3.2.50 \(\int (a+b \cosh ^{-1}(c x))^{5/2} \, dx\) [150]

Optimal. Leaf size=160 \[ \frac {15}{4} b^2 x \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c}-\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c} \]

[Out]

x*(a+b*arccosh(c*x))^(5/2)-15/16*b^(5/2)*exp(a/b)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c-15/16*b^(5/
2)*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c/exp(a/b)-5/2*b*(a+b*arccosh(c*x))^(3/2)*(c*x-1)^(1/2)*(c*
x+1)^(1/2)/c+15/4*b^2*x*(a+b*arccosh(c*x))^(1/2)

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Rubi [A]
time = 0.47, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5879, 5915, 5953, 3388, 2211, 2236, 2235} \begin {gather*} -\frac {15 \sqrt {\pi } b^{5/2} e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c}-\frac {15 \sqrt {\pi } b^{5/2} e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c}+\frac {15}{4} b^2 x \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])^(5/2),x]

[Out]

(15*b^2*x*Sqrt[a + b*ArcCosh[c*x]])/4 - (5*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(3/2))/(2*c) +
x*(a + b*ArcCosh[c*x])^(5/2) - (15*b^(5/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(16*c) - (1
5*b^(5/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(16*c*E^(a/b))

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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]

Rule 2235

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]

Rule 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[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]

Rule 5879

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[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]

Rule 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Dist[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] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

Rule 5953

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[Int[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]

Rubi steps

\begin {align*} \int \left (a+b \cosh ^{-1}(c x)\right )^{5/2} \, dx &=x \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {1}{2} (5 b c) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2}+\frac {1}{4} \left (15 b^2\right ) \int \sqrt {a+b \cosh ^{-1}(c x)} \, dx\\ &=\frac {15}{4} b^2 x \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {1}{8} \left (15 b^3 c\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}} \, dx\\ &=\frac {15}{4} b^2 x \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {\left (15 b^3\right ) \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 c}\\ &=\frac {15}{4} b^2 x \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {\left (15 b^3\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c}-\frac {\left (15 b^3\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c}\\ &=\frac {15}{4} b^2 x \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 c}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 c}\\ &=\frac {15}{4} b^2 x \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c}-\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(452\) vs. \(2(160)=320\).
time = 1.64, size = 452, normalized size = 2.82 \begin {gather*} \frac {4 b \sqrt {a+b \cosh ^{-1}(c x)} \left (2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (a-5 b \cosh ^{-1}(c x)\right )+b c x \left (15+4 \cosh ^{-1}(c x)^2\right )\right )+8 a^2 e^{-\frac {a}{b}} \sqrt {a+b \cosh ^{-1}(c x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\cosh ^{-1}(c x)\right )}{\sqrt {\frac {a}{b}+\cosh ^{-1}(c x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{\sqrt {-\frac {a+b \cosh ^{-1}(c 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 \cosh ^{-1}(c 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 \cosh ^{-1}(c 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 x}{1+c x}} (1+c x) \sqrt {a+b \cosh ^{-1}(c x)}+8 c x \cosh ^{-1}(c x) \sqrt {a+b \cosh ^{-1}(c x)}+\frac {(2 a+3 b) \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c 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 \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}\right )}{16 c} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c*x])^(5/2),x]

[Out]

(4*b*Sqrt[a + b*ArcCosh[c*x]]*(2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a - 5*b*ArcCosh[c*x]) + b*c*x*(15 + 4*A
rcCosh[c*x]^2)) + (8*a^2*Sqrt[a + b*ArcCosh[c*x]]*((E^((2*a)/b)*Gamma[3/2, a/b + ArcCosh[c*x]])/Sqrt[a/b + Arc
Cosh[c*x]] + Gamma[3/2, -((a + b*ArcCosh[c*x])/b)]/Sqrt[-((a + b*ArcCosh[c*x])/b)]))/E^(a/b) - Sqrt[b]*(4*a^2
+ 12*a*b + 15*b^2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[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*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]) + 4*a*b*(-12*Sqrt[(-1
+ c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[a + b*ArcCosh[c*x]] + 8*c*x*ArcCosh[c*x]*Sqrt[a + b*ArcCosh[c*x]] + ((2*a + 3
*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]))/Sqrt[b] + ((2*a - 3*b)*Sqrt[Pi]*E
rf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]))/Sqrt[b]))/(16*c)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {5}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))^(5/2),x)

[Out]

int((a+b*arccosh(c*x))^(5/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*arccosh(c*x) + a)^(5/2), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))**(5/2),x)

[Out]

Integral((a + b*acosh(c*x))**(5/2), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^(5/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c*x))^(5/2),x)

[Out]

int((a + b*acosh(c*x))^(5/2), x)

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