3.2.63 \(\int (a+b \cosh ^{-1}(c+d x))^{3/2} \, dx\) [163]

Optimal. Leaf size=157 \[ -\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{d}-\frac {3 b^{3/2} e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {3 b^{3/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d} \]

[Out]

(d*x+c)*(a+b*arccosh(d*x+c))^(3/2)/d-3/8*b^(3/2)*exp(a/b)*erf((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d+3
/8*b^(3/2)*erfi((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d/exp(a/b)-3/2*b*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*
(a+b*arccosh(d*x+c))^(1/2)/d

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Rubi [A]
time = 0.25, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5995, 5879, 5915, 5881, 3389, 2211, 2236, 2235} \begin {gather*} -\frac {3 \sqrt {\pi } b^{3/2} e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {3 \sqrt {\pi } b^{3/2} e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}-\frac {3 b \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c + d*x])^(3/2),x]

[Out]

(-3*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(2*d) + ((c + d*x)*(a + b*ArcCosh[c +
d*x])^(3/2))/d - (3*b^(3/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(8*d) + (3*b^(3/2)*Sq
rt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(8*d*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 3389

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

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 5881

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Sinh[-a/b + x/b], x], x
, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

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 5995

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcCosh[x])^n, x
], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

Rubi steps

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

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Mathematica [A]
time = 0.56, size = 290, normalized size = 1.85 \begin {gather*} \frac {-12 b \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}+8 b (c+d x) \cosh ^{-1}(c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}+4 a e^{-\frac {a}{b}} \sqrt {a+b \cosh ^{-1}(c+d x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{\sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \cosh ^{-1}(c+d x)}{b}}}\right )+\sqrt {b} (2 a+3 b) \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(2 a-3 b) \sqrt {b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )}{8 d} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c + d*x])^(3/2),x]

[Out]

(-12*b*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]] + 8*b*(c + d*x)*ArcCosh[c
+ d*x]*Sqrt[a + b*ArcCosh[c + d*x]] + (4*a*Sqrt[a + b*ArcCosh[c + d*x]]*((E^((2*a)/b)*Gamma[3/2, a/b + ArcCos
h[c + d*x]])/Sqrt[a/b + ArcCosh[c + d*x]] + Gamma[3/2, -((a + b*ArcCosh[c + d*x])/b)]/Sqrt[-((a + b*ArcCosh[c
+ d*x])/b)]))/E^(a/b) + Sqrt[b]*(2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - S
inh[a/b]) + (2*a - 3*b)*Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]))/(8
*d)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {3}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(d*x+c))^(3/2),x)

[Out]

int((a+b*arccosh(d*x+c))^(3/2),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(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*arccosh(d*x + c) + a)^(3/2), 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(d*x+c))^(3/2),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 + d x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(d*x+c))**(3/2),x)

[Out]

Integral((a + b*acosh(c + d*x))**(3/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(d*x + c) + a)^(3/2), x)

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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+d\,x\right )\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c + d*x))^(3/2),x)

[Out]

int((a + b*acosh(c + d*x))^(3/2), x)

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