3.2.57 \(\int (c e+d e x) \sqrt {a+b \cosh ^{-1}(c+d x)} \, dx\) [157]

Optimal. Leaf size=164 \[ -\frac {e \sqrt {a+b \cosh ^{-1}(c+d x)}}{4 d}+\frac {e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}-\frac {\sqrt {b} e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {b} e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d} \]

[Out]

-1/32*e*exp(2*a/b)*erf(2^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/d-1/32*e*erfi(2^(1
/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/d/exp(2*a/b)-1/4*e*(a+b*arccosh(d*x+c))^(1/2)
/d+1/2*e*(d*x+c)^2*(a+b*arccosh(d*x+c))^(1/2)/d

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Rubi [A]
time = 0.35, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5996, 12, 5884, 5953, 3393, 3388, 2211, 2236, 2235} \begin {gather*} -\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}+\frac {e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}-\frac {e \sqrt {a+b \cosh ^{-1}(c+d x)}}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)*Sqrt[a + b*ArcCosh[c + d*x]],x]

[Out]

-1/4*(e*Sqrt[a + b*ArcCosh[c + d*x]])/d + (e*(c + d*x)^2*Sqrt[a + b*ArcCosh[c + d*x]])/(2*d) - (Sqrt[b]*e*E^((
2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(16*d) - (Sqrt[b]*e*Sqrt[Pi/2]*Erfi[(S
qrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(16*d*E^((2*a)/b))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[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])
)

Rule 5884

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCosh[c*x])^n/(
m + 1)), x] - Dist[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]

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]

Rule 5996

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

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(437\) vs. \(2(164)=328\).
time = 1.99, size = 437, normalized size = 2.66 \begin {gather*} \frac {e \left (-32 c (c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}+8 \sqrt {a+b \cosh ^{-1}(c+d x)} \cosh \left (2 \cosh ^{-1}(c+d x)\right )+8 \sqrt {b} c \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )-\sqrt {b} \sqrt {2 \pi } \cosh \left (\frac {2 a}{b}\right ) \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+16 c 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 )-8 \sqrt {b} c \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+8 \sqrt {b} c \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 )+\sqrt {b} \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )-\sqrt {b} \sqrt {2 \pi } \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )\right )}{32 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)*Sqrt[a + b*ArcCosh[c + d*x]],x]

[Out]

(e*(-32*c*(c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]] + 8*Sqrt[a + b*ArcCosh[c + d*x]]*Cosh[2*ArcCosh[c + d*x]] + 8
*Sqrt[b]*c*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]] - Sqrt[b]*Sqrt[2*Pi]*Cosh[(2*a)/b]*Er
fi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]] + (16*c*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*ArcCosh[c + d*x])/b)]/Sqrt[-((a
+ b*ArcCosh[c + d*x])/b)]))/E^(a/b) - 8*Sqrt[b]*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*Sinh[a/
b] + 8*Sqrt[b]*c*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]) + Sqrt[b]*Sqrt[2*P
i]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*Sinh[(2*a)/b] - Sqrt[b]*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a
+ b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b])))/(32*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)*(a+b*arccosh(d*x+c))^(1/2),x)

[Out]

int((d*e*x+c*e)*(a+b*arccosh(d*x+c))^(1/2),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((d*e*x+c*e)*(a+b*arccosh(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate((d*x*e + c*e)*sqrt(b*arccosh(d*x + c) + a), 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((d*e*x+c*e)*(a+b*arccosh(d*x+c))^(1/2),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*} e \left (\int c \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int d x \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(a+b*acosh(d*x+c))**(1/2),x)

[Out]

e*(Integral(c*sqrt(a + b*acosh(c + d*x)), x) + Integral(d*x*sqrt(a + b*acosh(c + d*x)), 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((d*e*x+c*e)*(a+b*arccosh(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)*sqrt(b*arccosh(d*x + c) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)*(a + b*acosh(c + d*x))^(1/2),x)

[Out]

int((c*e + d*e*x)*(a + b*acosh(c + d*x))^(1/2), x)

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