Integrand size = 21, antiderivative size = 493 \[ \int \frac {x^3}{\sqrt {a+i a \sinh (e+f x)}} \, dx=\frac {4 i x^3 \text {arctanh}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f \sqrt {a+i a \sinh (e+f x)}}+\frac {12 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {12 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {48 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {48 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {96 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \operatorname {PolyLog}\left (4,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^4 \sqrt {a+i a \sinh (e+f x)}}-\frac {96 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \operatorname {PolyLog}\left (4,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^4 \sqrt {a+i a \sinh (e+f x)}} \]
-4*I*x^3*arctanh(exp(1/2*e+3/4*I*Pi+1/2*f*x))*cosh(1/2*e+1/4*I*Pi+1/2*f*x) /f/(a+I*a*sinh(f*x+e))^(1/2)+12*I*x^2*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog (2,exp(1/2*e+3/4*I*Pi+1/2*f*x))/f^2/(a+I*a*sinh(f*x+e))^(1/2)-12*I*x^2*cos h(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(2,-exp(1/2*e+3/4*I*Pi+1/2*f*x))/f^2/(a+I *a*sinh(f*x+e))^(1/2)-48*I*x*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(3,exp(1/ 2*e+3/4*I*Pi+1/2*f*x))/f^3/(a+I*a*sinh(f*x+e))^(1/2)+48*I*x*cosh(1/2*e+1/4 *I*Pi+1/2*f*x)*polylog(3,-exp(1/2*e+3/4*I*Pi+1/2*f*x))/f^3/(a+I*a*sinh(f*x +e))^(1/2)+96*I*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(4,exp(1/2*e+3/4*I*Pi+ 1/2*f*x))/f^4/(a+I*a*sinh(f*x+e))^(1/2)-96*I*cosh(1/2*e+1/4*I*Pi+1/2*f*x)* polylog(4,-exp(1/2*e+3/4*I*Pi+1/2*f*x))/f^4/(a+I*a*sinh(f*x+e))^(1/2)
Time = 0.81 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.67 \[ \int \frac {x^3}{\sqrt {a+i a \sinh (e+f x)}} \, dx=\frac {(1-i) (-1)^{3/4} \left (2 i e^3 \arctan \left (\sqrt [4]{-1} e^{\frac {1}{2} (e+f x)}\right )+e^3 \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+f^3 x^3 \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-e^3 \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-f^3 x^3 \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-6 f^2 x^2 \operatorname {PolyLog}\left (2,-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+6 f^2 x^2 \operatorname {PolyLog}\left (2,(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+24 f x \operatorname {PolyLog}\left (3,-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-24 f x \operatorname {PolyLog}\left (3,(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-48 \operatorname {PolyLog}\left (4,-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+48 \operatorname {PolyLog}\left (4,(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )\right ) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )}{f^4 \sqrt {a+i a \sinh (e+f x)}} \]
((1 - I)*(-1)^(3/4)*((2*I)*e^3*ArcTan[(-1)^(1/4)*E^((e + f*x)/2)] + e^3*Lo g[1 - (-1)^(3/4)*E^((e + f*x)/2)] + f^3*x^3*Log[1 - (-1)^(3/4)*E^((e + f*x )/2)] - e^3*Log[1 + (-1)^(3/4)*E^((e + f*x)/2)] - f^3*x^3*Log[1 + (-1)^(3/ 4)*E^((e + f*x)/2)] - 6*f^2*x^2*PolyLog[2, -((-1)^(3/4)*E^((e + f*x)/2))] + 6*f^2*x^2*PolyLog[2, (-1)^(3/4)*E^((e + f*x)/2)] + 24*f*x*PolyLog[3, -(( -1)^(3/4)*E^((e + f*x)/2))] - 24*f*x*PolyLog[3, (-1)^(3/4)*E^((e + f*x)/2) ] - 48*PolyLog[4, -((-1)^(3/4)*E^((e + f*x)/2))] + 48*PolyLog[4, (-1)^(3/4 )*E^((e + f*x)/2)])*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2]))/(f^4*Sqrt[a + I*a*Sinh[e + f*x]])
Time = 0.90 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.58, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3800, 3042, 4670, 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.
