cosh (c+d x)}}-\frac {48 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^3 \sqrt {a+a \cosh (c+d x)}}-\frac {96 i \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^4 \sqrt {a+a \cosh (c+d x)}}+\frac {96 i \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^4 \sqrt {a+a \cosh (c+d x)}} \]
4*x^3*arctan(exp(1/2*d*x+1/2*c))*cosh(1/2*d*x+1/2*c)/d/(a+a*cosh(d*x+c))^( 1/2)-12*I*x^2*cosh(1/2*d*x+1/2*c)*polylog(2,-I*exp(1/2*d*x+1/2*c))/d^2/(a+ a*cosh(d*x+c))^(1/2)+12*I*x^2*cosh(1/2*d*x+1/2*c)*polylog(2,I*exp(1/2*d*x+ 1/2*c))/d^2/(a+a*cosh(d*x+c))^(1/2)+48*I*x*cosh(1/2*d*x+1/2*c)*polylog(3,- I*exp(1/2*d*x+1/2*c))/d^3/(a+a*cosh(d*x+c))^(1/2)-48*I*x*cosh(1/2*d*x+1/2* c)*polylog(3,I*exp(1/2*d*x+1/2*c))/d^3/(a+a*cosh(d*x+c))^(1/2)-96*I*cosh(1 /2*d*x+1/2*c)*polylog(4,-I*exp(1/2*d*x+1/2*c))/d^4/(a+a*cosh(d*x+c))^(1/2) +96*I*cosh(1/2*d*x+1/2*c)*polylog(4,I*exp(1/2*d*x+1/2*c))/d^4/(a+a*cosh(d* x+c))^(1/2)
Time = 0.25 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.56 \[ \int \frac {x^3}{\sqrt {a+a \cosh (c+d x)}} \, dx=\frac {2 i \cosh \left (\frac {1}{2} (c+d x)\right ) \left (d^3 x^3 \log \left (1-i e^{\frac {1}{2} (c+d x)}\right )-d^3 x^3 \log \left (1+i e^{\frac {1}{2} (c+d x)}\right )-6 d^2 x^2 \operatorname {PolyLog}\left (2,-i e^{\frac {1}{2} (c+d x)}\right )+6 d^2 x^2 \operatorname {PolyLog}\left (2,i e^{\frac {1}{2} (c+d x)}\right )+24 d x \operatorname {PolyLog}\left (3,-i e^{\frac {1}{2} (c+d x)}\right )-24 d x \operatorname {PolyLog}\left (3,i e^{\frac {1}{2} (c+d x)}\right )-48 \operatorname {PolyLog}\left (4,-i e^{\frac {1}{2} (c+d x)}\right )+48 \operatorname {PolyLog}\left (4,i e^{\frac {1}{2} (c+d x)}\right )\right )}{d^4 \sqrt {a (1+\cosh (c+d x))}} \]
((2*I)*Cosh[(c + d*x)/2]*(d^3*x^3*Log[1 - I*E^((c + d*x)/2)] - d^3*x^3*Log [1 + I*E^((c + d*x)/2)] - 6*d^2*x^2*PolyLog[2, (-I)*E^((c + d*x)/2)] + 6*d ^2*x^2*PolyLog[2, I*E^((c + d*x)/2)] + 24*d*x*PolyLog[3, (-I)*E^((c + d*x) /2)] - 24*d*x*PolyLog[3, I*E^((c + d*x)/2)] - 48*PolyLog[4, (-I)*E^((c + d *x)/2)] + 48*PolyLog[4, I*E^((c + d*x)/2)]))/(d^4*Sqrt[a*(1 + Cosh[c + d*x ])])
Time = 0.87 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.62, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3800, 3042, 4668, 3011, 7163, 2720, 7143}
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.
