3.73 \(\int \frac{(f+g x) (a+b \cosh ^{-1}(c x))}{(d-c^2 d x^2)^{3/2}} \, dx\)

Optimal. Leaf size=142 \[ \frac{\left (c^2 f x+g\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{c x-1} \sqrt{c x+1} (c f-g) \tanh ^{-1}(c x)}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{b f \sqrt{c x-1} \sqrt{c x+1} \log (1-c x)}{c d \sqrt{d-c^2 d x^2}} \]

[Out]

((g + c^2*f*x)*(a + b*ArcCosh[c*x]))/(c^2*d*Sqrt[d - c^2*d*x^2]) - (b*(c*f - g)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*A
rcTanh[c*x])/(c^2*d*Sqrt[d - c^2*d*x^2]) - (b*f*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c*x])/(c*d*Sqrt[d - c^2*d
*x^2])

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Rubi [A]  time = 0.306586, antiderivative size = 178, normalized size of antiderivative = 1.25, number of steps used = 5, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {5836, 78, 37, 5820, 35, 206} \[ -\frac{(c f-g) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{f (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{c x-1} \sqrt{c x+1} (c f-g) \tanh ^{-1}(c x)}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{b f \sqrt{c x-1} \sqrt{c x+1} \log (1-c x)}{c d \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((c*f - g)*(a + b*ArcCosh[c*x]))/(c^2*d*Sqrt[d - c^2*d*x^2])) + (f*(1 + c*x)*(a + b*ArcCosh[c*x]))/(c*d*Sqrt
[d - c^2*d*x^2]) - (b*(c*f - g)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(c^2*d*Sqrt[d - c^2*d*x^2]) - (b*f*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c*x])/(c*d*Sqrt[d - c^2*d*x^2])

Rule 5836

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Dist[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f + g*x
)^m*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d
+ e, 0] && IntegerQ[m] && IntegerQ[p - 1/2]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 5820

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x
_))^(m_.), x_Symbol] :> With[{u = IntHide[(f + g*x)^m*(d1 + e1*x)^p*(d2 + e2*x)^p, x]}, Dist[a + b*ArcCosh[c*x
], u, x] - Dist[b*c, Int[Dist[1/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), u, x], x], x]] /; FreeQ[{a, b, c, d1, e1, d2,
e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ
[d2, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])

Rule 35

Int[1/(((a_) + (b_.)*(x_))*((c_) + (d_.)*(x_))), x_Symbol] :> Int[1/(a*c + b*d*x^2), x] /; FreeQ[{a, b, c, d},
x] && EqQ[b*c + a*d, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(f+g x) \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{(f+g x) \left (a+b \cosh ^{-1}(c x)\right )}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{(c f-g) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{f (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}}+\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \left (\frac{f}{c (1-c x)}-\frac{c f-g}{c^2 (1-c x) (1+c x)}\right ) \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{(c f-g) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{f (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}}-\frac{b f \sqrt{-1+c x} \sqrt{1+c x} \log (1-c x)}{c d \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f-g) \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{(1-c x) (1+c x)} \, dx}{c d \sqrt{d-c^2 d x^2}}\\ &=-\frac{(c f-g) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{f (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}}-\frac{b f \sqrt{-1+c x} \sqrt{1+c x} \log (1-c x)}{c d \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f-g) \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{1-c^2 x^2} \, dx}{c d \sqrt{d-c^2 d x^2}}\\ &=-\frac{(c f-g) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{f (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}}-\frac{b (c f-g) \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{b f \sqrt{-1+c x} \sqrt{1+c x} \log (1-c x)}{c d \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.375863, size = 123, normalized size = 0.87 \[ \frac{b \sqrt{d-c^2 d x^2} ((c f+g) \log (1-c x)+(c f-g) \log (c x+1))}{2 c^2 d^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{d-c^2 d x^2} \left (c^2 f x+g\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d^2 \left (c^2 x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((g + c^2*f*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(c^2*d^2*(-1 + c^2*x^2))) + (b*Sqrt[d - c^2*d*x^2]*
((c*f + g)*Log[1 - c*x] + (c*f - g)*Log[1 + c*x]))/(2*c^2*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Maple [B]  time = 0.227, size = 498, normalized size = 3.5 \begin{align*}{\frac{ag}{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}+{\frac{afx}{d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}-{\frac{bf{\rm arccosh} \left (cx\right )}{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}+{\frac{b{\rm arccosh} \left (cx\right ) \left ( cx+1 \right ) \left ( cx-1 \right ) g}{{c}^{2}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b{\rm arccosh} \left (cx\right ){x}^{2}g}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b{\rm arccosh} \left (cx\right )xf}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{bf}{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{bg}{{c}^{2}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{bf}{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{cx-1}\sqrt{cx+1}-1 \right ) }+{\frac{bg}{{c}^{2}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{cx-1}\sqrt{cx+1}-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*g/c^2/d/(-c^2*d*x^2+d)^(1/2)+a*f/d*x/(-c^2*d*x^2+d)^(1/2)-b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/
2)/d^2/c/(c^2*x^2-1)*f*arccosh(c*x)+b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/c^2/d^2/(c^2*x^2-1)*(c*x+1)*(c*x-1)*
g-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/(c^2*x^2-1)*x^2*g-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/(c^2*x
^2-1)*x*f+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d^2/(c^
2*x^2-1)*f-b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^2/d^2/
(c^2*x^2-1)*g+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/d^2/(c^2*x^2-1)*ln(c*x+(c*x-1)^(1/2)*(c*x
+1)^(1/2)-1)*f+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^2/d^2/(c^2*x^2-1)*ln(c*x+(c*x-1)^(1/2)*(
c*x+1)^(1/2)-1)*g

