Integrand size = 23, antiderivative size = 329 \[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=-\frac {b \left (19 c^2 d+9 e\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e^2 \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e^2 \sqrt {-c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}+\frac {b \left (45 c^4 d^2+10 c^2 d e+9 e^2\right ) x \arctan \left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{120 c^4 e^{5/2} \sqrt {-c^2 x^2}}+\frac {8 b c d^{5/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{15 e^3 \sqrt {-c^2 x^2}} \]
-2/3*d*(e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/e^3+1/5*(e*x^2+d)^(5/2)*(a+b*arc csch(c*x))/e^3+1/120*b*(45*c^4*d^2+10*c^2*d*e+9*e^2)*x*arctan(e^(1/2)*(-c^ 2*x^2-1)^(1/2)/c/(e*x^2+d)^(1/2))/c^4/e^(5/2)/(-c^2*x^2)^(1/2)+8/15*b*c*d^ (5/2)*x*arctan((e*x^2+d)^(1/2)/d^(1/2)/(-c^2*x^2-1)^(1/2))/e^3/(-c^2*x^2)^ (1/2)+1/20*b*x*(e*x^2+d)^(3/2)*(-c^2*x^2-1)^(1/2)/c/e^2/(-c^2*x^2)^(1/2)+d ^2*(a+b*arccsch(c*x))*(e*x^2+d)^(1/2)/e^3-1/120*b*(19*c^2*d+9*e)*x*(-c^2*x ^2-1)^(1/2)*(e*x^2+d)^(1/2)/c^3/e^2/(-c^2*x^2)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 3.03 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.85 \[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\frac {16 a \left (d+e x^2\right ) \left (8 d^2-4 d e x^2+3 e^2 x^4\right )+\frac {2 b e \sqrt {1+\frac {1}{c^2 x^2}} \left (d+e x^2\right ) \left (-9 e x+c^2 \left (-13 d x+6 e x^3\right )\right )}{c^3}+\frac {b \left (-64 c^2 d^3 \sqrt {1+\frac {d}{e x^2}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,-\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )+\frac {e \left (45 c^4 d^2+10 c^2 d e+9 e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x^4 \sqrt {1+\frac {e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,-c^2 x^2,-\frac {e x^2}{d}\right )}{\sqrt {1+c^2 x^2}}\right )}{c^3 x}+16 b \left (d+e x^2\right ) \left (8 d^2-4 d e x^2+3 e^2 x^4\right ) \text {csch}^{-1}(c x)}{240 e^3 \sqrt {d+e x^2}} \]
(16*a*(d + e*x^2)*(8*d^2 - 4*d*e*x^2 + 3*e^2*x^4) + (2*b*e*Sqrt[1 + 1/(c^2 *x^2)]*(d + e*x^2)*(-9*e*x + c^2*(-13*d*x + 6*e*x^3)))/c^3 + (b*(-64*c^2*d ^3*Sqrt[1 + d/(e*x^2)]*AppellF1[1, 1/2, 1/2, 2, -(1/(c^2*x^2)), -(d/(e*x^2 ))] + (e*(45*c^4*d^2 + 10*c^2*d*e + 9*e^2)*Sqrt[1 + 1/(c^2*x^2)]*x^4*Sqrt[ 1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, -(c^2*x^2), -((e*x^2)/d)])/Sqrt[1 + c^2*x^2]))/(c^3*x) + 16*b*(d + e*x^2)*(8*d^2 - 4*d*e*x^2 + 3*e^2*x^4)*Ar cCsch[c*x])/(240*e^3*Sqrt[d + e*x^2])
Time = 1.38 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.91, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6856, 27, 7282, 2118, 27, 171, 27, 175, 66, 104, 217, 218}
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.