\(\displaystyle \int \frac {x^3}{\sqrt {a+i a \sinh (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {x^3}{\sqrt {a+a \sin (i e+i f x)}}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \int x^3 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{\sqrt {a+i a \sinh (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \int x^3 \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx}{\sqrt {a+i a \sinh (e+f x)}}\) |
\(\Big \downarrow \) 4670 |
\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (\frac {6 i \int x^2 \log \left (1-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}-\frac {6 i \int x^2 \log \left (1+e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}+\frac {4 i x^3 \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}\right )}{\sqrt {a+i a \sinh (e+f x)}}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (-\frac {6 i \left (\frac {4 \int x \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}+\frac {6 i \left (\frac {4 \int x \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}+\frac {4 i x^3 \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}\right )}{\sqrt {a+i a \sinh (e+f x)}}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (-\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}-\frac {2 \int \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}\right )}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}+\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}-\frac {2 \int \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}\right )}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}+\frac {4 i x^3 \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}\right )}{\sqrt {a+i a \sinh (e+f x)}}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (-\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}-\frac {4 \int e^{\frac {1}{4} (i \pi -2 e)-\frac {f x}{2}} \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )de^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}}{f^2}\right )}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}+\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}-\frac {4 \int e^{\frac {1}{4} (i \pi -2 e)-\frac {f x}{2}} \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )de^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}}{f^2}\right )}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}+\frac {4 i x^3 \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}\right )}{\sqrt {a+i a \sinh (e+f x)}}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (\frac {4 i x^3 \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}-\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}-\frac {4 \operatorname {PolyLog}\left (4,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}\right )}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}+\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}-\frac {4 \operatorname {PolyLog}\left (4,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}\right )}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}\right )}{\sqrt {a+i a \sinh (e+f x)}}\) |
(Cosh[e/2 + (I/4)*Pi + (f*x)/2]*(((4*I)*x^3*ArcTanh[E^((2*e - I*Pi)/4 + (f *x)/2)])/f - ((6*I)*((-2*x^2*PolyLog[2, -E^((2*e - I*Pi)/4 + (f*x)/2)])/f + (4*((2*x*PolyLog[3, -E^((2*e - I*Pi)/4 + (f*x)/2)])/f - (4*PolyLog[4, -E ^((2*e - I*Pi)/4 + (f*x)/2)])/f^2))/f))/f + ((6*I)*((-2*x^2*PolyLog[2, E^( (2*e - I*Pi)/4 + (f*x)/2)])/f + (4*((2*x*PolyLog[3, E^((2*e - I*Pi)/4 + (f *x)/2)])/f - (4*PolyLog[4, E^((2*e - I*Pi)/4 + (f*x)/2)])/f^2))/f))/f))/Sq rt[a + I*a*Sinh[e + f*x]]
3.2.36.3.1 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_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && 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]
\[\int \frac {x^{3}}{\sqrt {a +i a \sinh \left (f x +e \right )}}d x\]
\[ \int \frac {x^3}{\sqrt {a+i a \sinh (e+f x)}} \, dx=\int { \frac {x^{3}}{\sqrt {i \, a \sinh \left (f x + e\right ) + a}} \,d x } \]
\[ \int \frac {x^3}{\sqrt {a+i a \sinh (e+f x)}} \, dx=\int \frac {x^{3}}{\sqrt {i a \left (\sinh {\left (e + f x \right )} - i\right )}}\, dx \]
\[ \int \frac {x^3}{\sqrt {a+i a \sinh (e+f x)}} \, dx=\int { \frac {x^{3}}{\sqrt {i \, a \sinh \left (f x + e\right ) + a}} \,d x } \]
\[ \int \frac {x^3}{\sqrt {a+i a \sinh (e+f x)}} \, dx=\int { \frac {x^{3}}{\sqrt {i \, a \sinh \left (f x + e\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {x^3}{\sqrt {a+i a \sinh (e+f x)}} \, dx=\int \frac {x^3}{\sqrt {a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}}} \,d x \]