Optimal. Leaf size=95 \[ -\frac {i \text {ArcTan}\left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3-2 i} d}+\frac {i \text {ArcTan}\left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d} \]
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Rubi [A]
time = 0.07, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3656, 924, 95,
211} \begin {gather*} \frac {i \text {ArcTan}\left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d}-\frac {i \text {ArcTan}\left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3-2 i} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 211
Rule 924
Rule 3656
Rubi steps
\begin {align*} \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {2-3 x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{2 \sqrt {2-3 x} (i-x) \sqrt {x}}+\frac {1}{2 \sqrt {2-3 x} \sqrt {x} (i+x)}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2-3 x} (i-x) \sqrt {x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2-3 x} \sqrt {x} (i+x)} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{i-(2-3 i) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{d}+\frac {\text {Subst}\left (\int \frac {1}{i+(2+3 i) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{d}\\ &=-\frac {i \tan ^{-1}\left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3-2 i} d}+\frac {i \tan ^{-1}\left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 103, normalized size = 1.08 \begin {gather*} \frac {i \left (\sqrt {3+2 i} \text {ArcTan}\left (\frac {\sqrt {\frac {3}{13}+\frac {2 i}{13}} \sqrt {2-3 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )+\sqrt {-3+2 i} \tanh ^{-1}\left (\frac {\sqrt {-\frac {3}{13}+\frac {2 i}{13}} \sqrt {2-3 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )\right )}{\sqrt {13} d} \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. \(433\) vs.
\(2(77)=154\).
time = 0.67, size = 434, normalized size = 4.57
method | result | size |
derivativedivides | \(\frac {\sqrt {2-3 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \left (\sqrt {13}\, \sqrt {2 \sqrt {13}+6}\, \arctanh \left (\frac {\left (\sqrt {13}+3\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{52 \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \sqrt {2 \sqrt {13}-6}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {2 \sqrt {13}-6}-3 \sqrt {2 \sqrt {13}+6}\, \arctanh \left (\frac {\left (\sqrt {13}+3\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{52 \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \sqrt {2 \sqrt {13}-6}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {2 \sqrt {13}-6}+12 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right ) \sqrt {13}-44 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \left (-2+3 \tan \left (d x +c \right )\right ) \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )}\) | \(434\) |
default | \(\frac {\sqrt {2-3 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \left (\sqrt {13}\, \sqrt {2 \sqrt {13}+6}\, \arctanh \left (\frac {\left (\sqrt {13}+3\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{52 \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \sqrt {2 \sqrt {13}-6}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {2 \sqrt {13}-6}-3 \sqrt {2 \sqrt {13}+6}\, \arctanh \left (\frac {\left (\sqrt {13}+3\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{52 \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \sqrt {2 \sqrt {13}-6}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {2 \sqrt {13}-6}+12 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right ) \sqrt {13}-44 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \left (-2+3 \tan \left (d x +c \right )\right ) \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )}\) | \(434\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\tan {\left (c + d x \right )}}}{\sqrt {2 - 3 \tan {\left (c + d x \right )}}}\, dx \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. 641 vs.
\(2 (67) = 134\).
time = 0.94, size = 641, normalized size = 6.75 \begin {gather*} -\frac {1}{2028} \, \sqrt {3} {\left (\frac {2 \, {\left (2 \, d^{2} \sqrt {1014 \, \sqrt {13} - 702} - 3 \, d \sqrt {1014 \, \sqrt {13} + 702} {\left | d \right |}\right )} \arctan \left (\frac {13 \, \left (\frac {9}{13}\right )^{\frac {3}{4}} {\left (2 \, \left (\frac {9}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {3}{26} \, \sqrt {13} + \frac {1}{2}} + \frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} - \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )}}{18 \, \sqrt {\frac {3}{26} \, \sqrt {13} + \frac {1}{2}}}\right )}{d^{3}} + \frac {2 \, {\left (2 \, d^{2} \sqrt {1014 \, \sqrt {13} - 702} - 3 \, d \sqrt {1014 \, \sqrt {13} + 702} {\left | d \right |}\right )} \arctan \left (-\frac {13 \, \left (\frac {9}{13}\right )^{\frac {3}{4}} {\left (2 \, \left (\frac {9}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {3}{26} \, \sqrt {13} + \frac {1}{2}} - \frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} + \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )}}{18 \, \sqrt {\frac {3}{26} \, \sqrt {13} + \frac {1}{2}}}\right )}{d^{3}} + \frac {{\left (2 \, d^{2} \sqrt {1014 \, \sqrt {13} + 702} + 3 \, d \sqrt {1014 \, \sqrt {13} - 702} {\left | d \right |}\right )} \log \left ({\left (\frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} - \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )}^{2} + 4 \, \left (\frac {9}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {3}{26} \, \sqrt {13} + \frac {1}{2}} {\left (\frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} - \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )} + 12 \, \sqrt {\frac {1}{13}}\right )}{d^{3}} - \frac {{\left (2 \, d^{2} \sqrt {1014 \, \sqrt {13} + 702} + 3 \, d \sqrt {1014 \, \sqrt {13} - 702} {\left | d \right |}\right )} \log \left ({\left (\frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} - \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )}^{2} - 4 \, \left (\frac {9}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {3}{26} \, \sqrt {13} + \frac {1}{2}} {\left (\frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} - \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )} + 12 \, \sqrt {\frac {1}{13}}\right )}{d^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.43, size = 205, normalized size = 2.16 \begin {gather*} \mathrm {atan}\left (\frac {\sqrt {2}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (6+4{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}\,\left (-6-4{}\mathrm {i}\right )}{3\,\mathrm {tan}\left (c+d\,x\right )+\sqrt {2}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}-2}\right )\,\sqrt {\frac {\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {\sqrt {2}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (6-4{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}\,\left (-6+4{}\mathrm {i}\right )}{3\,\mathrm {tan}\left (c+d\,x\right )+\sqrt {2}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}-2}\right )\,\sqrt {\frac {\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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