Optimal. Leaf size=162 \[ \frac {2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{6 c d \left (d-c^2 d x^2\right )^{3/2}} \]
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Rubi [A] time = 0.27, antiderivative size = 189, normalized size of antiderivative = 1.17, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5713, 5691, 5688, 260, 261} \[ \frac {2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{6 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 260
Rule 261
Rule 5688
Rule 5691
Rule 5713
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{6 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{1-c^2 x^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{6 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 132, normalized size = 0.81 \[ \frac {4 a c^3 x^3-6 a c x-2 b \sqrt {c x-1} \sqrt {c x+1} \left (c^2 x^2-1\right ) \log \left (1-c^2 x^2\right )+2 b c x \left (2 c^2 x^2-3\right ) \cosh ^{-1}(c x)-b \sqrt {c x-1} \sqrt {c x+1}}{6 c d^2 \left (c^2 x^2-1\right ) \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 1073, normalized size = 6.62 \[ \frac {a x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 a x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right )}{3 d^{3} c \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} \left (c x -1\right ) \left (c x +1\right ) x^{5}}{3 \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) d^{3}}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{6} x^{7}}{3 \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) d^{3}}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3} \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, x^{4}}{\left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) d^{3}}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} \mathrm {arccosh}\left (c x \right ) x^{5}}{\left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) d^{3}}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \left (c x -1\right ) \left (c x +1\right ) x^{3}}{3 \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) d^{3}}+\frac {7 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} x^{5}}{3 \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) d^{3}}-\frac {14 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, x^{2}}{3 \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) d^{3}}+\frac {17 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{3 \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) d^{3}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c x -1\right ) \left (c x +1\right ) x}{\left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) d^{3}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2}}{2 \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) d^{3}}-\frac {8 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} x^{3}}{3 \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) d^{3}}+\frac {8 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{3 \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) c \,d^{3}}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x}{\left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) d^{3}}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}}{3 \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) c \,d^{3}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x}{\left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) d^{3}}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )}{3 d^{3} c \left (c^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 157, normalized size = 0.97 \[ \frac {1}{6} \, b c {\left (\frac {\sqrt {-d}}{c^{4} d^{3} x^{2} - c^{2} d^{3}} + \frac {2 \, \sqrt {-d} \log \left (c x + 1\right )}{c^{2} d^{3}} + \frac {2 \, \sqrt {-d} \log \left (c x - 1\right )}{c^{2} d^{3}}\right )} + \frac {1}{3} \, b {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \operatorname {arcosh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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