Integrand size = 14, antiderivative size = 186 \[ \int (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\frac {15 b^2 (c+d x) \sqrt {a+b \text {arccosh}(c+d x)}}{4 d}-\frac {5 b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{3/2}}{2 d}+\frac {(c+d x) (a+b \text {arccosh}(c+d x))^{5/2}}{d}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{16 d} \]
(d*x+c)*(a+b*arccosh(d*x+c))^(5/2)/d-15/16*b^(5/2)*exp(a/b)*erf((a+b*arcco sh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d-15/16*b^(5/2)*erfi((a+b*arccosh(d*x+c ))^(1/2)/b^(1/2))*Pi^(1/2)/d/exp(a/b)-5/2*b*(a+b*arccosh(d*x+c))^(3/2)*(d* x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d+15/4*b^2*(d*x+c)*(a+b*arccosh(d*x+c))^(1/2) /d
Leaf count is larger than twice the leaf count of optimal. \(494\) vs. \(2(186)=372\).
Time = 2.42 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.66 \[ \int (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\frac {4 b \sqrt {a+b \text {arccosh}(c+d x)} \left (2 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) (a-5 b \text {arccosh}(c+d x))+b (c+d x) \left (15+4 \text {arccosh}(c+d x)^2\right )\right )+8 a^2 e^{-\frac {a}{b}} \sqrt {a+b \text {arccosh}(c+d x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )}{\sqrt {\frac {a}{b}+\text {arccosh}(c+d x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}}}\right )-\sqrt {b} \left (4 a^2+12 a b+15 b^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )-\sqrt {b} \left (4 a^2-12 a b+15 b^2\right ) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )+4 a b \left (-12 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \sqrt {a+b \text {arccosh}(c+d x)}+8 (c+d x) \text {arccosh}(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}+\frac {(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}+\frac {(2 a-3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}\right )}{16 d} \]
(4*b*Sqrt[a + b*ArcCosh[c + d*x]]*(2*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*(a - 5*b*ArcCosh[c + d*x]) + b*(c + d*x)*(15 + 4*ArcCosh[c + d *x]^2)) + (8*a^2*Sqrt[a + b*ArcCosh[c + d*x]]*((E^((2*a)/b)*Gamma[3/2, a/b + ArcCosh[c + d*x]])/Sqrt[a/b + ArcCosh[c + d*x]] + Gamma[3/2, -((a + b*A rcCosh[c + d*x])/b)]/Sqrt[-((a + b*ArcCosh[c + d*x])/b)]))/E^(a/b) - Sqrt[ b]*(4*a^2 + 12*a*b + 15*b^2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sq rt[b]]*(Cosh[a/b] - Sinh[a/b]) - Sqrt[b]*(4*a^2 - 12*a*b + 15*b^2)*Sqrt[Pi ]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]) + 4*a* b*(-12*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*Sqrt[a + b*ArcCosh [c + d*x]] + 8*(c + d*x)*ArcCosh[c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]] + ( (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]))/Sqrt[b] + ((2*a - 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]))/Sqrt[b]))/(16*d)
Time = 1.15 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6410, 6294, 6330, 6294, 6368, 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.
\(\displaystyle \int (a+b \text {arccosh}(c+d x))^{5/2} \, dx\) |
\(\Big \downarrow \) 6410 |
\(\displaystyle \frac {\int (a+b \text {arccosh}(c+d x))^{5/2}d(c+d x)}{d}\) |
\(\Big \downarrow \) 6294 |
\(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \int \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{3/2}}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 6330 |
\(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \int \sqrt {a+b \text {arccosh}(c+d x)}d(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 6294 |
\(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} b \int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)\right )\right )}{d}\) |
\(\Big \downarrow \) 6368 |
\(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )\right )}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )\right )}{d}\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arccosh}(c+d x)}+\frac {1}{2} \left (\frac {1}{2} i \int \frac {i e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))-\frac {1}{2} i \int -\frac {i e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )\right )\right )}{d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))-\frac {1}{2} \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )+(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (-\int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-\int e^{\frac {a+b \text {arccosh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}\right )+(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (-\int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )+(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )}{d}\) |
((c + d*x)*(a + b*ArcCosh[c + d*x])^(5/2) - (5*b*(Sqrt[-1 + c + d*x]*Sqrt[ 1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2) - (3*b*((c + d*x)*Sqrt[a + b*A rcCosh[c + d*x]] + (-1/2*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[ c + d*x]]/Sqrt[b]]) - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/ Sqrt[b]])/(2*E^(a/b)))/2))/2))/2)/d
3.2.68.3.1 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_Symbol] :> Simp[x*(a + b*A rcCosh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt [1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p _)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 *c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ p] Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && E qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x _))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p] Subst[In t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c *x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d , n}, x]
\[\int \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}d x\]
Exception generated. \[ \int (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int { {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int { {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (a+b \text {arccosh}(c+d x))^{5/2} \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]