3.3.2 \(\int \frac {a+b \cosh ^{-1}(c+d x)}{\sqrt {c e+d e x}} \, dx\) [202]

Optimal. Leaf size=104 \[ \frac {2 \sqrt {e (c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e}-\frac {4 b \sqrt {1-c-d x} \sqrt {e (c+d x)} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {1+c+d x}}{\sqrt {2}}\right )\right |2\right )}{d e \sqrt {-c-d x} \sqrt {-1+c+d x}} \]

[Out]

2*(a+b*arccosh(d*x+c))*(e*(d*x+c))^(1/2)/d/e-4*b*EllipticE(1/2*(d*x+c+1)^(1/2)*2^(1/2),2^(1/2))*(-d*x-c+1)^(1/
2)*(e*(d*x+c))^(1/2)/d/e/(-d*x-c)^(1/2)/(d*x+c-1)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5996, 5883, 115, 114} \begin {gather*} \frac {2 \sqrt {e (c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e}-\frac {4 b \sqrt {-c-d x+1} \sqrt {e (c+d x)} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {c+d x+1}}{\sqrt {2}}\right )\right |2\right )}{d e \sqrt {-c-d x} \sqrt {c+d x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e*(c + d*x)]*(a + b*ArcCosh[c + d*x]))/(d*e) - (4*b*Sqrt[1 - c - d*x]*Sqrt[e*(c + d*x)]*EllipticE[ArcS
in[Sqrt[1 + c + d*x]/Sqrt[2]], 2])/(d*e*Sqrt[-c - d*x]*Sqrt[-1 + c + d*x])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*Rt[-(b
*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /;
FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-(b*c - a*d)/d, 0] &&
!(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])

Rule 115

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[e + f*x
]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d*x]*Sqrt[b*((e + f*x)/(b*e - a*f))])), Int[Sqrt[b*(e/(b*e - a*f)
) + b*f*(x/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]

Rule 5883

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcC
osh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

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 \frac {a+b \cosh ^{-1}(c+d x)}{\sqrt {c e+d e x}} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{\sqrt {e x}} \, dx,x,c+d x\right )}{d}\\ &=\frac {2 \sqrt {e (c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e}-\frac {(2 b) \text {Subst}\left (\int \frac {\sqrt {e x}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d e}\\ &=\frac {2 \sqrt {e (c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e}-\frac {\left (\sqrt {2} b \sqrt {1-c-d x} \sqrt {e (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {-x}}{\sqrt {\frac {1}{2}-\frac {x}{2}} \sqrt {1+x}} \, dx,x,c+d x\right )}{d e \sqrt {-c-d x} \sqrt {-1+c+d x}}\\ &=\frac {2 \sqrt {e (c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e}-\frac {4 b \sqrt {1-c-d x} \sqrt {e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {1+c+d x}}{\sqrt {2}}\right )\right |2\right )}{d e \sqrt {-c-d x} \sqrt {-1+c+d x}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.14, size = 94, normalized size = 0.90 \begin {gather*} \frac {2 \sqrt {e (c+d x)} \left (3 \left (a+b \cosh ^{-1}(c+d x)\right )-\frac {2 b (c+d x) \sqrt {1-(c+d x)^2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};(c+d x)^2\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}\right )}{3 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e*(c + d*x)]*(3*(a + b*ArcCosh[c + d*x]) - (2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Hypergeometric2F1[1/2,
3/4, 7/4, (c + d*x)^2])/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])))/(3*d*e)

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Maple [C] Result contains complex when optimal does not.
time = 0.06, size = 138, normalized size = 1.33

method	result	size
derivativedivides	\(\frac {2 \sqrt {d e x +c e}\, a +2 b \left (\sqrt {d e x +c e}\, \mathrm {arccosh}\left (\frac {d e x +c e}{e}\right )-\frac {2 \left (\EllipticF \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-\EllipticE \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )\right ) \sqrt {\frac {-d e x -c e +e}{e}}}{\sqrt {-\frac {1}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(138\)
default	\(\frac {2 \sqrt {d e x +c e}\, a +2 b \left (\sqrt {d e x +c e}\, \mathrm {arccosh}\left (\frac {d e x +c e}{e}\right )-\frac {2 \left (\EllipticF \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-\EllipticE \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )\right ) \sqrt {\frac {-d e x -c e +e}{e}}}{\sqrt {-\frac {1}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(138\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/e*((d*e*x+c*e)^(1/2)*a+b*((d*e*x+c*e)^(1/2)*arccosh((d*e*x+c*e)/e)-2*(EllipticF((d*e*x+c*e)^(1/2)*(-1/e)^(
1/2),I)-EllipticE((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I))*((-d*e*x-c*e+e)/e)^(1/2)/(-1/e)^(1/2)/(-(-d*e*x-c*e+e)/e)
^(1/2)))

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

[Out]

(2*(d*x*e^(1/2) + c*e^(1/2))*e^(-1)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)/(sqrt(d*x + c)*d) - (I*
(log(I*sqrt(d*x + c) + 1) - log(-I*sqrt(d*x + c) + 1))*e^(-1/2) - e^(-1/2)*log(sqrt(d*x + c) + 1) + e^(-1/2)*l
og(sqrt(d*x + c) - 1) + 4*e^(1/2*log(d*x + c) - 1/2))/d + integrate(2*(d*x*e^(1/2) + c*e^(1/2))/((d^3*x^3*e +
3*c*d^2*x^2*e + (3*c^2*d - d)*x*e + (d^2*x^2*e + 2*c*d*x*e + (c^2 - 1)*e)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1)
+ (c^3 - c)*e)*sqrt(d*x + c)), x))*b + 2*sqrt(d*x*e + c*e)*a*e^(-1)/d

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 128, normalized size = 1.23 \begin {gather*} \frac {2 \, {\left (\sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )} b d \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )} a d + 2 \, \sqrt {d^{3} \cosh \left (1\right ) + d^{3} \sinh \left (1\right )} b {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )\right )}}{d^{2} \cosh \left (1\right ) + d^{2} \sinh \left (1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(sqrt((d*x + c)*cosh(1) + (d*x + c)*sinh(1))*b*d*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) + sqrt((d*
x + c)*cosh(1) + (d*x + c)*sinh(1))*a*d + 2*sqrt(d^3*cosh(1) + d^3*sinh(1))*b*weierstrassZeta(4/d^2, 0, weiers
trassPInverse(4/d^2, 0, (d*x + c)/d)))/(d^2*cosh(1) + d^2*sinh(1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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