Optimal. Leaf size=459 \[ -\frac {(1-c x) (c f-g)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {(c x+1) (c f+g)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f-g)^2 \log \left (\frac {2}{c x+1}\right )}{2 c^3 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f+g)^2 \log \left (\sqrt {-\frac {1-c x}{c x+1}}\right )}{c^3 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f+g)^2 \log \left (\frac {2}{c x+1}\right )}{2 c^3 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 1.27, antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {5836, 5834, 37, 5848, 12, 6719, 260, 266, 36, 31, 29, 5676} \[ -\frac {(1-c x) (c f-g)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {(c x+1) (c f+g)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f-g)^2 \log \left (\frac {2}{c x+1}\right )}{2 c^3 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f+g)^2 \log \left (\sqrt {-\frac {1-c x}{c x+1}}\right )}{c^3 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f+g)^2 \log \left (\frac {2}{c x+1}\right )}{2 c^3 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 31
Rule 36
Rule 37
Rule 260
Rule 266
Rule 5676
Rule 5834
Rule 5836
Rule 5848
Rule 6719
Rubi steps
\begin {align*} \int \frac {(f+g x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {(f+g x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \left (-\frac {(c f-g)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {-1+c x} (1+c x)^{3/2}}+\frac {(c f+g)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 (-1+c x)^{3/2} \sqrt {1+c x}}+\frac {g^2 \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\left ((c f-g)^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{2 c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (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}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left ((c f+g)^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{(-1+c x)^{3/2} \sqrt {1+c x}} \, dx}{2 c^2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {(c f-g)^2 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^2 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g)^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {\frac {-1+c x}{1+c x}}}{c \sqrt {1-c^2 x^2}} \, dx}{2 c d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{c \sqrt {\frac {-1+c x}{1+c x}} \sqrt {1-c^2 x^2}} \, dx}{2 c d \sqrt {d-c^2 d x^2}}\\ &=-\frac {(c f-g)^2 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^2 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g)^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {\frac {-1+c x}{1+c x}}}{\sqrt {1-c^2 x^2}} \, dx}{2 c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {\frac {-1+c x}{1+c x}} \sqrt {1-c^2 x^2}} \, dx}{2 c^2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {(c f-g)^2 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^2 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (b (c f-g)^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \sqrt {-\frac {x^2}{\left (-1+x^2\right )^2}} \, dx,x,\sqrt {\frac {-1+c x}{1+c x}}\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (b (c f+g)^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-\frac {x^2}{\left (-1+x^2\right )^2}}}{x^2} \, dx,x,\sqrt {\frac {-1+c x}{1+c x}}\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {(c f-g)^2 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^2 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g)^2 \sqrt {-(-1+c x) (1+c x)} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{-1+x^2} \, dx,x,\sqrt {\frac {-1+c x}{1+c x}}\right )}{c^3 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^2 \sqrt {-(-1+c x) (1+c x)} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (-1+x^2\right )} \, dx,x,\sqrt {\frac {-1+c x}{1+c x}}\right )}{c^3 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}\\ &=-\frac {(c f-g)^2 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^2 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^2 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^3 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^2 \sqrt {-(-1+c x) (1+c x)} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) x} \, dx,x,\frac {-1+c x}{1+c x}\right )}{2 c^3 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}\\ &=-\frac {(c f-g)^2 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^2 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^2 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^3 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^2 \sqrt {-(-1+c x) (1+c x)} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,\frac {-1+c x}{1+c x}\right )}{2 c^3 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (b (c f+g)^2 \sqrt {-(-1+c x) (1+c x)} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\frac {-1+c x}{1+c x}\right )}{2 c^3 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}\\ &=-\frac {(c f-g)^2 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^2 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^2 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (-\frac {1-c x}{1+c x}\right )}{2 c^3 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^2 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^3 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^2 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^3 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 1.08, size = 281, normalized size = 0.61 \[ \frac {2 \sqrt {d} \left (a c \left (c^2 f^2 x+2 f g+g^2 x\right )-b \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (c^2 f^2+g^2\right ) \log \left (\sqrt {\frac {c x-1}{c x+1}} (c x+1)\right )-2 b c f g \sqrt {\frac {c x-1}{c x+1}} (c x+1) \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )\right )+2 a g^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )+2 b c \sqrt {d} \cosh ^{-1}(c x) \left (c^2 f^2 x+2 f g+g^2 x\right )-b \sqrt {d} g^2 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2}{2 c^3 d^{3/2} \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (a g^{2} x^{2} + 2 \, a f g x + a f^{2} + {\left (b g^{2} x^{2} + 2 \, b f g x + b f^{2}\right )} \operatorname {arcosh}\left (c x\right )\right )}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (g x + f\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.87, size = 879, normalized size = 1.92 \[ \frac {a \,g^{2} x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \,g^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{2} d \sqrt {c^{2} d}}+\frac {2 a f g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {a \,f^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, g^{2} \mathrm {arccosh}\left (c x \right )^{2}}{2 d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) f^{2}}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) g^{2}}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \left (c x +1\right ) \left (c x -1\right ) f g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x^{2} f g}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x \,f^{2}}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x \,g^{2}}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right ) f^{2}}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right ) f g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right ) g^{2}}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f^{2}}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g^{2}}{d^{2} c^{3} \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 \[ -\frac {b c f^{2} \sqrt {-\frac {1}{c^{4} d}} \log \left (x^{2} - \frac {1}{c^{2}}\right )}{2 \, d} + a g^{2} {\left (\frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {\arcsin \left (c x\right )}{c^{3} d^{\frac {3}{2}}}\right )} + \frac {b f^{2} x \operatorname {arcosh}\left (c x\right )}{\sqrt {-c^{2} d x^{2} + d} d} + \frac {a f^{2} x}{\sqrt {-c^{2} d x^{2} + d} d} + \frac {2 \, a f g}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} + \int \frac {b g^{2} x^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} + \frac {2 \, b f g x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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