Optimal. Leaf size=95 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {-2+3 \tan (c+d x)}}\right )}{\sqrt {3-2 i} d}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {-2+3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d} \]
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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,
214} \begin {gather*} \frac {i \tanh ^{-1}\left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {3 \tan (c+d x)-2}}\right )}{\sqrt {3+2 i} d}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {3 \tan (c+d x)-2}}\right )}{\sqrt {3-2 i} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 214
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 (i-x) \sqrt {x} \sqrt {-2+3 x}}+\frac {1}{2 \sqrt {x} (i+x) \sqrt {-2+3 x}}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{(i-x) \sqrt {x} \sqrt {-2+3 x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} (i+x) \sqrt {-2+3 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 \tanh ^{-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 \tanh ^{-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 = 95, normalized size = 1.00 \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 \tanh ^{-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 {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(478\) vs.
\(2(77)=154\).
time = 0.85, size = 479, normalized size = 5.04
method | result | size |
derivativedivides | \(-\frac {\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 {2 \sqrt {13}-6}\, \arctan \left (\frac {\sqrt {2 \sqrt {13}-6}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (39+11 \sqrt {13}\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 (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )}{416 \tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}\right ) \sqrt {13}\, \sqrt {2 \sqrt {13}+6}-3 \sqrt {2 \sqrt {13}-6}\, \arctan \left (\frac {\sqrt {2 \sqrt {13}-6}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (39+11 \sqrt {13}\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 (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )}{416 \tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}\right ) \sqrt {2 \sqrt {13}+6}-12 \arctanh \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 \arctanh \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 )}\, \sqrt {-2+3 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )}\) | \(479\) |
default | \(-\frac {\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 {2 \sqrt {13}-6}\, \arctan \left (\frac {\sqrt {2 \sqrt {13}-6}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (39+11 \sqrt {13}\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 (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )}{416 \tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}\right ) \sqrt {13}\, \sqrt {2 \sqrt {13}+6}-3 \sqrt {2 \sqrt {13}-6}\, \arctan \left (\frac {\sqrt {2 \sqrt {13}-6}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (39+11 \sqrt {13}\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 (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )}{416 \tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}\right ) \sqrt {2 \sqrt {13}+6}-12 \arctanh \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 \arctanh \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 )}\, \sqrt {-2+3 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )}\) | \(479\) |
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 {3 \tan {\left (c + d x \right )} - 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.64, size = 191, normalized size = 2.01 \begin {gather*} -\mathrm {atan}\left (\frac {12\,d\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (\frac {\sqrt {2}\,\sqrt {3}}{3}-\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{\sqrt {3\,\mathrm {tan}\left (c+d\,x\right )-2}\,\left (\frac {3\,{\left (\frac {\sqrt {2}\,\sqrt {3}}{3}-\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}^2}{3\,\mathrm {tan}\left (c+d\,x\right )-2}+1\right )}\right )\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {12\,d\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (\frac {\sqrt {2}\,\sqrt {3}}{3}-\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{\sqrt {3\,\mathrm {tan}\left (c+d\,x\right )-2}\,\left (\frac {3\,{\left (\frac {\sqrt {2}\,\sqrt {3}}{3}-\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}^2}{3\,\mathrm {tan}\left (c+d\,x\right )-2}+1\right )}\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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