Integrand size = 26, antiderivative size = 264 \[ \int \frac {x^7 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{3 c^9 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {b \left (1-c^2 x^2\right )^{3/2} \sqrt {1+c^2 x^2}}{18 c^9 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {b \left (1-c^2 x^2\right )^{5/2} \sqrt {1+c^2 x^2}}{30 c^9 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^8}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{6 c^8}+\frac {b \sqrt {1+c^2 x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{3 c^9 \sqrt {1+\frac {1}{c^2 x^2}} x} \]
1/6*(-c^4*x^4+1)^(3/2)*(a+b*arccsch(c*x))/c^8+1/18*b*(-c^2*x^2+1)^(3/2)*(c ^2*x^2+1)^(1/2)/c^9/x/(1+1/c^2/x^2)^(1/2)-1/30*b*(-c^2*x^2+1)^(5/2)*(c^2*x ^2+1)^(1/2)/c^9/x/(1+1/c^2/x^2)^(1/2)+1/3*b*arctanh((-c^2*x^2+1)^(1/2))*(c ^2*x^2+1)^(1/2)/c^9/x/(1+1/c^2/x^2)^(1/2)-1/3*b*(-c^2*x^2+1)^(1/2)*(c^2*x^ 2+1)^(1/2)/c^9/x/(1+1/c^2/x^2)^(1/2)-1/2*(a+b*arccsch(c*x))*(-c^4*x^4+1)^( 1/2)/c^8
Time = 0.52 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.68 \[ \int \frac {x^7 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {15 a \sqrt {1-c^4 x^4} \left (2+c^4 x^4\right )+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {1-c^4 x^4} \left (28-c^2 x^2+3 c^4 x^4\right )}{1+c^2 x^2}+15 b \sqrt {1-c^4 x^4} \left (2+c^4 x^4\right ) \text {csch}^{-1}(c x)+30 b \log \left (x+c^2 x^3\right )-30 b \log \left (1+c^2 x^2+c \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {1-c^4 x^4}\right )}{90 c^8} \]
-1/90*(15*a*Sqrt[1 - c^4*x^4]*(2 + c^4*x^4) + (b*c*Sqrt[1 + 1/(c^2*x^2)]*x *Sqrt[1 - c^4*x^4]*(28 - c^2*x^2 + 3*c^4*x^4))/(1 + c^2*x^2) + 15*b*Sqrt[1 - c^4*x^4]*(2 + c^4*x^4)*ArcCsch[c*x] + 30*b*Log[x + c^2*x^3] - 30*b*Log[ 1 + c^2*x^2 + c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 - c^4*x^4]])/c^8
Time = 1.14 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.59, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {6864, 27, 7272, 1388, 1579, 517, 25, 1584, 2009}
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.
\(\displaystyle \int \frac {x^7 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx\) |
\(\Big \downarrow \) 6864 |
\(\displaystyle \frac {b \int -\frac {\sqrt {1-c^4 x^4} \left (c^4 x^4+2\right )}{6 c^8 \sqrt {1+\frac {1}{c^2 x^2}} x^2}dx}{c}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^8}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {b \int \frac {\sqrt {1-c^4 x^4} \left (c^4 x^4+2\right )}{\sqrt {1+\frac {1}{c^2 x^2}} x^2}dx}{6 c^9}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^8}\) |
\(\Big \downarrow \) 7272 |
\(\displaystyle -\frac {b \sqrt {c^2 x^2+1} \int \frac {\sqrt {1-c^4 x^4} \left (c^4 x^4+2\right )}{x \sqrt {c^2 x^2+1}}dx}{6 c^9 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^8}\) |
\(\Big \downarrow \) 1388 |
\(\displaystyle -\frac {b \sqrt {c^2 x^2+1} \int \frac {\sqrt {1-c^2 x^2} \left (c^4 x^4+2\right )}{x}dx}{6 c^9 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^8}\) |
\(\Big \downarrow \) 1579 |
\(\displaystyle -\frac {b \sqrt {c^2 x^2+1} \int \frac {\sqrt {1-c^2 x^2} \left (c^4 x^4+2\right )}{x^2}dx^2}{12 c^9 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^8}\) |
\(\Big \downarrow \) 517 |
\(\displaystyle -\frac {b \sqrt {c^2 x^2+1} \int -\frac {x^4 \left (c^4 x^8-2 c^4 x^4+3 c^4\right )}{1-x^4}d\sqrt {1-c^2 x^2}}{6 c^{13} x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^8}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {b \sqrt {c^2 x^2+1} \int \frac {x^4 \left (c^4 x^8-2 c^4 x^4+3 c^4\right )}{1-x^4}d\sqrt {1-c^2 x^2}}{6 c^{13} x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^8}\) |
\(\Big \downarrow \) 1584 |
