\(\int \frac {(f x)^m (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^2} \, dx\) [149]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{2 d^2 f \left (1-c^2 x^2\right )}-\frac {b c (f x)^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{2 d^2 f^2 (2+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(1-m) \text {Int}\left (\frac {(f x)^m (a+b \text {arccosh}(c x))}{d-c^2 d x^2},x\right )}{2 d} \]

[Out]

1/2*(f*x)^(1+m)*(a+b*arccosh(c*x))/d^2/f/(-c^2*x^2+1)-1/2*b*c*(f*x)^(2+m)*hypergeom([3/2, 1+1/2*m],[2+1/2*m],c
^2*x^2)*(-c^2*x^2+1)^(1/2)/d^2/f^2/(2+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/2*(1-m)*Unintegrable((f*x)^m*(a+b*arcco
sh(c*x))/(-c^2*d*x^2+d),x)/d

Rubi [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx \]

[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*} \text {integral}& = \frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{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 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx}{2 d} \\ & = \frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{2 d^2 f \left (1-c^2 x^2\right )}+\frac {(1-m) \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{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} (a+b \text {arccosh}(c x))}{2 d^2 f \left (1-c^2 x^2\right )}+\frac {(1-m) \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{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} (a+b \text {arccosh}(c x))}{2 d^2 f \left (1-c^2 x^2\right )}-\frac {b c (f x)^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\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 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx}{2 d} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 9.49 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 1.35 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

\[\int \frac {\left (f x \right )^{m} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{\left (-c^{2} d \,x^{2}+d \right )^{2}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 64.46 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.15 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a \left (f x\right )^{m}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]

[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

Maxima [N/A]

Not integrable

Time = 0.76 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 3.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]

[In]

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

[Out]

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