\(\int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx\) [235]

Optimal result

Integrand size = 31, antiderivative size = 31 \[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {2 b c (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f (2+m)}-\frac {2 b^2 c^2 (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{f^3 (2+m)^2 (3+m) (1-c x) (1+c x)}+\frac {d \text {Int}\left (\frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}},x\right )}{2+m} \]

[Out]

(f*x)^(1+m)*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/f/(2+m)-2*b*c*(f*x)^(2+m)*(a+b*arccosh(c*x))*(-c^2*d*x^2
+d)^(1/2)/f^2/(2+m)^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2*b^2*c^2*(f*x)^(3+m)*hypergeom([1/2, 3/2+1/2*m],[5/2+1/2*m]
,c^2*x^2)*(-c^2*x^2+1)^(1/2)*(-c^2*d*x^2+d)^(1/2)/f^3/(2+m)^2/(3+m)/(-c*x+1)/(c*x+1)+d*Unintegrable((f*x)^m*(a
+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x)/(2+m)

Rubi [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 31, 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 = {} \[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f (2+m)}+\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{2+m}-\frac {\left (2 b c \sqrt {d-c^2 d x^2}\right ) \int (f x)^{1+m} (a+b \text {arccosh}(c x)) \, dx}{f (2+m) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {2 b c (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f (2+m)}+\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{2+m}+\frac {\left (2 b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m}}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{f^2 (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {2 b c (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f (2+m)}+\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{2+m}+\frac {\left (2 b^2 c^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m}}{\sqrt {-1+c^2 x^2}} \, dx}{f^2 (2+m)^2 (-1+c x) (1+c x)} \\ & = -\frac {2 b c (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f (2+m)}+\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{2+m}+\frac {\left (2 b^2 c^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^{2+m}}{\sqrt {1-c^2 x^2}} \, dx}{f^2 (2+m)^2 (-1+c x) (1+c x)} \\ & = -\frac {2 b c (f x)^{2+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f^2 (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{f (2+m)}-\frac {2 b^2 c^2 (f x)^{3+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{f^3 (2+m)^2 (3+m) (1-c x) (1+c x)}+\frac {d \int \frac {(f x)^m (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{2+m} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.76 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 3.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94

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

[In]

int((f*x)^m*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2,x)

[Out]

int((f*x)^m*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2,x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{m} \,d x } \]

[In]

integrate((f*x)^m*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 72.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int \left (f x\right )^{m} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{m} \,d x } \]

[In]

integrate((f*x)^m*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((f*x)^m*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 3.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2}\,{\left (f\,x\right )}^m \,d x \]

[In]

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

[Out]

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