\(\displaystyle \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx\) |
\(\Big \downarrow \) 6856 |
\(\displaystyle -\frac {b c x \int \frac {\sqrt {e x^2+d} \left (3 e^2 x^4-4 d e x^2+8 d^2\right )}{15 e^3 x \sqrt {-c^2 x^2-1}}dx}{\sqrt {-c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {b c x \int \frac {\sqrt {e x^2+d} \left (3 e^2 x^4-4 d e x^2+8 d^2\right )}{x \sqrt {-c^2 x^2-1}}dx}{15 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
\(\Big \downarrow \) 7282 |
\(\displaystyle -\frac {b c x \int \frac {\sqrt {e x^2+d} \left (3 e^2 x^4-4 d e x^2+8 d^2\right )}{x^2 \sqrt {-c^2 x^2-1}}dx^2}{30 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
\(\Big \downarrow \) 2118 |
\(\displaystyle -\frac {b c x \left (-\frac {\int -\frac {e \sqrt {e x^2+d} \left (32 c^2 d^2-e \left (19 d c^2+9 e\right ) x^2\right )}{2 x^2 \sqrt {-c^2 x^2-1}}dx^2}{2 c^2 e}-\frac {3 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {b c x \left (\frac {\int \frac {\sqrt {e x^2+d} \left (32 c^2 d^2-e \left (19 d c^2+9 e\right ) x^2\right )}{x^2 \sqrt {-c^2 x^2-1}}dx^2}{4 c^2}-\frac {3 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
\(\Big \downarrow \) 171 |
\(\displaystyle -\frac {b c x \left (\frac {\frac {e \sqrt {-c^2 x^2-1} \left (19 c^2 d+9 e\right ) \sqrt {d+e x^2}}{c^2}-\frac {\int -\frac {64 d^3 c^4+e \left (45 d^2 c^4+10 d e c^2+9 e^2\right ) x^2}{2 x^2 \sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{c^2}}{4 c^2}-\frac {3 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {b c x \left (\frac {\frac {\int \frac {64 d^3 c^4+e \left (45 d^2 c^4+10 d e c^2+9 e^2\right ) x^2}{x^2 \sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{2 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (19 c^2 d+9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}-\frac {3 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
\(\Big \downarrow \) 175 |
\(\displaystyle -\frac {b c x \left (\frac {\frac {64 c^4 d^3 \int \frac {1}{x^2 \sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx^2+e \left (45 c^4 d^2+10 c^2 d e+9 e^2\right ) \int \frac {1}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{2 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (19 c^2 d+9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}-\frac {3 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle -\frac {b c x \left (\frac {\frac {64 c^4 d^3 \int \frac {1}{x^2 \sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx^2+2 e \left (45 c^4 d^2+10 c^2 d e+9 e^2\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {-c^2 x^2-1}}{\sqrt {e x^2+d}}}{2 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (19 c^2 d+9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}-\frac {3 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle -\frac {b c x \left (\frac {\frac {128 c^4 d^3 \int \frac {1}{-x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {-c^2 x^2-1}}+2 e \left (45 c^4 d^2+10 c^2 d e+9 e^2\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {-c^2 x^2-1}}{\sqrt {e x^2+d}}}{2 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (19 c^2 d+9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}-\frac {3 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {b c x \left (\frac {\frac {2 e \left (45 c^4 d^2+10 c^2 d e+9 e^2\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {-c^2 x^2-1}}{\sqrt {e x^2+d}}-128 c^4 d^{5/2} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{2 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (19 c^2 d+9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}-\frac {3 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {-c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {d^2 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {b c x \left (\frac {\frac {-128 c^4 d^{5/2} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )-\frac {2 \sqrt {e} \left (45 c^4 d^2+10 c^2 d e+9 e^2\right ) \arctan \left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{c}}{2 c^2}+\frac {e \sqrt {-c^2 x^2-1} \left (19 c^2 d+9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}-\frac {3 e \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {-c^2 x^2}}\) |
