3.3.3 \(\int \frac {a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^{3/2}} \, dx\) [203]

Optimal. Leaf size=84 \[ -\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}}+\frac {4 b \sqrt {1-c-d x} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{d e^{3/2} \sqrt {-1+c+d x}} \]

[Out]

4*b*EllipticF((e*(d*x+c))^(1/2)/e^(1/2),I)*(-d*x-c+1)^(1/2)/d/e^(3/2)/(d*x+c-1)^(1/2)-2*(a+b*arccosh(d*x+c))/d
/e/(e*(d*x+c))^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 84, 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, 118, 117} \begin {gather*} \frac {4 b \sqrt {-c-d x+1} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{d e^{3/2} \sqrt {c+d x-1}}-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2/(b*Sqrt[e]))*Rt
[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] &&
GtQ[c, 0] && GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])

Rule 118

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[1 + d*(x/c)]*
(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x])), Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 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)}{(c e+d e x)^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{(e x)^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d e}\\ &=-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}}+\frac {\left (2 b \sqrt {1-c-d x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d e \sqrt {-1+c+d x}}\\ &=-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}}+\frac {4 b \sqrt {1-c-d x} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{d e^{3/2} \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 = 0.11, size = 92, normalized size = 1.10 \begin {gather*} \frac {2 \left (-a-b \cosh ^{-1}(c+d x)+\frac {2 b (c+d x) \sqrt {1-(c+d x)^2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(c+d x)^2\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}\right )}{d e \sqrt {e (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-a - b*ArcCosh[c + d*x] + (2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^
2])/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])))/(d*e*Sqrt[e*(c + d*x)])

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Maple [A]
time = 0.05, size = 119, normalized size = 1.42

method	result	size
derivativedivides	\(\frac {-\frac {2 a}{\sqrt {d e x +c e}}+2 b \left (-\frac {\mathrm {arccosh}\left (\frac {d e x +c e}{e}\right )}{\sqrt {d e x +c e}}+\frac {2 \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) \sqrt {\frac {-d e x -c e +e}{e}}}{e \sqrt {-\frac {1}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(119\)
default	\(\frac {-\frac {2 a}{\sqrt {d e x +c e}}+2 b \left (-\frac {\mathrm {arccosh}\left (\frac {d e x +c e}{e}\right )}{\sqrt {d e x +c e}}+\frac {2 \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) \sqrt {\frac {-d e x -c e +e}{e}}}{e \sqrt {-\frac {1}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(119\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/e*(-a/(d*e*x+c*e)^(1/2)+b*(-1/(d*e*x+c*e)^(1/2)*arccosh((d*e*x+c*e)/e)+2/e*EllipticF((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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.12, size = 169, normalized size = 2.01 \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^{2} \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} - 2 \, \sqrt {d^{3} \cosh \left (1\right ) + d^{3} \sinh \left (1\right )} {\left (b d x + b c\right )} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )}}{{\left (d^{4} x + c d^{3}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (d^{4} x + c d^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (d^{4} x + c d^{3}\right )} \sinh \left (1\right )^{2}} \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)^(3/2),x, algorithm="fricas")

[Out]

-2*(sqrt((d*x + c)*cosh(1) + (d*x + c)*sinh(1))*b*d^2*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 - 2*sqrt(d^3*cosh(1) + d^3*sinh(1))*(b*d*x + b*c)*weierstrassPInv
erse(4/d^2, 0, (d*x + c)/d))/((d^4*x + c*d^3)*cosh(1)^2 + 2*(d^4*x + c*d^3)*cosh(1)*sinh(1) + (d^4*x + c*d^3)*
sinh(1)^2)

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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 )}}{\left (e \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))/(d*e*x+c*e)**(3/2),x)

[Out]

Integral((a + b*acosh(c + d*x))/(e*(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))/(d*e*x+c*e)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(d*x + c) + a)/(d*e*x + c*e)^(3/2), 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 )}{{\left (c\,e+d\,e\,x\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))/(c*e + d*e*x)^(3/2),x)

[Out]

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

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