Optimal. Leaf size=87 \[ \frac {b c d \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}-\frac {d \left (a+b \sec ^{-1}(c x)\right )}{x}+e x \left (a+b \sec ^{-1}(c x)\right )-\frac {b e x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {c^2 x^2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 5346, 462,
223, 212} \begin {gather*} -\frac {d \left (a+b \sec ^{-1}(c x)\right )}{x}+e x \left (a+b \sec ^{-1}(c x)\right )+\frac {b c d \sqrt {c^2 x^2-1}}{\sqrt {c^2 x^2}}-\frac {b e x \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{\sqrt {c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 212
Rule 223
Rule 462
Rule 5346
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d \left (a+b \sec ^{-1}(c x)\right )}{x}+e x \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c x) \int \frac {-d+e x^2}{x^2 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {b c d \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}-\frac {d \left (a+b \sec ^{-1}(c x)\right )}{x}+e x \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c e x) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {b c d \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}-\frac {d \left (a+b \sec ^{-1}(c x)\right )}{x}+e x \left (a+b \sec ^{-1}(c x)\right )-\frac {(b c e x) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {c^2 x^2}}\\ &=\frac {b c d \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}-\frac {d \left (a+b \sec ^{-1}(c x)\right )}{x}+e x \left (a+b \sec ^{-1}(c x)\right )-\frac {b e x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 104, normalized size = 1.20 \begin {gather*} -\frac {a d}{x}+a e x+b c d \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}-\frac {b d \sec ^{-1}(c x)}{x}+b e x \sec ^{-1}(c x)-\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {-1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 137, normalized size = 1.57
method | result | size |
derivativedivides | \(c \left (\frac {a \left (e c x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \,\mathrm {arcsec}\left (c x \right ) e x}{c}-\frac {b \,\mathrm {arcsec}\left (c x \right ) d}{c x}+\frac {b \left (c^{2} x^{2}-1\right ) d}{c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )\) | \(137\) |
default | \(c \left (\frac {a \left (e c x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \,\mathrm {arcsec}\left (c x \right ) e x}{c}-\frac {b \,\mathrm {arcsec}\left (c x \right ) d}{c x}+\frac {b \left (c^{2} x^{2}-1\right ) d}{c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )\) | \(137\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 91, normalized size = 1.05 \begin {gather*} {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsec}\left (c x\right )}{x}\right )} b d + a x e + \frac {{\left (2 \, c x \operatorname {arcsec}\left (c x\right ) - \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) + \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b e}{2 \, c} - \frac {a d}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 5.01, size = 128, normalized size = 1.47 \begin {gather*} \frac {b c^{2} d x + a c x^{2} e + b x e \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {c^{2} x^{2} - 1} b c d - a c d + {\left (b c d x - b c d + {\left (b c x^{2} - b c x\right )} e\right )} \operatorname {arcsec}\left (c x\right ) - 2 \, {\left (b c d x - b c x e\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.01, size = 73, normalized size = 0.84 \begin {gather*} - \frac {a d}{x} + a e x + b c d \sqrt {1 - \frac {1}{c^{2} x^{2}}} - \frac {b d \operatorname {asec}{\left (c x \right )}}{x} + b e x \operatorname {asec}{\left (c x \right )} - \frac {b e \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{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. 1088 vs.
\(2 (79) = 158\).
time = 0.81, size = 1088, normalized size = 12.51 \begin {gather*} -{\left (\frac {b c^{2} d \arccos \left (\frac {1}{c x}\right )}{c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}} + \frac {a c^{2} d}{c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}} + \frac {2 \, b c^{2} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arccos \left (\frac {1}{c x}\right )}{{\left (c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}\right )} {\left (\frac {1}{c x} + 1\right )}^{2}} - \frac {2 \, b c^{2} d \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{{\left (c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}\right )} {\left (\frac {1}{c x} + 1\right )}} + \frac {2 \, a c^{2} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )}}{{\left (c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}\right )} {\left (\frac {1}{c x} + 1\right )}^{2}} - \frac {b e \arccos \left (\frac {1}{c x}\right )}{c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}} + \frac {b c^{2} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arccos \left (\frac {1}{c x}\right )}{{\left (c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}\right )} {\left (\frac {1}{c x} + 1\right )}^{4}} + \frac {b e \log \left ({\left | \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {1}{c x} + 1 \right |}\right )}{c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}} - \frac {b e \log \left ({\left | \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {1}{c x} - 1 \right |}\right )}{c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}} + \frac {2 \, b c^{2} d {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}}}{{\left (c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}\right )} {\left (\frac {1}{c x} + 1\right )}^{3}} - \frac {a e}{c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}} + \frac {a c^{2} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}\right )} {\left (\frac {1}{c x} + 1\right )}^{4}} + \frac {2 \, b e {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arccos \left (\frac {1}{c x}\right )}{{\left (c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}\right )} {\left (\frac {1}{c x} + 1\right )}^{2}} + \frac {2 \, a e {\left (\frac {1}{c^{2} x^{2}} - 1\right )}}{{\left (c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}\right )} {\left (\frac {1}{c x} + 1\right )}^{2}} - \frac {b e {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arccos \left (\frac {1}{c x}\right )}{{\left (c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}\right )} {\left (\frac {1}{c x} + 1\right )}^{4}} - \frac {b e {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \log \left ({\left | \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {1}{c x} + 1 \right |}\right )}{{\left (c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}\right )} {\left (\frac {1}{c x} + 1\right )}^{4}} + \frac {b e {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \log \left ({\left | \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {1}{c x} - 1 \right |}\right )}{{\left (c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}\right )} {\left (\frac {1}{c x} + 1\right )}^{4}} - \frac {a e {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (c^{2} - \frac {c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2}}{{\left (\frac {1}{c x} + 1\right )}^{4}}\right )} {\left (\frac {1}{c x} + 1\right )}^{4}}\right )} c \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.79, size = 72, normalized size = 0.83 \begin {gather*} a\,e\,x-\frac {d\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )-b\,c\,x\,\sqrt {1-\frac {1}{c^2\,x^2}}\right )}{x}-\frac {b\,e\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {1}{c^2\,x^2}}}\right )}{c}+b\,e\,x\,\mathrm {acos}\left (\frac {1}{c\,x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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