Optimal. Leaf size=142 \[ \frac {\left (c^2 f x+g\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} (c f-g) \tanh ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b f \sqrt {c x-1} \sqrt {c x+1} \log (1-c x)}{c d \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.31, antiderivative size = 178, normalized size of antiderivative = 1.25, number of steps used = 5, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {5836, 78, 37, 5820, 35, 206} \[ -\frac {(c f-g) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} (c f-g) \tanh ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b f \sqrt {c x-1} \sqrt {c x+1} \log (1-c x)}{c d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 35
Rule 37
Rule 78
Rule 206
Rule 5820
Rule 5836
Rubi steps
\begin {align*} \int \frac {(f+g x) \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) \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 {(c f-g) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \left (\frac {f}{c (1-c x)}-\frac {c f-g}{c^2 (1-c x) (1+c x)}\right ) \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {(c f-g) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {b f \sqrt {-1+c x} \sqrt {1+c x} \log (1-c x)}{c d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g) \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(1-c x) (1+c x)} \, dx}{c d \sqrt {d-c^2 d x^2}}\\ &=-\frac {(c f-g) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {b f \sqrt {-1+c x} \sqrt {1+c x} \log (1-c x)}{c d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g) \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{1-c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}}\\ &=-\frac {(c f-g) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {b (c f-g) \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b f \sqrt {-1+c x} \sqrt {1+c x} \log (1-c x)}{c d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 123, normalized size = 0.87 \[ \frac {b \sqrt {d-c^2 d x^2} ((c f+g) \log (1-c x)+(c f-g) \log (c x+1))}{2 c^2 d^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \left (c^2 f x+g\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d^2 \left (c^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.25, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (a g x + a f + {\left (b g x + b f\right )} \operatorname {arcosh}\left (c x\right )\right )}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.66, size = 498, normalized size = 3.51 \[ \frac {a g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {a f x}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, f \,\mathrm {arccosh}\left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \left (c x -1\right ) \left (c x +1\right ) g}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x^{2} g}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x f}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f}{c \,d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right ) f}{c \,d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right ) g}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )} \]
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 \[ -\frac {b c f \sqrt {-\frac {1}{c^{4} d}} \log \left (x^{2} - \frac {1}{c^{2}}\right )}{2 \, d} + b g {\left (\frac {\frac {{\left (c \sqrt {d} x + \sqrt {c x + 1} \sqrt {c x - 1} \sqrt {d}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {-c x + 1}} + \frac {\sqrt {c x + 1} \sqrt {c x - 1} \sqrt {d}}{\sqrt {-c x + 1}}}{\sqrt {c x + 1} c^{3} d^{2} x + {\left (c x + 1\right )} \sqrt {c x - 1} c^{2} d^{2}} - \int \frac {c^{2} x^{3} + c x^{2} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )} - x}{\sqrt {-c x + 1} {\left ({\left (c^{2} d^{\frac {3}{2}} x^{2} - d^{\frac {3}{2}}\right )} e^{\left (\frac {3}{2} \, \log \left (c x + 1\right ) + \log \left (c x - 1\right )\right )} + 2 \, {\left (c^{3} d^{\frac {3}{2}} x^{3} - c d^{\frac {3}{2}} x\right )} e^{\left (\log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )} + {\left (c^{4} d^{\frac {3}{2}} x^{4} - c^{2} d^{\frac {3}{2}} x^{2}\right )} \sqrt {c x + 1}\right )}}\,{d x}\right )} + \frac {b f x \operatorname {arcosh}\left (c x\right )}{\sqrt {-c^{2} d x^{2} + d} d} + \frac {a f x}{\sqrt {-c^{2} d x^{2} + d} d} + \frac {a g}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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