Optimal. Leaf size=556 \[ \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac {4 b \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {16 a b x \sqrt {c x-1} \sqrt {c x+1}}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac {2 b x^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {94 b^2 (1-c x) (c x+1)}{27 c^6 d \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x^2 (1-c x) (c x+1)}{27 c^4 d \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 1.37, antiderivative size = 578, normalized size of antiderivative = 1.04, number of steps used = 23, number of rules used = 14, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {5798, 5752, 5759, 5718, 5654, 74, 5662, 100, 12, 5766, 5694, 4182, 2279, 2391} \[ \frac {2 b^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {16 a b x \sqrt {c x-1} \sqrt {c x+1}}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {4 x^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b x \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {8 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d \sqrt {d-c^2 d x^2}}+\frac {4 b \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x^2 (1-c x) (c x+1)}{27 c^4 d \sqrt {d-c^2 d x^2}}+\frac {94 b^2 (1-c x) (c x+1)}{27 c^6 d \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 74
Rule 100
Rule 2279
Rule 2391
Rule 4182
Rule 5654
Rule 5662
Rule 5694
Rule 5718
Rule 5752
Rule 5759
Rule 5766
Rule 5798
Rubi steps
\begin {align*} \int \frac {x^5 \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^5 \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^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3 \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^4 \left (a+b \cosh ^{-1}(c x)\right )}{-1+c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (8 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 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^2 \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 (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 b^2 x^2 (1-c x) (1+c x)}{9 c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d \sqrt {d-c^2 d x^2}}+\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 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 {a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {\left (16 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c^2 d \sqrt {d-c^2 d x^2}}\\ &=\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 (1-c x) (1+c x)}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x^2 (1-c x) (1+c x)}{27 c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d \sqrt {d-c^2 d x^2}}+\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 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) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \cosh ^{-1}(c x) \, dx}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{27 c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c^4 d \sqrt {d-c^2 d x^2}}\\ &=\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {22 b^2 (1-c x) (1+c x)}{9 c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x^2 (1-c x) (1+c x)}{27 c^4 d \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d \sqrt {d-c^2 d x^2}}+\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d \sqrt {d-c^2 d x^2}}+\frac {4 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^6 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^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^6 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^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{27 c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^4 d \sqrt {d-c^2 d x^2}}\\ &=\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {94 b^2 (1-c x) (1+c x)}{27 c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x^2 (1-c x) (1+c x)}{27 c^4 d \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d \sqrt {d-c^2 d x^2}}+\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d \sqrt {d-c^2 d x^2}}+\frac {4 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^6 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 \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^6 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 \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}\\ &=\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {94 b^2 (1-c x) (1+c x)}{27 c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x^2 (1-c x) (1+c x)}{27 c^4 d \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d \sqrt {d-c^2 d x^2}}+\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d \sqrt {d-c^2 d x^2}}+\frac {4 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 3.84, size = 358, normalized size = 0.64 \[ \frac {-36 a^2 \left (c^4 x^4+4 c^2 x^2-8\right )+3 a b \left (-60 \cosh \left (2 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)-3 \cosh \left (4 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)+135 \cosh ^{-1}(c x)+62 \sinh \left (2 \cosh ^{-1}(c x)\right )+\sinh \left (4 \cosh ^{-1}(c x)\right )-72 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )\right )-b^2 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (216 \text {Li}_2\left (-e^{-\cosh ^{-1}(c x)}\right )-216 \text {Li}_2\left (e^{-\cosh ^{-1}(c x)}\right )+378 \sqrt {\frac {c x-1}{c x+1}} (c x+1)+189 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2-378 c x \cosh ^{-1}(c x)-6 \cosh \left (3 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)+216 \cosh ^{-1}(c x) \log \left (1-e^{-\cosh ^{-1}(c x)}\right )-216 \cosh ^{-1}(c x) \log \left (e^{-\cosh ^{-1}(c x)}+1\right )+9 \cosh ^{-1}(c x)^2 \sinh \left (3 \cosh ^{-1}(c x)\right )+2 \sinh \left (3 \cosh ^{-1}(c x)\right )+54 \cosh ^{-1}(c x)^2 \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )-54 \cosh ^{-1}(c x)^2 \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{108 c^6 d \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{5} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b x^{5} \operatorname {arcosh}\left (c x\right ) + a^{2} x^{5}\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
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 = 0.86, size = 1099, normalized size = 1.98 \[ \text {result too large to display} \]
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 \[ -\frac {1}{3} \, a^{2} {\left (\frac {x^{4}}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} + \frac {4 \, x^{2}}{\sqrt {-c^{2} d x^{2} + d} c^{4} d} - \frac {8}{\sqrt {-c^{2} d x^{2} + d} c^{6} d}\right )} + \frac {{\left (b^{2} c^{4} \sqrt {d} x^{4} + 4 \, b^{2} c^{2} \sqrt {d} x^{2} - 8 \, b^{2} \sqrt {d}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{3 \, {\left (c^{8} d^{2} x^{2} - c^{6} d^{2}\right )}} + \int -\frac {2 \, {\left ({\left (4 \, b^{2} c^{3} x^{3} - {\left (3 \, a b c^{5} - b^{2} c^{5}\right )} x^{5} - 8 \, b^{2} c x\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (3 \, b^{2} c^{4} x^{4} - {\left (3 \, a b c^{6} - b^{2} c^{6}\right )} x^{6} - 12 \, b^{2} c^{2} x^{2} + 8 \, b^{2}\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{3 \, {\left (c^{10} d^{\frac {3}{2}} x^{5} - 2 \, c^{8} d^{\frac {3}{2}} x^{3} + c^{6} d^{\frac {3}{2}} x + {\left (c^{9} d^{\frac {3}{2}} x^{4} - 2 \, c^{7} d^{\frac {3}{2}} x^{2} + c^{5} d^{\frac {3}{2}}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )}}\,{d x} \]
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 \[ \int \frac {x^5\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{5} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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