Optimal. Leaf size=307 \[ \frac {c^4 d^2 (f x)^{m+5} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (m+5)}-\frac {2 c^2 d^2 (f x)^{m+3} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {d^2 (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+1)}-\frac {b c d^2 \left (15 m^2+100 m+149\right ) \sqrt {1-c^2 x^2} (f x)^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{f^2 (m+1) (m+2) (m+3)^2 (m+5)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 \left (m^2+13 m+38\right ) \left (1-c^2 x^2\right ) (f x)^{m+2}}{f^2 (m+3)^2 (m+5)^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d^2 \left (1-c^2 x^2\right ) (f x)^{m+4}}{f^4 (m+5)^2 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.50, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {270, 5731, 12, 520, 1267, 459, 365, 364} \[ -\frac {2 c^2 d^2 (f x)^{m+3} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {c^4 d^2 (f x)^{m+5} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (m+5)}+\frac {d^2 (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+1)}-\frac {b c d^2 \left (15 m^2+100 m+149\right ) \sqrt {1-c^2 x^2} (f x)^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{f^2 (m+1) (m+2) (m+3)^2 (m+5)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 \left (m^2+13 m+38\right ) \left (1-c^2 x^2\right ) (f x)^{m+2}}{f^2 (m+3)^2 (m+5)^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d^2 \left (1-c^2 x^2\right ) (f x)^{m+4}}{f^4 (m+5)^2 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 364
Rule 365
Rule 459
Rule 520
Rule 1267
Rule 5731
Rubi steps
\begin {align*} \int (f x)^m \left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-(b c) \int \frac {d^2 (f x)^{1+m} \left (\frac {1}{1+m}-\frac {2 c^2 x^2}{3+m}+\frac {c^4 x^4}{5+m}\right )}{f \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {\left (b c d^2\right ) \int \frac {(f x)^{1+m} \left (\frac {1}{1+m}-\frac {2 c^2 x^2}{3+m}+\frac {c^4 x^4}{5+m}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{f}\\ &=\frac {d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m} \left (\frac {1}{1+m}-\frac {2 c^2 x^2}{3+m}+\frac {c^4 x^4}{5+m}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{f \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c^3 d^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {\left (b d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m} \left (\frac {c^2 (5+m)}{1+m}-\frac {c^4 \left (38+13 m+m^2\right ) x^2}{(3+m) (5+m)}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{c f (5+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 \left (38+13 m+m^2\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {\left (b c d^2 \left (149+100 m+15 m^2\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\sqrt {-1+c^2 x^2}} \, dx}{f (1+m) (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 \left (38+13 m+m^2\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {\left (b c d^2 \left (149+100 m+15 m^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\sqrt {1-c^2 x^2}} \, dx}{f (1+m) (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 \left (38+13 m+m^2\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {b c d^2 \left (149+100 m+15 m^2\right ) (f x)^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m) (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 290, normalized size = 0.94 \[ d^2 x (f x)^m \left (\frac {c^4 x^4 \left (a+b \cosh ^{-1}(c x)\right )}{m+5}-\frac {2 c^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{m+3}+\frac {a+b \cosh ^{-1}(c x)}{m+1}-\frac {b c x \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{\left (m^2+3 m+2\right ) \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 x^5 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+6}{2};\frac {m+8}{2};c^2 x^2\right )}{(m+5) (m+6) \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c^3 x^3 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+4}{2};\frac {m+6}{2};c^2 x^2\right )}{\left (m^2+7 m+12\right ) \sqrt {c x-1} \sqrt {c x+1}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a c^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} + {\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \operatorname {arcosh}\left (c x\right )\right )} \left (f x\right )^{m}, x\right ) \]
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 [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )\, dx \]
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 \[ \frac {a c^{4} d^{2} f^{m} x^{5} x^{m}}{m + 5} - \frac {2 \, a c^{2} d^{2} f^{m} x^{3} x^{m}}{m + 3} + \frac {\left (f x\right )^{m + 1} a d^{2}}{f {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 4 \, m + 3\right )} b c^{4} d^{2} f^{m} x^{5} - 2 \, {\left (m^{2} + 6 \, m + 5\right )} b c^{2} d^{2} f^{m} x^{3} + {\left (m^{2} + 8 \, m + 15\right )} b d^{2} f^{m} x\right )} x^{m} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{m^{3} + 9 \, m^{2} + 23 \, m + 15} + \int \frac {{\left ({\left (m^{2} + 4 \, m + 3\right )} b c^{5} d^{2} f^{m} x^{5} - 2 \, {\left (m^{2} + 6 \, m + 5\right )} b c^{3} d^{2} f^{m} x^{3} + {\left (m^{2} + 8 \, m + 15\right )} b c d^{2} f^{m} x\right )} x^{m}}{{\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} c^{3} x^{3} - {\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} c x + {\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} c^{2} x^{2} - m^{3} - 9 \, m^{2} - 23 \, m - 15\right )} \sqrt {c x + 1} \sqrt {c x - 1}}\,{d x} - \int \frac {{\left ({\left (m^{2} + 4 \, m + 3\right )} b c^{6} d^{2} f^{m} x^{6} - 2 \, {\left (m^{2} + 6 \, m + 5\right )} b c^{4} d^{2} f^{m} x^{4} + {\left (m^{2} + 8 \, m + 15\right )} b c^{2} d^{2} f^{m} x^{2}\right )} x^{m}}{{\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} c^{2} x^{2} - m^{3} - 9 \, m^{2} - 23 \, m - 15}\,{d x} \]
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 \[ \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2\,{\left (f\,x\right )}^m \,d x \]
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 \[ d^{2} \left (\int a \left (f x\right )^{m}\, dx + \int b \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 2 a c^{2} x^{2} \left (f x\right )^{m}\right )\, dx + \int a c^{4} x^{4} \left (f x\right )^{m}\, dx + \int \left (- 2 b c^{2} x^{2} \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{4} x^{4} \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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