Optimal. Leaf size=40 \[ \frac {1}{3} \log (4-3 \tan (x))+\frac {8}{3 \sqrt {4-3 \tan (x)}}+\frac {2}{3} \sqrt {4-3 \tan (x)} \]
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Rubi [A]
time = 0.10, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4427, 45}
\begin {gather*} \frac {2}{3} \sqrt {4-3 \tan (x)}+\frac {8}{3 \sqrt {4-3 \tan (x)}}+\frac {1}{3} \log (4-3 \tan (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 4427
Rubi steps
\begin {align*} \int \frac {\sec ^2(x) \left (-\sqrt {4-3 \tan (x)}+3 \tan (x)\right )}{(4-3 \tan (x))^{3/2}} \, dx &=\text {Subst}\left (\int \left (\frac {3 x}{(4-3 x)^{3/2}}+\frac {1}{-4+3 x}\right ) \, dx,x,\tan (x)\right )\\ &=\frac {1}{3} \log (4-3 \tan (x))+3 \text {Subst}\left (\int \frac {x}{(4-3 x)^{3/2}} \, dx,x,\tan (x)\right )\\ &=\frac {1}{3} \log (4-3 \tan (x))+3 \text {Subst}\left (\int \left (\frac {4}{3 (4-3 x)^{3/2}}-\frac {1}{3 \sqrt {4-3 x}}\right ) \, dx,x,\tan (x)\right )\\ &=\frac {1}{3} \log (4-3 \tan (x))+\frac {8}{3 \sqrt {4-3 \tan (x)}}+\frac {2}{3} \sqrt {4-3 \tan (x)}\\ \end {align*}
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Mathematica [A]
time = 0.86, size = 38, normalized size = 0.95 \begin {gather*} \frac {16+\log (4-3 \tan (x)) \sqrt {4-3 \tan (x)}-6 \tan (x)}{3 \sqrt {4-3 \tan (x)}} \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. \(218\) vs.
\(2(30)=60\).
time = 0.47, size = 219, normalized size = 5.48
method | result | size |
default | \(\frac {\left (\cos \left (x \right )-1\right )^{2} \left (1+\cos \left (x \right )\right )^{2} \left (16 \sqrt {\frac {4 \cos \left (x \right )-3 \sin \left (x \right )}{\cos \left (x \right )}}\, \cos \left (x \right )-4 \cos \left (x \right ) \ln \left (-\frac {-1+\cos \left (x \right )-\sin \left (x \right )}{\sin \left (x \right )}\right )+4 \cos \left (x \right ) \ln \left (-\frac {\sin \left (x \right )-2+2 \cos \left (x \right )}{\sin \left (x \right )}\right )-4 \cos \left (x \right ) \ln \left (-\frac {-1+\cos \left (x \right )+\sin \left (x \right )}{\sin \left (x \right )}\right )+4 \cos \left (x \right ) \ln \left (-\frac {-2 \sin \left (x \right )-1+\cos \left (x \right )}{\sin \left (x \right )}\right )-6 \sin \left (x \right ) \sqrt {\frac {4 \cos \left (x \right )-3 \sin \left (x \right )}{\cos \left (x \right )}}+3 \sin \left (x \right ) \ln \left (-\frac {-1+\cos \left (x \right )-\sin \left (x \right )}{\sin \left (x \right )}\right )-3 \sin \left (x \right ) \ln \left (-\frac {\sin \left (x \right )-2+2 \cos \left (x \right )}{\sin \left (x \right )}\right )+3 \sin \left (x \right ) \ln \left (-\frac {-1+\cos \left (x \right )+\sin \left (x \right )}{\sin \left (x \right )}\right )-3 \sin \left (x \right ) \ln \left (-\frac {-2 \sin \left (x \right )-1+\cos \left (x \right )}{\sin \left (x \right )}\right )\right )}{3 \left (4 \cos \left (x \right )-3 \sin \left (x \right )\right ) \sin \left (x \right )^{4}}\) | \(219\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.16, size = 30, normalized size = 0.75 \begin {gather*} \frac {2}{3} \, \sqrt {-3 \, \tan \left (x\right ) + 4} + \frac {8}{3 \, \sqrt {-3 \, \tan \left (x\right ) + 4}} + \frac {1}{3} \, \log \left (-3 \, \tan \left (x\right ) + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs.
\(2 (30) = 60\).
time = 0.86, size = 82, normalized size = 2.05 \begin {gather*} \frac {{\left (4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )\right )} \log \left (\frac {7}{4} \, \cos \left (x\right )^{2} - 6 \, \cos \left (x\right ) \sin \left (x\right ) + \frac {9}{4}\right ) - {\left (4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )\right )} \log \left (\cos \left (x\right )^{2}\right ) + 4 \, \sqrt {\frac {4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )}{\cos \left (x\right )}} {\left (8 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )\right )}}{6 \, {\left (4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )\right )}} \end {gather*}
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 \begin {gather*} - \int \frac {\sqrt {4 - 3 \tan {\left (x \right )}}}{- 3 \sqrt {4 - 3 \tan {\left (x \right )}} \cos ^{2}{\left (x \right )} \tan {\left (x \right )} + 4 \sqrt {4 - 3 \tan {\left (x \right )}} \cos ^{2}{\left (x \right )}}\, dx - \int \left (- \frac {3 \tan {\left (x \right )}}{- 3 \sqrt {4 - 3 \tan {\left (x \right )}} \cos ^{2}{\left (x \right )} \tan {\left (x \right )} + 4 \sqrt {4 - 3 \tan {\left (x \right )}} \cos ^{2}{\left (x \right )}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.26, size = 31, normalized size = 0.78 \begin {gather*} \frac {2}{3} \, \sqrt {-3 \, \tan \left (x\right ) + 4} + \frac {8}{3 \, \sqrt {-3 \, \tan \left (x\right ) + 4}} + \frac {1}{3} \, \log \left ({\left | -3 \, \tan \left (x\right ) + 4 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.42, size = 105, normalized size = 2.62 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^{x\,2{}\mathrm {i}}\,\left (-\frac {16}{3}-4{}\mathrm {i}\right )-\frac {16}{3}+4{}\mathrm {i}\right )}{3}-\frac {\ln \left ({\mathrm {e}}^{x\,2{}\mathrm {i}}\,\left (\frac {16}{3}-4{}\mathrm {i}\right )+\frac {16}{3}-4{}\mathrm {i}\right )}{3}+\frac {2\,{\mathrm {e}}^{x\,1{}\mathrm {i}}\,\cos \left (x\right )\,\left (\frac {32\,{\mathrm {e}}^{x\,1{}\mathrm {i}}\,\cos \left (x\right )}{3}-4\,{\mathrm {e}}^{x\,1{}\mathrm {i}}\,\sin \left (x\right )\right )\,\sqrt {4-\frac {3\,\sin \left (x\right )}{\cos \left (x\right )}}}{8\,{\mathrm {e}}^{x\,2{}\mathrm {i}}+8\,\cos \left (2\,x\right )\,{\mathrm {e}}^{x\,2{}\mathrm {i}}-6\,\sin \left (2\,x\right )\,{\mathrm {e}}^{x\,2{}\mathrm {i}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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