\(\displaystyle \int \frac {x^3}{\sqrt {a \cosh (c+d x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {x^3}{\sqrt {a+a \sin \left (i c+i d x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \int x^3 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right )dx}{\sqrt {a \cosh (c+d x)+a}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \int x^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{2}\right )dx}{\sqrt {a \cosh (c+d x)+a}}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-\frac {6 i \int x^2 \log \left (1-i e^{\frac {c}{2}+\frac {d x}{2}}\right )dx}{d}+\frac {6 i \int x^2 \log \left (1+i e^{\frac {c}{2}+\frac {d x}{2}}\right )dx}{d}+\frac {4 x^3 \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}\right )}{\sqrt {a \cosh (c+d x)+a}}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {6 i \left (\frac {4 \int x \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )dx}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}\right )}{d}-\frac {6 i \left (\frac {4 \int x \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right )dx}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}\right )}{d}+\frac {4 x^3 \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}\right )}{\sqrt {a \cosh (c+d x)+a}}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}-\frac {2 \int \operatorname {PolyLog}\left (3,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )dx}{d}\right )}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}\right )}{d}-\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}-\frac {2 \int \operatorname {PolyLog}\left (3,i e^{\frac {c}{2}+\frac {d x}{2}}\right )dx}{d}\right )}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}\right )}{d}+\frac {4 x^3 \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}\right )}{\sqrt {a \cosh (c+d x)+a}}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}-\frac {4 \int e^{-\frac {c}{2}-\frac {d x}{2}} \operatorname {PolyLog}\left (3,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )de^{\frac {c}{2}+\frac {d x}{2}}}{d^2}\right )}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}\right )}{d}-\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}-\frac {4 \int e^{-\frac {c}{2}-\frac {d x}{2}} \operatorname {PolyLog}\left (3,i e^{\frac {c}{2}+\frac {d x}{2}}\right )de^{\frac {c}{2}+\frac {d x}{2}}}{d^2}\right )}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}\right )}{d}+\frac {4 x^3 \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}\right )}{\sqrt {a \cosh (c+d x)+a}}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {4 x^3 \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}+\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}-\frac {4 \operatorname {PolyLog}\left (4,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2}\right )}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}\right )}{d}-\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}-\frac {4 \operatorname {PolyLog}\left (4,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2}\right )}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d}\right )}{d}\right )}{\sqrt {a \cosh (c+d x)+a}}\) |
(Cosh[c/2 + (d*x)/2]*((4*x^3*ArcTan[E^(c/2 + (d*x)/2)])/d + ((6*I)*((-2*x^ 2*PolyLog[2, (-I)*E^(c/2 + (d*x)/2)])/d + (4*((2*x*PolyLog[3, (-I)*E^(c/2 + (d*x)/2)])/d - (4*PolyLog[4, (-I)*E^(c/2 + (d*x)/2)])/d^2))/d))/d - ((6* I)*((-2*x^2*PolyLog[2, I*E^(c/2 + (d*x)/2)])/d + (4*((2*x*PolyLog[3, I*E^( c/2 + (d*x)/2)])/d - (4*PolyLog[4, I*E^(c/2 + (d*x)/2)])/d^2))/d))/d))/Sqr t[a + a*Cosh[c + d*x]]
3.2.39.3.1 Defintions of rubi rules used
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[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(2*a)^IntPart[n]*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e /2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n])) Int[(c + d*x)^m*Sin[e/2 + a *(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
\[\int \frac {x^{3}}{\sqrt {a +a \cosh \left (d x +c \right )}}d x\]
\[ \int \frac {x^3}{\sqrt {a+a \cosh (c+d x)}} \, dx=\int { \frac {x^{3}}{\sqrt {a \cosh \left (d x + c\right ) + a}} \,d x } \]
\[ \int \frac {x^3}{\sqrt {a+a \cosh (c+d x)}} \, dx=\int \frac {x^{3}}{\sqrt {a \left (\cosh {\left (c + d x \right )} + 1\right )}}\, dx \]
\[ \int \frac {x^3}{\sqrt {a+a \cosh (c+d x)}} \, dx=\int { \frac {x^{3}}{\sqrt {a \cosh \left (d x + c\right ) + a}} \,d x } \]
2*sqrt(2)*d^3*integrate(x^3*e^(1/2*d*x + 1/2*c)/(sqrt(a)*d^3*e^(2*d*x + 2* c) + 2*sqrt(a)*d^3*e^(d*x + c) + sqrt(a)*d^3), x) + 12*sqrt(2)*d^2*integra te(x^2*e^(1/2*d*x + 1/2*c)/(sqrt(a)*d^3*e^(2*d*x + 2*c) + 2*sqrt(a)*d^3*e^ (d*x + c) + sqrt(a)*d^3), x) + 48*sqrt(2)*d*integrate(x*e^(1/2*d*x + 1/2*c )/(sqrt(a)*d^3*e^(2*d*x + 2*c) + 2*sqrt(a)*d^3*e^(d*x + c) + sqrt(a)*d^3), x) + 96*sqrt(2)*(e^(1/2*d*x + 1/2*c)/((sqrt(a)*d^3*e^(d*x + c) + sqrt(a)* d^3)*d) + arctan(e^(1/2*d*x + 1/2*c))/(sqrt(a)*d^4)) - 2*(sqrt(2)*sqrt(a)* d^3*x^3*e^(1/2*c) + 6*sqrt(2)*sqrt(a)*d^2*x^2*e^(1/2*c) + 24*sqrt(2)*sqrt( a)*d*x*e^(1/2*c) + 48*sqrt(2)*sqrt(a)*e^(1/2*c))*e^(1/2*d*x)/(a*d^4*e^(d*x + c) + a*d^4)
\[ \int \frac {x^3}{\sqrt {a+a \cosh (c+d x)}} \, dx=\int { \frac {x^{3}}{\sqrt {a \cosh \left (d x + c\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {x^3}{\sqrt {a+a \cosh (c+d x)}} \, dx=\int \frac {x^3}{\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]