Optimal. Leaf size=478 \[ -\frac {3 b f^2 g x \sqrt {-1+c x} \sqrt {1+c x}}{c \sqrt {d-c^2 d x^2}}-\frac {2 b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{4 c \sqrt {d-c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{9 c \sqrt {d-c^2 d x^2}}-\frac {3 f^2 g (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {2 g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}+\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.85, antiderivative size = 478, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {5972, 5975,
5893, 5915, 8, 5939, 30} \begin {gather*} \frac {f^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {3 f^2 g (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt {d-c^2 d x^2}}-\frac {2 g^3 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt {d-c^2 d x^2}}+\frac {3 f g^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d-c^2 d x^2}}-\frac {3 b f^2 g x \sqrt {c x-1} \sqrt {c x+1}}{c \sqrt {d-c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{4 c \sqrt {d-c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {c x-1} \sqrt {c x+1}}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 b g^3 x \sqrt {c x-1} \sqrt {c x+1}}{3 c^3 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 30
Rule 5893
Rule 5915
Rule 5939
Rule 5972
Rule 5975
Rubi steps
\begin {align*} \int \frac {(f+g x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {(f+g x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \left (\frac {f^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 f^2 g x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 f g^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {g^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\left (f^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}+\frac {\left (3 f^2 g \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}+\frac {\left (3 f g^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}+\frac {\left (g^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {3 f^2 g (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {\left (3 b f^2 g \sqrt {-1+c x} \sqrt {1+c x}\right ) \int 1 \, dx}{c \sqrt {d-c^2 d x^2}}+\frac {\left (3 f g^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (3 b f g^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x \, dx}{2 c \sqrt {d-c^2 d x^2}}+\frac {\left (2 g^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b g^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^2 \, dx}{3 c \sqrt {d-c^2 d x^2}}\\ &=-\frac {3 b f^2 g x \sqrt {-1+c x} \sqrt {1+c x}}{c \sqrt {d-c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{4 c \sqrt {d-c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{9 c \sqrt {d-c^2 d x^2}}-\frac {3 f^2 g (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {2 g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}+\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b g^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int 1 \, dx}{3 c^3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {3 b f^2 g x \sqrt {-1+c x} \sqrt {1+c x}}{c \sqrt {d-c^2 d x^2}}-\frac {2 b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{4 c \sqrt {d-c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{9 c \sqrt {d-c^2 d x^2}}-\frac {3 f^2 g (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {2 g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}+\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d-c^2 d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.15, size = 371, normalized size = 0.78 \begin {gather*} \frac {18 b c \sqrt {d} f \left (2 c^2 f^2+3 g^2\right ) (-1+c x) \cosh ^{-1}(c x)^2-4 \left (\sqrt {d} g \left (-1+c^2 x^2\right ) \left (2 b \left (7 g^2-c g^2 x+c^2 \left (27 f^2+g^2 x^2\right )\right )-3 a \sqrt {\frac {-1+c x}{1+c x}} \left (4 g^2+c^2 \left (18 f^2+9 f g x+2 g^2 x^2\right )\right )\right )+9 a c f \left (2 c^2 f^2+3 g^2\right ) \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )\right )-27 b c \sqrt {d} f g^2 (-1+c x) \cosh \left (2 \cosh ^{-1}(c x)\right )+6 b \sqrt {d} g (-1+c x) \cosh ^{-1}(c x) \left (4 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (2 g^2+c^2 \left (9 f^2+g^2 x^2\right )\right )+9 c f g \sinh \left (2 \cosh ^{-1}(c x)\right )\right )}{72 c^4 \sqrt {d} \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 9.06, size = 836, normalized size = 1.75
method | result | size |
default | \(-\frac {a \,g^{3} x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 a \,g^{3} \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}-\frac {3 a f \,g^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {3 a f \,g^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}-\frac {3 a \,f^{2} g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+\frac {a \,f^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, f \mathrm {arccosh}\left (c x \right )^{2} \left (2 c^{2} f^{2}+3 g^{2}\right )}{4 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +1\right ) g^{3} \left (-1+3 \,\mathrm {arccosh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) f \,g^{2} \left (2 \,\mathrm {arccosh}\left (c x \right )-1\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) g \left (4 \,\mathrm {arccosh}\left (c x \right ) c^{2} f^{2}-4 c^{2} f^{2}+\mathrm {arccosh}\left (c x \right ) g^{2}-g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) g \left (4 \,\mathrm {arccosh}\left (c x \right ) c^{2} f^{2}+4 c^{2} f^{2}+\mathrm {arccosh}\left (c x \right ) g^{2}+g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) f \,g^{2} \left (2 \,\mathrm {arccosh}\left (c x \right )+1\right )}{16 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -5 c^{2} x^{2}+1\right ) g^{3} \left (1+3 \,\mathrm {arccosh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}\right )\) | \(836\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________