Optimal. Leaf size=252 \[ -\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x^2 \sqrt {-1+c^2 x^2}}{1680 c^5 \sqrt {c^2 x^2}}-\frac {b e \left (84 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}-\frac {b e^2 x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{1680 c^6 \sqrt {c^2 x^2}} \]
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Rubi [A]
time = 0.17, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {276, 5346, 12,
1281, 470, 327, 223, 212} \begin {gather*} \frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b e^2 x^6 \sqrt {c^2 x^2-1}}{42 c \sqrt {c^2 x^2}}-\frac {b e x^4 \sqrt {c^2 x^2-1} \left (84 c^2 d+25 e\right )}{840 c^3 \sqrt {c^2 x^2}}-\frac {b x \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{1680 c^6 \sqrt {c^2 x^2}}-\frac {b x^2 \sqrt {c^2 x^2-1} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^5 \sqrt {c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 212
Rule 223
Rule 276
Rule 327
Rule 470
Rule 1281
Rule 5346
Rubi steps
\begin {align*} \int x^2 \left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^2 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{105 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^2 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{105 \sqrt {c^2 x^2}}\\ &=-\frac {b e^2 x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {(b x) \int \frac {x^2 \left (210 c^2 d^2+3 e \left (84 c^2 d+25 e\right ) x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{630 c \sqrt {c^2 x^2}}\\ &=-\frac {b e \left (84 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}-\frac {b e^2 x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )--\frac {\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) x\right ) \int \frac {x^2}{\sqrt {-1+c^2 x^2}} \, dx}{2520 c^3 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x^2 \sqrt {-1+c^2 x^2}}{1680 c^5 \sqrt {c^2 x^2}}-\frac {b e \left (84 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}-\frac {b e^2 x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )--\frac {\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{5040 c^5 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x^2 \sqrt {-1+c^2 x^2}}{1680 c^5 \sqrt {c^2 x^2}}-\frac {b e \left (84 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}-\frac {b e^2 x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )--\frac {\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) x\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{5040 c^5 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x^2 \sqrt {-1+c^2 x^2}}{1680 c^5 \sqrt {c^2 x^2}}-\frac {b e \left (84 c^2 d+25 e\right ) x^4 \sqrt {-1+c^2 x^2}}{840 c^3 \sqrt {c^2 x^2}}-\frac {b e^2 x^6 \sqrt {-1+c^2 x^2}}{42 c \sqrt {c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{1680 c^6 \sqrt {c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 186, normalized size = 0.74 \begin {gather*} \frac {c^2 x^2 \left (16 a c^5 x \left (35 d^2+42 d e x^2+15 e^2 x^4\right )-b \sqrt {1-\frac {1}{c^2 x^2}} \left (75 e^2+2 c^2 e \left (126 d+25 e x^2\right )+8 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )\right )\right )+16 b c^7 x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right ) \sec ^{-1}(c x)-b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{1680 c^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(474\) vs.
\(2(222)=444\).
