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

Optimal. Leaf size=293 \[ \frac{(1-m) (3-m) \text{Unintegrable}\left (\frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2},x\right )}{8 d^2}-\frac{b c (3-m) \sqrt{1-c^2 x^2} (f x)^{m+2} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{8 d^3 f^2 (m+2) \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \sqrt{1-c^2 x^2} (f x)^{m+2} \text{Hypergeometric2F1}\left (\frac{5}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{4 d^3 f^2 (m+2) \sqrt{c x-1} \sqrt{c x+1}}+\frac{(3-m) (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 f \left (1-c^2 x^2\right )}+\frac{(f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 f \left (1-c^2 x^2\right )^2} \]

[Out]

((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(4*d^3*f*(1 - c^2*x^2)^2) + ((3 - m)*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/
(8*d^3*f*(1 - c^2*x^2)) - (b*c*(3 - m)*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[3/2, (2 + m)/2, (4 +
m)/2, c^2*x^2])/(8*d^3*f^2*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hyperg
eometric2F1[5/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(4*d^3*f^2*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((1 - m)*(
3 - m)*Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2), x])/(8*d^2)

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Rubi [A]  time = 0.335463, 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 = {} \[ \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(4*d^3*f*(1 - c^2*x^2)^2) + ((3 - m)*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/
(8*d^3*f*(1 - c^2*x^2)) - (b*c*(3 - m)*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[3/2, (2 + m)/2, (4 +
m)/2, c^2*x^2])/(8*d^3*f^2*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hyperg
eometric2F1[5/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(4*d^3*f^2*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((1 - m)*(
3 - m)*Defer[Int][((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2), x])/(8*d^2)

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 )^3} \, dx &=\frac{(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 f \left (1-c^2 x^2\right )^2}-\frac{(b c) \int \frac{(f x)^{1+m}}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 d^3 f}+\frac{(3-m) \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d}\\ &=\frac{(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 f \left (1-c^2 x^2\right )^2}+\frac{(3-m) (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 f \left (1-c^2 x^2\right )}+\frac{(b c (3-m)) \int \frac{(f x)^{1+m}}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{8 d^3 f}+\frac{((1-m) (3-m)) \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{8 d^2}-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{(f x)^{1+m}}{\left (-1+c^2 x^2\right )^{5/2}} \, dx}{4 d^3 f \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 f \left (1-c^2 x^2\right )^2}+\frac{(3-m) (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 f \left (1-c^2 x^2\right )}+\frac{((1-m) (3-m)) \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{8 d^2}-\frac{\left (b c \sqrt{1-c^2 x^2}\right ) \int \frac{(f x)^{1+m}}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{4 d^3 f \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c (3-m) \sqrt{-1+c^2 x^2}\right ) \int \frac{(f x)^{1+m}}{\left (-1+c^2 x^2\right )^{3/2}} \, dx}{8 d^3 f \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 f \left (1-c^2 x^2\right )^2}+\frac{(3-m) (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 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{5}{2},\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{4 d^3 f^2 (2+m) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{((1-m) (3-m)) \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{8 d^2}-\frac{\left (b c (3-m) \sqrt{1-c^2 x^2}\right ) \int \frac{(f x)^{1+m}}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{8 d^3 f \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 f \left (1-c^2 x^2\right )^2}+\frac{(3-m) (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 f \left (1-c^2 x^2\right )}-\frac{b c (3-m) (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 )}{8 d^3 f^2 (2+m) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c (f x)^{2+m} \sqrt{1-c^2 x^2} \, _2F_1\left (\frac{5}{2},\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{4 d^3 f^2 (2+m) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{((1-m) (3-m)) \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{8 d^2}\\ \end{align*}

Mathematica [A]  time = 6.52498, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.569, 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 ) ^{3}}}\, 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)^3,x)

[Out]

int((f*x)^m*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^3,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 )}^{3}}\,{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)^3,x, algorithm="maxima")

[Out]

-integrate((b*arccosh(c*x) + a)*(f*x)^m/(c^2*d*x^2 - d)^3, 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^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, 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)^3,x, algorithm="fricas")

[Out]

integral(-(b*arccosh(c*x) + a)*(f*x)^m/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**3,x)

[Out]

Timed out

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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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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