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b c f \sqrt{-\frac{1}{c^{4} d}} \log \left (x^{2} - \frac{1}{c^{2}}\right )}{2 \, d} + b g{\left (\frac{\frac{{\left (c \sqrt{d} x + \sqrt{c x + 1} \sqrt{c x - 1} \sqrt{d}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{\sqrt{-c x + 1}} + \frac{\sqrt{c x + 1} \sqrt{c x - 1} \sqrt{d}}{\sqrt{-c x + 1}}}{\sqrt{c x + 1} c^{3} d^{2} x +{\left (c x + 1\right )} \sqrt{c x - 1} c^{2} d^{2}} - \int \frac{c^{2} x^{3} + c x^{2} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (c x - 1\right )\right )} - x}{\sqrt{-c x + 1}{\left ({\left (c^{2} d^{\frac{3}{2}} x^{2} - d^{\frac{3}{2}}\right )} e^{\left (\frac{3}{2} \, \log \left (c x + 1\right ) + \log \left (c x - 1\right )\right )} + 2 \,{\left (c^{3} d^{\frac{3}{2}} x^{3} - c d^{\frac{3}{2}} x\right )} e^{\left (\log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (c x - 1\right )\right )} +{\left (c^{4} d^{\frac{3}{2}} x^{4} - c^{2} d^{\frac{3}{2}} x^{2}\right )} \sqrt{c x + 1}\right )}}\,{d x}\right )} + \frac{b f x \operatorname{arcosh}\left (c x\right )}{\sqrt{-c^{2} d x^{2} + d} d} + \frac{a f x}{\sqrt{-c^{2} d x^{2} + d} d} + \frac{a g}{\sqrt{-c^{2} d x^{2} + d} c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

-1/2*b*c*f*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2)/d + b*g*(((c*sqrt(d)*x + sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(d))*log
(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c*x + 1) + sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(d)/sqrt(-c*x + 1))/(sqrt
(c*x + 1)*c^3*d^2*x + (c*x + 1)*sqrt(c*x - 1)*c^2*d^2) - integrate((c^2*x^3 + c*x^2*e^(1/2*log(c*x + 1) + 1/2*
log(c*x - 1)) - x)/(sqrt(-c*x + 1)*((c^2*d^(3/2)*x^2 - d^(3/2))*e^(3/2*log(c*x + 1) + log(c*x - 1)) + 2*(c^3*d
^(3/2)*x^3 - c*d^(3/2)*x)*e^(log(c*x + 1) + 1/2*log(c*x - 1)) + (c^4*d^(3/2)*x^4 - c^2*d^(3/2)*x^2)*sqrt(c*x +
1))), x)) + b*f*x*arccosh(c*x)/(sqrt(-c^2*d*x^2 + d)*d) + a*f*x/(sqrt(-c^2*d*x^2 + d)*d) + a*g/(sqrt(-c^2*d*x
^2 + d)*c^2*d)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} d x^{2} + d}{\left (a g x + a f +{\left (b g x + b f\right )} \operatorname{arcosh}\left (c x\right )\right )}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(a*g*x + a*f + (b*g*x + b*f)*arccosh(c*x))/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2),
x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acosh}{\left (c x \right )}\right ) \left (f + g x\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acosh(c*x))*(f + g*x)/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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