Optimal. Leaf size=584 \[ \frac {3 d^2 \text {Int}\left (\frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}},x\right )}{m^2+6 m+8}-\frac {2 b c d \sqrt {d-c^2 d x^2} (f x)^{m+2} \left (a+b \cosh ^{-1}(c x)\right )}{f^2 (m+2) (m+4) \sqrt {c x-1} \sqrt {c x+1}}-\frac {6 b c d \sqrt {d-c^2 d x^2} (f x)^{m+2} \left (a+b \cosh ^{-1}(c x)\right )}{f^2 (m+2)^2 (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \sqrt {d-c^2 d x^2} (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{f \left (m^2+6 m+8\right )}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{f (m+4)}+\frac {2 b c^3 d \sqrt {d-c^2 d x^2} (f x)^{m+4} \left (a+b \cosh ^{-1}(c x)\right )}{f^4 (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b^2 c^2 d (3 m+10) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (f x)^{m+3} \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};c^2 x^2\right )}{f^3 (m+2) (m+3) (m+4)^3 (1-c x) (c x+1)}-\frac {6 b^2 c^2 d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (f x)^{m+3} \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};c^2 x^2\right )}{f^3 (m+2)^2 (m+3) (m+4) (1-c x) (c x+1)}-\frac {2 b^2 c^2 d \sqrt {d-c^2 d x^2} (f x)^{m+3}}{f^3 (m+4)^3} \]
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Rubi [A] time = 0.52, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int (f x)^m (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 0, normalized size = 0.00 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c^{2} d x^{2} - a^{2} d + {\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} - a b d\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \left (f x\right )^{m}, x\right ) \]
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 [A] time = 1.75, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}\,{\left (f\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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