Optimal. Leaf size=440 \[ \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^4 d^2}+\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{4 c^5 d \sqrt {d-c^2 d x^2}}+\frac {b^2 x (1-c x) (c x+1)}{4 c^4 d \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 1.21, antiderivative size = 451, normalized size of antiderivative = 1.02, number of steps used = 15, number of rules used = 13, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {5798, 5752, 5759, 5676, 5662, 90, 52, 5766, 5715, 3716, 2190, 2279, 2391} \[ -\frac {b^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {3 x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^4 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {b^2 x (1-c x) (c x+1)}{4 c^4 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{4 c^5 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 52
Rule 90
Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5662
Rule 5676
Rule 5715
Rule 5752
Rule 5759
Rule 5766
Rule 5798
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{\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 )^2}{(-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 )^2}{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 )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{-1+c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{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}{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 {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{-1+c^2 x^2} \, dx}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (3 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x (1-c x) (1+c x)}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{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}{2 c^4 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c^2 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 x (1-c x) (1+c x)}{4 c^4 d \sqrt {d-c^2 d x^2}}+\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{2 c^5 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c^5 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}{2 c^4 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 c^4 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 x (1-c x) (1+c x)}{4 c^4 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{4 c^5 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c^5 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}{2 c^4 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 x (1-c x) (1+c x)}{4 c^4 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{4 c^5 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c^5 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}{2 c^4 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 x (1-c x) (1+c x)}{4 c^4 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{4 c^5 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c^5 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}{2 c^4 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 2.00, size = 343, normalized size = 0.78 \[ \frac {-4 a^2 c \sqrt {d} x \left (c^2 x^2-3\right )+12 a^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )+2 a b \sqrt {d} \left (8 c x \cosh ^{-1}(c x)-\sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (8 \log \left (\sqrt {\frac {c x-1}{c x+1}} (c x+1)\right )+6 \cosh ^{-1}(c x)^2-\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )+b^2 \sqrt {d} \left (8 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {Li}_2\left (e^{-2 \cosh ^{-1}(c x)}\right )+8 c x \cosh ^{-1}(c x)^2-\sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (4 \cosh ^{-1}(c x)^3-2 \cosh ^{-1}(c x) \left (\cosh \left (2 \cosh ^{-1}(c x)\right )-8 \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )\right )+2 \cosh ^{-1}(c x)^2 \left (\sinh \left (2 \cosh ^{-1}(c x)\right )+4\right )+\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )}{8 c^5 d^{3/2} \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname {arcosh}\left (c x\right ) + a^{2} x^{4}\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
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 = 1.01, size = 1141, normalized size = 2.59 \[ \text {result too large to display} \]
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 \[ -\frac {1}{2} \, a^{2} {\left (\frac {x^{3}}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {3 \, x}{\sqrt {-c^{2} d x^{2} + d} c^{4} d} + \frac {3 \, \arcsin \left (c x\right )}{c^{5} d^{\frac {3}{2}}}\right )} + \int \frac {b^{2} x^{4} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} + \frac {2 \, a b x^{4} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^4\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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