3.149 \(\int \frac{(f x)^m (a+b \cosh ^{-1}(c x))}{(d-c^2 d x^2)^2} \, dx\)

Optimal. Leaf size=160 \[ \frac{(1-m) \text{Unintegrable}\left (\frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2},x\right )}{2 d}-\frac{b c \sqrt{1-c^2 x^2} (f x)^{m+2} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{2 d^2 f^2 (m+2) \sqrt{c x-1} \sqrt{c x+1}}+\frac{(f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 f \left (1-c^2 x^2\right )} \]

[Out]

((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(2*d^2*f*(1 - c^2*x^2)) - (b*c*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeome
tric2F1[3/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(2*d^2*f^2*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((1 - m)*Unint
egrable[((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2), x])/(2*d)

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Rubi [A]  time = 0.207492, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(2*d^2*f*(1 - c^2*x^2)) - (b*c*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeome
tric2F1[3/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(2*d^2*f^2*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((1 - m)*Defer
[Int][((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2), x])/(2*d)

Rubi steps

\begin{align*} \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac{(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 f \left (1-c^2 x^2\right )}+\frac{(b c) \int \frac{(f x)^{1+m}}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2 f}+\frac{(1-m) \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 d}\\ &=\frac{(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 f \left (1-c^2 x^2\right )}+\frac{(1-m) \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 d}+\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{(f x)^{1+m}}{\left (-1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2 f \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 f \left (1-c^2 x^2\right )}+\frac{(1-m) \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 d}-\frac{\left (b c \sqrt{1-c^2 x^2}\right ) \int \frac{(f x)^{1+m}}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2 f \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 f \left (1-c^2 x^2\right )}-\frac{b c (f x)^{2+m} \sqrt{1-c^2 x^2} \, _2F_1\left (\frac{3}{2},\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{2 d^2 f^2 (2+m) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{(1-m) \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 d}\\ \end{align*}

Mathematica [A]  time = 5.89835, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.562, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{m} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) }{ \left ( -{c}^{2}d{x}^{2}+d \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{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((f*x)^m*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a \left (f x\right )^{m}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac{b \left (f x\right )^{m} \operatorname{acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a*(f*x)**m/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b*(f*x)**m*acosh(c*x)/(c**4*x**4 - 2*c**2*x*
*2 + 1), x))/d**2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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