Integrand size = 14, antiderivative size = 228 \[ \int x (a+b \text {arccosh}(c x))^{5/2} \, dx=-\frac {15 b^2 \sqrt {a+b \text {arccosh}(c x)}}{64 c^2}+\frac {15}{32} b^2 x^2 \sqrt {a+b \text {arccosh}(c x)}-\frac {5 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{3/2}}{8 c}-\frac {(a+b \text {arccosh}(c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{5/2}-\frac {15 b^{5/2} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{256 c^2}-\frac {15 b^{5/2} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{256 c^2} \] Output:
-15/64*b^2*(a+b*arccosh(c*x))^(1/2)/c^2+15/32*b^2*x^2*(a+b*arccosh(c*x))^( 1/2)-5/8*b*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^(3/2)/c-1/4*(a +b*arccosh(c*x))^(5/2)/c^2+1/2*x^2*(a+b*arccosh(c*x))^(5/2)-15/512*b^(5/2) *exp(2*a/b)*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2)) /c^2-15/512*b^(5/2)*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*(a+b*arccosh(c*x))^(1/2) /b^(1/2))/c^2/exp(2*a/b)
Time = 1.06 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.91 \[ \int x (a+b \text {arccosh}(c x))^{5/2} \, dx=\frac {-15 b^{5/2} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right )-15 b^{5/2} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+8 \sqrt {a+b \text {arccosh}(c x)} \left (\left (16 a^2+15 b^2\right ) \cosh (2 \text {arccosh}(c x))+16 b^2 \text {arccosh}(c x)^2 \cosh (2 \text {arccosh}(c x))-20 a b \sinh (2 \text {arccosh}(c x))+4 b \text {arccosh}(c x) (8 a \cosh (2 \text {arccosh}(c x))-5 b \sinh (2 \text {arccosh}(c x)))\right )}{512 c^2} \] Input:
Integrate[x*(a + b*ArcCosh[c*x])^(5/2),x]
Output:
(-15*b^(5/2)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*( Cosh[(2*a)/b] - Sinh[(2*a)/b]) - 15*b^(5/2)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + 8*Sqrt[a + b*ArcCosh[c*x]]*((16*a^2 + 15*b^2)*Cosh[2*ArcCosh[c*x]] + 16*b^2*ArcCosh[c *x]^2*Cosh[2*ArcCosh[c*x]] - 20*a*b*Sinh[2*ArcCosh[c*x]] + 4*b*ArcCosh[c*x ]*(8*a*Cosh[2*ArcCosh[c*x]] - 5*b*Sinh[2*ArcCosh[c*x]])))/(512*c^2)
Time = 3.30 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6299, 6354, 6299, 6308, 6368, 3042, 3793, 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.
\(\displaystyle \int x (a+b \text {arccosh}(c x))^{5/2} \, dx\) |
\(\Big \downarrow \) 6299 |
\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{4} b c \int \frac {x^2 (a+b \text {arccosh}(c x))^{3/2}}{\sqrt {c x-1} \sqrt {c x+1}}dx\) |
\(\Big \downarrow \) 6354 |
\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{4} b c \left (\frac {\int \frac {(a+b \text {arccosh}(c x))^{3/2}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}-\frac {3 b \int x \sqrt {a+b \text {arccosh}(c x)}dx}{4 c}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{2 c^2}\right )\) |
\(\Big \downarrow \) 6299 |
\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{4} b c \left (\frac {\int \frac {(a+b \text {arccosh}(c x))^{3/2}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}-\frac {3 b \left (\frac {1}{2} x^2 \sqrt {a+b \text {arccosh}(c x)}-\frac {1}{4} b c \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}dx\right )}{4 c}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{2 c^2}\right )\) |
\(\Big \downarrow \) 6308 |
\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{4} b c \left (-\frac {3 b \left (\frac {1}{2} x^2 \sqrt {a+b \text {arccosh}(c x)}-\frac {1}{4} b c \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}dx\right )}{4 c}+\frac {(a+b \text {arccosh}(c x))^{5/2}}{5 b c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{2 c^2}\right )\) |
\(\Big \downarrow \) 6368 |
