Optimal. Leaf size=478 \[ \frac {f^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {3 f^2 g (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt {d-c^2 d x^2}}-\frac {2 g^3 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt {d-c^2 d x^2}}+\frac {3 f g^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d-c^2 d x^2}}-\frac {3 b f^2 g x \sqrt {c x-1} \sqrt {c x+1}}{c \sqrt {d-c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{4 c \sqrt {d-c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {c x-1} \sqrt {c x+1}}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 b g^3 x \sqrt {c x-1} \sqrt {c x+1}}{3 c^3 \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 1.29, 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 = {5836, 5822, 5676, 5718, 8, 5759, 30} \[ -\frac {3 f^2 g (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {f^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}+\frac {3 f g^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {2 g^3 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt {d-c^2 d x^2}}-\frac {g^3 x^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt {d-c^2 d x^2}}-\frac {3 b f^2 g x \sqrt {c x-1} \sqrt {c x+1}}{c \sqrt {d-c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{4 c \sqrt {d-c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {c x-1} \sqrt {c x+1}}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 b g^3 x \sqrt {c x-1} \sqrt {c x+1}}{3 c^3 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5676
Rule 5718
Rule 5759
Rule 5822
Rule 5836
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*}
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Mathematica [A] time = 2.09, size = 405, normalized size = 0.85 \[ \frac {-\frac {36 a c f \left (2 c^2 f^2+3 g^2\right ) \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )}{\sqrt {d}}-\frac {12 a \sqrt {d-c^2 d x^2} \left (c^2 g \left (18 f^2+9 f g x+2 g^2 x^2\right )+4 g^3\right )}{d}+\frac {216 b c^2 f^2 g \sqrt {d-c^2 d x^2} \left (c x-\sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)\right )}{d \sqrt {c x-1} \sqrt {c x+1}}+\frac {27 b c f g^2 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+\sinh \left (2 \cosh ^{-1}(c x)\right )\right )-\cosh \left (2 \cosh ^{-1}(c x)\right )\right )}{\sqrt {d-c^2 d x^2}}+\frac {8 b g^3 \sqrt {d-c^2 d x^2} \left (c x \left (c^2 x^2+6\right )-3 \sqrt {c x-1} \sqrt {c x+1} \left (c^2 x^2+2\right ) \cosh ^{-1}(c x)\right )}{d \sqrt {c x-1} \sqrt {c x+1}}+\frac {36 b c^3 f^3 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2}{\sqrt {d-c^2 d x^2}}}{72 c^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a g^{3} x^{3} + 3 \, a f g^{2} x^{2} + 3 \, a f^{2} g x + a f^{3} + {\left (b g^{3} x^{3} + 3 \, b f g^{2} x^{2} + 3 \, b f^{2} g x + b f^{3}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{2} d x^{2} - d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.04, size = 859, normalized size = 1.80 \[ -\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}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, f^{3} \mathrm {arccosh}\left (c x \right )^{2}}{2 d c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3}}{9 c d \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \sqrt {c x +1}\, \sqrt {c x -1}\, x}{3 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{2 d \left (c^{2} x^{2}-1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \mathrm {arccosh}\left (c x \right ) x}{2 d \,c^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{8 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,\mathrm {arccosh}\left (c x \right ) x^{2} f^{2}}{d \left (c^{2} x^{2}-1\right )}-\frac {3 b \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} g^{2}}{4 d \,c^{3} \left (c^{2} x^{2}-1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2}}{4 d c \left (c^{2} x^{2}-1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \sqrt {c x +1}\, \sqrt {c x -1}\, x \,f^{2}}{c d \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \mathrm {arccosh}\left (c x \right )}{3 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \mathrm {arccosh}\left (c x \right ) x^{4}}{3 d \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \mathrm {arccosh}\left (c x \right ) x^{2}}{3 c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,\mathrm {arccosh}\left (c x \right ) f^{2}}{c^{2} d \left (c^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, a g^{3} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} - \frac {3}{2} \, a f g^{2} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x}{c^{2} d} - \frac {\arcsin \left (c x\right )}{c^{3} \sqrt {d}}\right )} + \frac {a f^{3} \arcsin \left (c x\right )}{c \sqrt {d}} - \frac {3 \, \sqrt {-c^{2} d x^{2} + d} a f^{2} g}{c^{2} d} + \int \frac {b g^{3} x^{3} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {-c^{2} d x^{2} + d}} + \frac {3 \, b f g^{2} x^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {-c^{2} d x^{2} + d}} + \frac {3 \, b f^{2} g x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {-c^{2} d x^{2} + d}} + \frac {b f^{3} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {-c^{2} d x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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