Optimal. Leaf size=549 \[ -\frac {(1-c x) (c f-g)^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c x+1) (c f+g)^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f-g)^3 \log \left (\frac {2}{c x+1}\right )}{2 c^4 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f+g)^3 \log \left (\sqrt {-\frac {1-c x}{c x+1}}\right )}{c^4 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f+g)^3 \log \left (\frac {2}{c x+1}\right )}{2 c^4 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2}}+\frac {b g^3 x \sqrt {c x-1} \sqrt {c x+1}}{c^3 d \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 1.60, antiderivative size = 549, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 14, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {5836, 5834, 37, 5848, 12, 6719, 260, 266, 36, 31, 29, 5676, 5718, 8} \[ -\frac {3 f g^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {(1-c x) (c f-g)^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c x+1) (c f+g)^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f-g)^3 \log \left (\frac {2}{c x+1}\right )}{2 c^4 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f+g)^3 \log \left (\sqrt {-\frac {1-c x}{c x+1}}\right )}{c^4 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f+g)^3 \log \left (\frac {2}{c x+1}\right )}{2 c^4 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2}}+\frac {b g^3 x \sqrt {c x-1} \sqrt {c x+1}}{c^3 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 29
Rule 31
Rule 36
Rule 37
Rule 260
Rule 266
Rule 5676
Rule 5718
Rule 5834
Rule 5836
Rule 5848
Rule 6719
Rubi steps
\begin {align*} \int \frac {(f+g x)^3 \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)^3 \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)^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 \sqrt {-1+c x} (1+c x)^{3/2}}+\frac {(c f+g)^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 (-1+c x)^{3/2} \sqrt {1+c x}}+\frac {3 f g^2 \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {g^3 x \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)^3 \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^3 d \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}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (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}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left ((c f+g)^3 \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^3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {(c f-g)^3 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \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}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (b g^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int 1 \, dx}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g)^3 \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^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^3 \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^2 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \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}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g)^3 \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^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g)^3 \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^3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \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}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (b (c f-g)^3 \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^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (b (c f+g)^3 \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^4 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \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}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g)^3 \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^4 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)^3 \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^4 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}\\ &=\frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \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}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^4 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)^3 \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^4 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}\\ &=\frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \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}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^4 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)^3 \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^4 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)^3 \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^4 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}\\ &=\frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \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}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^3 \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^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}+\frac {b (c f-g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^4 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 = 1.95, size = 353, normalized size = 0.64 \[ \frac {-2 \sqrt {d} \left (-a \left (c^4 f^3 x+c^2 g \left (3 f^2+3 f g x-g^2 x^2\right )+2 g^3\right )+b c f \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (c^2 f^2+3 g^2\right ) \log \left (\sqrt {\frac {c x-1}{c x+1}} (c x+1)\right )+b g \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (3 c^2 f^2+g^2\right ) \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )\right )+6 a c f g^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )+b \sqrt {d} \cosh ^{-1}(c x) \left (2 c^4 f^3 x+6 c^2 f g (f+g x)-g^3 \cosh \left (2 \cosh ^{-1}(c x)\right )+3 g^3\right )-3 b c \sqrt {d} f g^2 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2+b \sqrt {d} g^3 \sinh \left (2 \cosh ^{-1}(c x)\right )}{2 c^4 d^{3/2} \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a g^{3} x^{3} + 3 \, a f g^{2} x^{2} + 3 \, a f^{2} g x + a f^{3} + {\left (b g^{3} x^{3} + 3 \, b f g^{2} x^{2} + 3 \, b f^{2} g x + b f^{3}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.92, size = 1238, normalized size = 2.26 \[ \text {result too large to display} \]
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 \[ -\frac {b c f^{3} \sqrt {-\frac {1}{c^{4} d}} \log \left (x^{2} - \frac {1}{c^{2}}\right )}{2 \, d} - a g^{3} {\left (\frac {x^{2}}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {2}{\sqrt {-c^{2} d x^{2} + d} c^{4} d}\right )} + 3 \, a f g^{2} {\left (\frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {\arcsin \left (c x\right )}{c^{3} d^{\frac {3}{2}}}\right )} + \frac {b f^{3} x \operatorname {arcosh}\left (c x\right )}{\sqrt {-c^{2} d x^{2} + d} d} + \frac {a f^{3} x}{\sqrt {-c^{2} d x^{2} + d} d} + \frac {3 \, a f^{2} g}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} + \int \frac {b g^{3} x^{3} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} + \frac {3 \, b f g^{2} x^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} + \frac {3 \, b f^{2} g x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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