\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{4} b c \left (-\frac {3 b \left (\frac {1}{2} x^2 \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{4 c^2}\right )}{4 c}+\frac {(a+b \text {arccosh}(c x))^{5/2}}{5 b c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{2 c^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{4} b c \left (-\frac {3 b \left (\frac {1}{2} x^2 \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )^2}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{4 c^2}\right )}{4 c}+\frac {(a+b \text {arccosh}(c x))^{5/2}}{5 b c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{2 c^2}\right )\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{4} b c \left (-\frac {3 b \left (\frac {1}{2} x^2 \sqrt {a+b \text {arccosh}(c x)}-\frac {\int \left (\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 \sqrt {a+b \text {arccosh}(c x)}}+\frac {1}{2 \sqrt {a+b \text {arccosh}(c x)}}\right )d(a+b \text {arccosh}(c x))}{4 c^2}\right )}{4 c}+\frac {(a+b \text {arccosh}(c x))^{5/2}}{5 b c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{2 c^2}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} x^2 (a+b \text {arccosh}(c x))^{5/2}-\frac {5}{4} b c \left (\frac {(a+b \text {arccosh}(c x))^{5/2}}{5 b c^3}-\frac {3 b \left (\frac {1}{2} x^2 \sqrt {a+b \text {arccosh}(c x)}-\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\sqrt {a+b \text {arccosh}(c x)}}{4 c^2}\right )}{4 c}+\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^{3/2}}{2 c^2}\right )\) |
Input:
Int[x*(a + b*ArcCosh[c*x])^(5/2),x]
Output:
(x^2*(a + b*ArcCosh[c*x])^(5/2))/2 - (5*b*c*((x*Sqrt[-1 + c*x]*Sqrt[1 + c* x]*(a + b*ArcCosh[c*x])^(3/2))/(2*c^2) + (a + b*ArcCosh[c*x])^(5/2)/(5*b*c ^3) - (3*b*((x^2*Sqrt[a + b*ArcCosh[c*x]])/2 - (Sqrt[a + b*ArcCosh[c*x]] + (Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sq rt[b]])/4 + (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sq rt[b]])/(4*E^((2*a)/b)))/(4*c^2)))/(4*c)))/4
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^(m + 1)*((a + b*ArcCosh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 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] && GtQ[n, 0]
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_)*((d1_) + (e 1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f *(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/( -1 + c*x)^p] Int[(f*x)^(m - 1)*(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, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N eQ[m + 2*p + 1, 0]
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]
\[\int x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {5}{2}}d x\]
Input:
int(x*(a+b*arccosh(c*x))^(5/2),x)
Output:
int(x*(a+b*arccosh(c*x))^(5/2),x)
Exception generated. \[ \int x (a+b \text {arccosh}(c x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(a+b*arccosh(c*x))^(5/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int x (a+b \text {arccosh}(c x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(x*(a+b*acosh(c*x))**(5/2),x)
Output:
Timed out
\[ \int x (a+b \text {arccosh}(c x))^{5/2} \, dx=\int { {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {5}{2}} x \,d x } \] Input:
integrate(x*(a+b*arccosh(c*x))^(5/2),x, algorithm="maxima")
Output:
integrate((b*arccosh(c*x) + a)^(5/2)*x, x)
Exception generated. \[ \int x (a+b \text {arccosh}(c x))^{5/2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x*(a+b*arccosh(c*x))^(5/2),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int x (a+b \text {arccosh}(c x))^{5/2} \, dx=\int x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{5/2} \,d x \] Input:
int(x*(a + b*acosh(c*x))^(5/2),x)
Output:
int(x*(a + b*acosh(c*x))^(5/2), x)
\[ \int x (a+b \text {arccosh}(c x))^{5/2} \, dx=2 \left (\int \sqrt {\mathit {acosh} \left (c x \right ) b +a}\, \mathit {acosh} \left (c x \right ) x d x \right ) a b +\left (\int \sqrt {\mathit {acosh} \left (c x \right ) b +a}\, \mathit {acosh} \left (c x \right )^{2} x d x \right ) b^{2}+\left (\int \sqrt {\mathit {acosh} \left (c x \right ) b +a}\, x d x \right ) a^{2} \] Input:
int(x*(a+b*acosh(c*x))^(5/2),x)
Output:
2*int(sqrt(acosh(c*x)*b + a)*acosh(c*x)*x,x)*a*b + int(sqrt(acosh(c*x)*b + a)*acosh(c*x)**2*x,x)*b**2 + int(sqrt(acosh(c*x)*b + a)*x,x)*a**2