Optimal. Leaf size=371 \[ \frac {d^2 (f x)^{m+1} \left (a+b \csc ^{-1}(c x)\right )}{f (m+1)}+\frac {2 d e (f x)^{m+3} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (m+5)}+\frac {b e^2 x \sqrt {c^2 x^2-1} (f x)^{m+3}}{c f^3 (m+4) (m+5) \sqrt {c^2 x^2}}+\frac {b e x \sqrt {c^2 x^2-1} (f x)^{m+1} \left (2 c^2 d \left (m^2+9 m+20\right )+e (m+3)^2\right )}{c^3 f (m+2) (m+3) (m+4) (m+5) \sqrt {c^2 x^2}}+\frac {b x \sqrt {1-c^2 x^2} (f x)^{m+1} \left (c^4 d^2 (m+2) (m+3) (m+4) (m+5)+e (m+1)^2 \left (2 c^2 d \left (m^2+9 m+20\right )+e (m+3)^2\right )\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right )}{c^3 f (m+1)^2 (m+2) (m+3) (m+4) (m+5) \sqrt {c^2 x^2} \sqrt {c^2 x^2-1}} \]
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Rubi [A] time = 0.43, antiderivative size = 352, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {270, 5239, 12, 1267, 459, 365, 364} \[ \frac {d^2 (f x)^{m+1} \left (a+b \csc ^{-1}(c x)\right )}{f (m+1)}+\frac {2 d e (f x)^{m+3} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (m+5)}+\frac {b c x \sqrt {1-c^2 x^2} (f x)^{m+1} \left (\frac {e \left (2 c^2 d \left (m^2+9 m+20\right )+e (m+3)^2\right )}{c^4 (m+2) (m+3) (m+4) (m+5)}+\frac {d^2}{(m+1)^2}\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right )}{f \sqrt {c^2 x^2} \sqrt {c^2 x^2-1}}+\frac {b e x \sqrt {c^2 x^2-1} (f x)^{m+1} \left (2 c^2 d \left (m^2+9 m+20\right )+e (m+3)^2\right )}{c^3 f (m+2) (m+3) (m+4) (m+5) \sqrt {c^2 x^2}}+\frac {b e^2 x \sqrt {c^2 x^2-1} (f x)^{m+3}}{c f^3 (m+4) (m+5) \sqrt {c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 364
Rule 365
Rule 459
Rule 1267
Rule 5239
Rubi steps
\begin {align*} \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {d^2 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {(b c x) \int \frac {(f x)^m \left (d^2 \left (15+8 m+m^2\right )+2 d e \left (5+6 m+m^2\right ) x^2+e^2 \left (3+4 m+m^2\right ) x^4\right )}{(1+m) (3+m) (5+m) \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {d^2 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {(b c x) \int \frac {(f x)^m \left (d^2 \left (15+8 m+m^2\right )+2 d e \left (5+6 m+m^2\right ) x^2+e^2 \left (3+4 m+m^2\right ) x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{\left (15+23 m+9 m^2+m^3\right ) \sqrt {c^2 x^2}}\\ &=\frac {b e^2 x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c f^3 (4+m) (5+m) \sqrt {c^2 x^2}}+\frac {d^2 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {(b x) \int \frac {(f x)^m \left (c^2 d^2 (3+m) (4+m) (5+m)+e (1+m) \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right ) x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{c (4+m) \left (15+23 m+9 m^2+m^3\right ) \sqrt {c^2 x^2}}\\ &=\frac {b e \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right ) x (f x)^{1+m} \sqrt {-1+c^2 x^2}}{c^3 f (2+m) (4+m) \left (15+8 m+m^2\right ) \sqrt {c^2 x^2}}+\frac {b e^2 x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c f^3 (4+m) (5+m) \sqrt {c^2 x^2}}+\frac {d^2 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+-\frac {\left (b \left (-c^4 d^2 (2+m) (3+m) (4+m) (5+m)-e (1+m)^2 \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right )\right ) x\right ) \int \frac {(f x)^m}{\sqrt {-1+c^2 x^2}} \, dx}{c^3 (2+m) (4+m) \left (15+23 m+9 m^2+m^3\right ) \sqrt {c^2 x^2}}\\ &=\frac {b e \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right ) x (f x)^{1+m} \sqrt {-1+c^2 x^2}}{c^3 f (2+m) (4+m) \left (15+8 m+m^2\right ) \sqrt {c^2 x^2}}+\frac {b e^2 x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c f^3 (4+m) (5+m) \sqrt {c^2 x^2}}+\frac {d^2 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+-\frac {\left (b \left (-c^4 d^2 (2+m) (3+m) (4+m) (5+m)-e (1+m)^2 \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right )\right ) x \sqrt {1-c^2 x^2}\right ) \int \frac {(f x)^m}{\sqrt {1-c^2 x^2}} \, dx}{c^3 (2+m) (4+m) \left (15+23 m+9 m^2+m^3\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}\\ &=\frac {b e \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right ) x (f x)^{1+m} \sqrt {-1+c^2 x^2}}{c^3 f (2+m) (4+m) \left (15+8 m+m^2\right ) \sqrt {c^2 x^2}}+\frac {b e^2 x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c f^3 (4+m) (5+m) \sqrt {c^2 x^2}}+\frac {d^2 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac {2 d e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {b \left (c^4 d^2 (2+m) (3+m) (4+m) (5+m)+e (1+m)^2 \left (e (3+m)^2+2 c^2 d \left (20+9 m+m^2\right )\right )\right ) x (f x)^{1+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{c^3 f (1+m) (2+m) (4+m) \left (15+23 m+9 m^2+m^3\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}\\ \end {align*}
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Mathematica [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} + {\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \operatorname {arccsc}\left (c x\right )\right )} \left (f x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 9.32, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \left (e \,x^{2}+d \right )^{2} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )\, dx \]
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 \[ \frac {a e^{2} f^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a d e f^{m} x^{3} x^{m}}{m + 3} + \frac {\left (f x\right )^{m + 1} a d^{2}}{f {\left (m + 1\right )}} + \frac {{\left ({\left (b e^{2} f^{m} m^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 4 \, b e^{2} f^{m} m \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 3 \, b e^{2} f^{m} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} x^{5} + 2 \, {\left (b d e f^{m} m^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 6 \, b d e f^{m} m \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 5 \, b d e f^{m} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} x^{3} + {\left (b d^{2} f^{m} m^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 8 \, b d^{2} f^{m} m \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 15 \, b d^{2} f^{m} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} x\right )} x^{m} - {\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} \int \frac {{\left (b d^{2} f^{m} m^{2} + 8 \, b d^{2} f^{m} m + {\left (b e^{2} f^{m} m^{2} + 4 \, b e^{2} f^{m} m + 3 \, b e^{2} f^{m}\right )} x^{4} + 15 \, b d^{2} f^{m} + 2 \, {\left (b d e f^{m} m^{2} + 6 \, b d e f^{m} m + 5 \, b d e f^{m}\right )} x^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1} x^{m}}{m^{3} - {\left (c^{2} m^{3} + 9 \, c^{2} m^{2} + 23 \, c^{2} m + 15 \, c^{2}\right )} x^{2} + 9 \, m^{2} + 23 \, m + 15}\,{d x}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]
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 \[ \int {\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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 \[ \int \left (f x\right )^{m} \left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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