Optimal. Leaf size=114 \[ -\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^2}{112 c^5}-\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{168 c^3}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {5 b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{112 c^7} \]
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Rubi [A]
time = 0.04, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5328, 272, 44,
65, 214} \begin {gather*} \frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b x^6 \sqrt {1-\frac {1}{c^2 x^2}}}{42 c}-\frac {5 b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{112 c^7}-\frac {5 b x^2 \sqrt {1-\frac {1}{c^2 x^2}}}{112 c^5}-\frac {5 b x^4 \sqrt {1-\frac {1}{c^2 x^2}}}{168 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 214
Rule 272
Rule 5328
Rubi steps
\begin {align*} \int x^6 \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {b \int \frac {x^5}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{7 c}\\ &=\frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )+\frac {b \text {Subst}\left (\int \frac {1}{x^4 \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{14 c}\\ &=-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )+\frac {(5 b) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{84 c^3}\\ &=-\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{168 c^3}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )+\frac {(5 b) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{112 c^5}\\ &=-\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^2}{112 c^5}-\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{168 c^3}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )+\frac {(5 b) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{224 c^7}\\ &=-\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^2}{112 c^5}-\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{168 c^3}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {(5 b) \text {Subst}\left (\int \frac {1}{c^2-c^2 x^2} \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right )}{112 c^5}\\ &=-\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^2}{112 c^5}-\frac {5 b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{168 c^3}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \sec ^{-1}(c x)\right )-\frac {5 b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{112 c^7}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 107, normalized size = 0.94 \begin {gather*} \frac {a x^7}{7}+b \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}} \left (-\frac {5 x^2}{112 c^5}-\frac {5 x^4}{168 c^3}-\frac {x^6}{42 c}\right )+\frac {1}{7} b x^7 \sec ^{-1}(c x)-\frac {5 b \log \left (x \left (1+\sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}\right )\right )}{112 c^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 184, normalized size = 1.61
method | result | size |
derivativedivides | \(\frac {\frac {c^{7} x^{7} a}{7}+\frac {b \,c^{7} x^{7} \mathrm {arcsec}\left (c x \right )}{7}-\frac {b \left (c^{2} x^{2}-1\right ) c^{4} x^{4}}{42 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}{168 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right )}{112 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}}{c^{7}}\) | \(184\) |
default | \(\frac {\frac {c^{7} x^{7} a}{7}+\frac {b \,c^{7} x^{7} \mathrm {arcsec}\left (c x \right )}{7}-\frac {b \left (c^{2} x^{2}-1\right ) c^{4} x^{4}}{42 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}{168 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \left (c^{2} x^{2}-1\right )}{112 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {5 b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{112 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}}{c^{7}}\) | \(184\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 162, normalized size = 1.42 \begin {gather*} \frac {1}{7} \, a x^{7} + \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.63, size = 116, normalized size = 1.02 \begin {gather*} \frac {48 \, a c^{7} x^{7} + 96 \, b c^{7} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 48 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \operatorname {arcsec}\left (c x\right ) + 15 \, b \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (8 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 15 \, b c x\right )} \sqrt {c^{2} x^{2} - 1}}{336 \, c^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 13.06, size = 221, normalized size = 1.94 \begin {gather*} \frac {a x^{7}}{7} + \frac {b x^{7} \operatorname {asec}{\left (c x \right )}}{7} - \frac {b \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. 8644 vs.
\(2 (96) = 192\).
time = 1.23, size = 8644, normalized size = 75.82 \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 = -0.01 \begin {gather*} \int x^6\,\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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