Integrand size = 14, antiderivative size = 229 \[ \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=-\frac {2 x \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {8 x^2}{15 b^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {32 x \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c \sqrt {a+b \text {arccosh}(c x)}}+\frac {8 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^2}+\frac {8 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^2} \]
4/15/b^2/c^2/(a+b*arccosh(c*x))^(3/2)-8/15*x^2/b^2/(a+b*arccosh(c*x))^(3/2 )+8/15*exp(2*a/b)*erf(2^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi ^(1/2)/b^(7/2)/c^2+8/15*erfi(2^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*2^( 1/2)*Pi^(1/2)/b^(7/2)/c^2/exp(2*a/b)-2/5*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c /(a+b*arccosh(c*x))^(5/2)-32/15*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b^3/c/(a+b*a rccosh(c*x))^(1/2)
\[ \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx \]
Time = 2.03 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6301, 6308, 6366, 6300, 25, 3042, 3788, 26, 2611, 2633, 2634}
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.
| \(\displaystyle \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6301 |
| \(\displaystyle \frac {4 c \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{5/2}}dx}{5 b}-\frac {2 \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{5/2}}dx}{5 b c}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle \frac {4 c \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{5/2}}dx}{5 b}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 6366 |
| \(\displaystyle \frac {4 c \left (\frac {4 \int \frac {x}{(a+b \text {arccosh}(c x))^{3/2}}dx}{3 b c}-\frac {2 x^2}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 6300 |
| \(\displaystyle \frac {4 c \left (\frac {4 \left (-\frac {2 \int -\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^2}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}-\frac {2 x^2}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 c \left (\frac {4 \left (\frac {2 \int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^2}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}-\frac {2 x^2}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 c \left (-\frac {2 x^2}{3 b c (a+b \text {arccosh}(c x))^{3/2}}+\frac {4 \left (-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {2 \int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^2}\right )}{3 b c}\right )}{5 b}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 3788 |
| \(\displaystyle \frac {4 c \left (-\frac {2 x^2}{3 b c (a+b \text {arccosh}(c x))^{3/2}}+\frac {4 \left (-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {2 \left (\frac {1}{2} i \int \frac {i e^{2 \text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} i \int -\frac {i e^{-2 \text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))\right )}{b^2 c^2}\right )}{3 b c}\right )}{5 b}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {4 c \left (\frac {4 \left (-\frac {2 \left (-\frac {1}{2} \int \frac {e^{-2 \text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} \int \frac {e^{2 \text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))\right )}{b^2 c^2}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}-\frac {2 x^2}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {4 c \left (\frac {4 \left (-\frac {2 \left (-\int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}}d\sqrt {a+b \text {arccosh}(c x)}-\int e^{\frac {2 (a+b \text {arccosh}(c x))}{b}-\frac {2 a}{b}}d\sqrt {a+b \text {arccosh}(c x)}\right )}{b^2 c^2}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}-\frac {2 x^2}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {4 c \left (\frac {4 \left (-\frac {2 \left (-\int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}}d\sqrt {a+b \text {arccosh}(c x)}-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^2}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}-\frac {2 x^2}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {4 c \left (\frac {4 \left (-\frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^2}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\right )}{3 b c}-\frac {2 x^2}{3 b c (a+b \text {arccosh}(c x))^{3/2}}\right )}{5 b}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}\) |
(-2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(5*b*c*(a + b*ArcCosh[c*x])^(5/2)) + 4 /(15*b^2*c^2*(a + b*ArcCosh[c*x])^(3/2)) + (4*c*((-2*x^2)/(3*b*c*(a + b*Ar cCosh[c*x])^(3/2)) + (4*((-2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*Sqrt[a + b*ArcCosh[c*x]]) - (2*(-1/2*(Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]* Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]) - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sq rt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(2*E^((2*a)/b))))/(b^2*c^2)))/(3*b*c)))/ (5*b)
3.2.61.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [I/2 Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e , f, m}, x] && IntegerQ[2*k]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) )), x] + Simp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) )), x] + (-Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcCosh[c*x ])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]) ), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 ] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1 _) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x ]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Simp[f*(m/(b*c*(n + 1)))*Simp [Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Int[ (f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e 1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]
\[\int \frac {x}{\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {7}{2}}}d x\]
Exception generated. \[ \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\int { \frac {x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\int { \frac {x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{7/2}} \,d x \]