Optimal. Leaf size=226 \[ \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^5 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d^2}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{2 c^5 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1}}{4 c^3 d \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.68, antiderivative size = 237, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5798, 5752, 5759, 5676, 30, 266, 43} \[ \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {3 x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^5 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1}}{4 c^3 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{2 c^5 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 43
Rule 266
Rule 5676
Rule 5752
Rule 5759
Rule 5798
Rubi steps
\begin {align*} \int \frac {x^4 \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 {x^4 \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 {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 \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 (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{-1+c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}}\\ &=\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (3 \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}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (3 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x \, dx}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {x}{-1+c^2 x} \, dx,x,x^2\right )}{2 c d \sqrt {d-c^2 d x^2}}\\ &=\frac {3 b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{4 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^5 d \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}+\frac {1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{2 c d \sqrt {d-c^2 d x^2}}\\ &=\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{4 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^5 d \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 )}{2 c^5 d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 1.48, size = 192, normalized size = 0.85 \[ \frac {-4 a c d x \left (c^2 x^2-3\right )+12 a \sqrt {d} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )+b d \left (8 c x \cosh ^{-1}(c x)-\sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (8 \log \left (\sqrt {\frac {c x-1}{c x+1}} (c x+1)\right )+6 \cosh ^{-1}(c x)^2-\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )}{8 c^5 d^2 \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} \operatorname {arcosh}\left (c x\right ) + a x^{4}\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.76, size = 445, normalized size = 1.97 \[ -\frac {a \,x^{3}}{2 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a x}{2 c^{4} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{4} d \sqrt {c^{2} d}}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right )^{2}}{4 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x^{3}}{2 d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2}}{4 d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right )}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x}{2 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}}{8 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )} \]
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 \[ -\frac {1}{2} \, a {\left (\frac {x^{3}}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {3 \, x}{\sqrt {-c^{2} d x^{2} + d} c^{4} d} + \frac {3 \, \arcsin \left (c x\right )}{c^{5} d^{\frac {3}{2}}}\right )} + b \int \frac {x^{4} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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