time = 0.25, size = 475, normalized size = 1.88
method | result | size |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \,\mathrm {arcsec}\left (c x \right ) d^{2} c^{3} x^{3}}{3}+\frac {2 b \,c^{3} \mathrm {arcsec}\left (c x \right ) d e \,x^{5}}{5}+\frac {b \,c^{3} \mathrm {arcsec}\left (c x \right ) e^{2} x^{7}}{7}-\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) x^{2} d e}{10 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) x^{4} e^{2}}{42 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {3 b \left (c^{2} x^{2}-1\right ) d e}{20 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right ) x^{2} e^{2}}{168 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{20 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {5 b \left (c^{2} x^{2}-1\right ) e^{2}}{112 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{3}}\) | \(475\) |
default | \(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \,\mathrm {arcsec}\left (c x \right ) d^{2} c^{3} x^{3}}{3}+\frac {2 b \,c^{3} \mathrm {arcsec}\left (c x \right ) d e \,x^{5}}{5}+\frac {b \,c^{3} \mathrm {arcsec}\left (c x \right ) e^{2} x^{7}}{7}-\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) x^{2} d e}{10 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) x^{4} e^{2}}{42 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {3 b \left (c^{2} x^{2}-1\right ) d e}{20 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right ) x^{2} e^{2}}{168 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {3 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{20 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {5 b \left (c^{2} x^{2}-1\right ) e^{2}}{112 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{3}}\) | \(475\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 405, normalized size = 1.61 \begin {gather*} \frac {1}{7} \, a x^{7} e^{2} + \frac {2}{5} \, a d x^{5} e + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arcsec}\left (c x\right ) - \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d^{2} + \frac {1}{40} \, {\left (16 \, x^{5} \operatorname {arcsec}\left (c x\right ) + \frac {\frac {2 \, {\left (3 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b d e + \frac {1}{672} \, {\left (96 \, x^{7} \operatorname {arcsec}\left (c x\right ) - \frac {\frac {2 \, {\left (15 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac {15 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b e^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.51, size = 272, normalized size = 1.08 \begin {gather*} \frac {240 \, a c^{7} x^{7} e^{2} + 672 \, a c^{7} d x^{5} e + 560 \, a c^{7} d^{2} x^{3} + 16 \, {\left (35 \, b c^{7} d^{2} x^{3} - 35 \, b c^{7} d^{2} + 15 \, {\left (b c^{7} x^{7} - b c^{7}\right )} e^{2} + 42 \, {\left (b c^{7} d x^{5} - b c^{7} d\right )} e\right )} \operatorname {arcsec}\left (c x\right ) + 32 \, {\left (35 \, b c^{7} d^{2} + 42 \, b c^{7} d e + 15 \, b c^{7} e^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (280 \, b c^{4} d^{2} + 252 \, b c^{2} d e + 75 \, b e^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (280 \, b c^{5} d^{2} x + 5 \, {\left (8 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 15 \, b c x\right )} e^{2} + 84 \, {\left (2 \, b c^{5} d x^{3} + 3 \, b c^{3} d x\right )} e\right )} \sqrt {c^{2} x^{2} - 1}}{1680 \, c^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 16.98, size = 542, normalized size = 2.15 \begin {gather*} \frac {a d^{2} x^{3}}{3} + \frac {2 a d e x^{5}}{5} + \frac {a e^{2} x^{7}}{7} + \frac {b d^{2} x^{3} \operatorname {asec}{\left (c x \right )}}{3} + \frac {2 b d e x^{5} \operatorname {asec}{\left (c x \right )}}{5} + \frac {b e^{2} x^{7} \operatorname {asec}{\left (c x \right )}}{7} - \frac {b d^{2} \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{3 c} - \frac {2 b d e \left (\begin {cases} \frac {c x^{5}}{4 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{3}}{8 c \sqrt {c^{2} x^{2} - 1}} - \frac {3 x}{8 c^{3} \sqrt {c^{2} x^{2} - 1}} + \frac {3 \operatorname {acosh}{\left (c x \right )}}{8 c^{4}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{5}}{4 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{3}}{8 c \sqrt {- c^{2} x^{2} + 1}} + \frac {3 i x}{8 c^{3} \sqrt {- c^{2} x^{2} + 1}} - \frac {3 i \operatorname {asin}{\left (c x \right )}}{8 c^{4}} & \text {otherwise} \end {cases}\right )}{5 c} - \frac {b e^{2} \left (\begin {cases} \frac {c x^{7}}{6 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{5}}{24 c \sqrt {c^{2} x^{2} - 1}} + \frac {5 x^{3}}{48 c^{3} \sqrt {c^{2} x^{2} - 1}} - \frac {5 x}{16 c^{5} \sqrt {c^{2} x^{2} - 1}} + \frac {5 \operatorname {acosh}{\left (c x \right )}}{16 c^{6}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{7}}{6 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{5}}{24 c \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i x^{3}}{48 c^{3} \sqrt {- c^{2} x^{2} + 1}} + \frac {5 i x}{16 c^{5} \sqrt {- c^{2} x^{2} + 1}} - \frac {5 i \operatorname {asin}{\left (c x \right )}}{16 c^{6}} & \text {otherwise} \end {cases}\right )}{7 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4760 vs.
\(2 (222) = 444\).
time = 5.95, size = 4760, normalized size = 18.89 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int x^2\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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