(d^2*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/e^3 - (2*d*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^3) + ((d + e*x^2)^(5/2)*(a + b*ArcCsch[c*x]))/(5* e^3) - (b*c*x*((-3*e*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(3/2))/(2*c^2) + ((e*( 19*c^2*d + 9*e)*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/c^2 + ((-2*Sqrt[e]*(45 *c^4*d^2 + 10*c^2*d*e + 9*e^2)*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt [d + e*x^2])])/c - 128*c^4*d^(5/2)*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(2*c^2))/(4*c^2)))/(30*e^3*Sqrt[-(c^2*x^2)])
3.2.38.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + q + 1)) Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x]
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*( x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Si mp[(a + b*ArcCsch[c*x]) u, x] - Simp[b*c*(x/Sqrt[(-c^2)*x^2]) Int[Simpl ifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I LtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 /lst[[2]] Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( lst[[3]]*x)^lst[[2]]], x] /; !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ u] && !RationalFunctionQ[u, x]
\[\int \frac {x^{5} \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}d x\]
Time = 0.74 (sec) , antiderivative size = 1633, normalized size of antiderivative = 4.96 \[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\text {Too large to display} \]
[1/480*(64*b*c^5*d^(5/2)*log(((c^4*d^2 + 6*c^2*d*e + e^2)*x^4 + 8*(c^2*d^2 + d*e)*x^2 - 4*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(d)*sqrt ((c^2*x^2 + 1)/(c^2*x^2)) + 8*d^2)/x^4) + (45*b*c^4*d^2 + 10*b*c^2*d*e + 9 *b*e^2)*sqrt(e)*log(8*c^4*e^2*x^4 + c^4*d^2 + 6*c^2*d*e + 8*(c^4*d*e + c^2 *e^2)*x^2 + 4*(2*c^4*e*x^3 + (c^4*d + c^2*e)*x)*sqrt(e*x^2 + d)*sqrt(e)*sq rt((c^2*x^2 + 1)/(c^2*x^2)) + e^2) + 32*(3*b*c^5*e^2*x^4 - 4*b*c^5*d*e*x^2 + 8*b*c^5*d^2)*sqrt(e*x^2 + d)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1 )/(c*x)) + 4*(24*a*c^5*e^2*x^4 - 32*a*c^5*d*e*x^2 + 64*a*c^5*d^2 + (6*b*c^ 4*e^2*x^3 - (13*b*c^4*d*e + 9*b*c^2*e^2)*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2))) *sqrt(e*x^2 + d))/(c^5*e^3), 1/240*(32*b*c^5*d^(5/2)*log(((c^4*d^2 + 6*c^2 *d*e + e^2)*x^4 + 8*(c^2*d^2 + d*e)*x^2 - 4*((c^3*d + c*e)*x^3 + 2*c*d*x)* sqrt(e*x^2 + d)*sqrt(d)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 8*d^2)/x^4) - (45* b*c^4*d^2 + 10*b*c^2*d*e + 9*b*e^2)*sqrt(-e)*arctan(1/2*(2*c^2*e*x^3 + (c^ 2*d + e)*x)*sqrt(e*x^2 + d)*sqrt(-e)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^2*e^ 2*x^4 + (c^2*d*e + e^2)*x^2 + d*e)) + 16*(3*b*c^5*e^2*x^4 - 4*b*c^5*d*e*x^ 2 + 8*b*c^5*d^2)*sqrt(e*x^2 + d)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 2*(24*a*c^5*e^2*x^4 - 32*a*c^5*d*e*x^2 + 64*a*c^5*d^2 + (6*b*c ^4*e^2*x^3 - (13*b*c^4*d*e + 9*b*c^2*e^2)*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) )*sqrt(e*x^2 + d))/(c^5*e^3), 1/480*(128*b*c^5*sqrt(-d)*d^2*arctan(1/2*((c ^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(-d)*sqrt((c^2*x^2 + 1)/...
\[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \]
Exception generated. \[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{5}}{\sqrt {e x^{2} + d}} \,d x } \]
Timed out. \[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {e\,x^2+d}} \,d x \]