Optimal. Leaf size=459 \[ -\frac {(c f-g)^2 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^2 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^2 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (\sqrt {-\frac {1-c x}{1+c x}}\right )}{c^3 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {b (c f-g)^2 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{1+c x}\right )}{2 c^3 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {b (c f+g)^2 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{1+c x}\right )}{2 c^3 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.83, antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 13, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {5972, 5981,
37, 5987, 12, 1986, 15, 266, 272, 36, 31, 29, 5893} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 15
Rule 29
Rule 31
Rule 36
Rule 37
Rule 266
Rule 272
Rule 1986
Rule 5893
Rule 5972
Rule 5981
Rule 5987
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.62, size = 281, normalized size = 0.61 \begin {gather*} \frac {2 b c \sqrt {d} \left (2 f g+c^2 f^2 x+g^2 x\right ) \cosh ^{-1}(c x)-b \sqrt {d} g^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)^2+2 a g^2 \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 )+2 \sqrt {d} \left (a c \left (2 f g+c^2 f^2 x+g^2 x\right )-b \left (c^2 f^2+g^2\right ) \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )-2 b c f g \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )\right )}{2 c^3 d^{3/2} \sqrt {d-c^2 d x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(878\) vs.
\(2(404)=808\).
time = 8.74, size = 879, normalized size = 1.92
method | result | size |
default | \(\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 (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 )}+\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 )}\) | \(879\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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 )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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