\(\displaystyle \frac {b \sqrt {c^2 x^2+1} \int \left (-c^4 x^8+c^4 x^4-2 c^4+\frac {2 c^4}{1-x^4}\right )d\sqrt {1-c^2 x^2}}{6 c^{13} x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^8}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^8}-\frac {b \sqrt {c^2 x^2+1} \left (-2 c^4 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {c^4 x^{10}}{5}-\frac {c^4 x^6}{3}+2 c^4 \sqrt {1-c^2 x^2}\right )}{6 c^{13} x \sqrt {\frac {1}{c^2 x^2}+1}}\) |
-1/2*(Sqrt[1 - c^4*x^4]*(a + b*ArcCsch[c*x]))/c^8 + ((1 - c^4*x^4)^(3/2)*( a + b*ArcCsch[c*x]))/(6*c^8) - (b*Sqrt[1 + c^2*x^2]*(-1/3*(c^4*x^6) + (c^4 *x^10)/5 + 2*c^4*Sqrt[1 - c^2*x^2] - 2*c^4*ArcTanh[Sqrt[1 - c^2*x^2]]))/(6 *c^13*Sqrt[1 + 1/(c^2*x^2)]*x)
3.2.75.3.1 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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*(e^m/d^(m + 2*p + 1)) Subst[Int[x^(2*n + 1)*(-c + x^ 2)^m*(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4)^p, x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && ILtQ[m, 0] && IntegerQ[n + 1/2]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHid e[u, x]}, Simp[(a + b*ArcCsch[c*x]) v, x] + Simp[b/c Int[SimplifyIntegr and[v/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x] ] /; FreeQ[{a, b, c}, x]
Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[b^IntPart[p]*(( a + b*x^n)^FracPart[p]/(x^(n*FracPart[p])*(1 + a*(1/(x^n*b)))^FracPart[p])) Int[u*x^(n*p)*(1 + a*(1/(x^n*b)))^p, x], x] /; FreeQ[{a, b, p}, x] && ! IntegerQ[p] && ILtQ[n, 0] && !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]
\[\int \frac {x^{7} \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )}{\sqrt {-c^{4} x^{4}+1}}d x\]
Time = 0.27 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.23 \[ \int \frac {x^7 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {15 \, {\left (b c^{6} x^{6} + b c^{4} x^{4} + 2 \, b c^{2} x^{2} + 2 \, b\right )} \sqrt {-c^{4} x^{4} + 1} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (3 \, b c^{5} x^{5} - b c^{3} x^{3} + 28 \, b c x\right )} \sqrt {-c^{4} x^{4} + 1} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - 15 \, {\left (b c^{2} x^{2} + b\right )} \log \left (\frac {c^{2} x^{2} + \sqrt {-c^{4} x^{4} + 1} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c^{2} x^{2} + 1}\right ) + 15 \, {\left (b c^{2} x^{2} + b\right )} \log \left (-\frac {c^{2} x^{2} - \sqrt {-c^{4} x^{4} + 1} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c^{2} x^{2} + 1}\right ) + 15 \, {\left (a c^{6} x^{6} + a c^{4} x^{4} + 2 \, a c^{2} x^{2} + 2 \, a\right )} \sqrt {-c^{4} x^{4} + 1}}{90 \, {\left (c^{10} x^{2} + c^{8}\right )}} \]
-1/90*(15*(b*c^6*x^6 + b*c^4*x^4 + 2*b*c^2*x^2 + 2*b)*sqrt(-c^4*x^4 + 1)*l og((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + (3*b*c^5*x^5 - b*c^3*x ^3 + 28*b*c*x)*sqrt(-c^4*x^4 + 1)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - 15*(b*c^ 2*x^2 + b)*log((c^2*x^2 + sqrt(-c^4*x^4 + 1)*c*x*sqrt((c^2*x^2 + 1)/(c^2*x ^2)) + 1)/(c^2*x^2 + 1)) + 15*(b*c^2*x^2 + b)*log(-(c^2*x^2 - sqrt(-c^4*x^ 4 + 1)*c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c^2*x^2 + 1)) + 15*(a*c^6*x ^6 + a*c^4*x^4 + 2*a*c^2*x^2 + 2*a)*sqrt(-c^4*x^4 + 1))/(c^10*x^2 + c^8)
Timed out. \[ \int \frac {x^7 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\text {Timed out} \]
\[ \int \frac {x^7 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{7}}{\sqrt {-c^{4} x^{4} + 1}} \,d x } \]
1/6*a*((-c^4*x^4 + 1)^(3/2)/c^8 - 3*sqrt(-c^4*x^4 + 1)/c^8) + 1/6*b*((c^8* x^8 + c^4*x^4 - 2)*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(c^2*x^2 + 1)*sqrt(c*x + 1)*sqrt(-c*x + 1)*c^8) - 6*integrate((x^7*log(c) + x^7*log(x))*e^(-1/2*l og(c^2*x^2 + 1) - 1/2*log(c*x + 1) - 1/2*log(-c*x + 1)), x) - 6*integrate( 1/6*(c^6*x^7 - c^4*x^5 + 2*c^2*x^3 - 2*x)/(sqrt(c^2*x^2 + 1)*sqrt(c*x + 1) *sqrt(-c*x + 1)*c^6 + sqrt(c*x + 1)*sqrt(-c*x + 1)*c^6), x))
Exception generated. \[ \int \frac {x^7 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^7 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int \frac {x^7\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {1-c^4\,x^4}} \,d x \]