Optimal. Leaf size=89 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3-2 i} d}+\frac {\tan ^{-1}\left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d} \]
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Rubi [A] time = 0.11, antiderivative size = 89, 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 = {3575, 912, 93, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3-2 i} d}+\frac {\tan ^{-1}\left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d} \]
Antiderivative was successfully verified.
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Rule 93
Rule 205
Rule 912
Rule 3575
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {2-3 x} \sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {i}{2 \sqrt {2-3 x} (i-x) \sqrt {x}}+\frac {i}{2 \sqrt {2-3 x} \sqrt {x} (i+x)}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {i \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-3 x} (i-x) \sqrt {x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {i \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-3 x} \sqrt {x} (i+x)} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {i \operatorname {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 \operatorname {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 {\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 {\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 = 0.17, size = 101, normalized size = 1.13 \[ \frac {\sqrt {-3+2 i} \tanh ^{-1}\left (\frac {\sqrt {-\frac {3}{13}+\frac {2 i}{13}} \sqrt {2-3 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )-\sqrt {3+2 i} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{13}+\frac {2 i}{13}} \sqrt {2-3 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )}{\sqrt {13} d} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2} \sqrt {\tan \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.29, size = 435, normalized size = 4.89 \[ \frac {\sqrt {2-3 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \left (3 \arctanh \left (\frac {\left (\sqrt {13}+3\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{52 \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \sqrt {-6+2 \sqrt {13}}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {13}\, \sqrt {2 \sqrt {13}+6}\, \sqrt {-6+2 \sqrt {13}}-11 \arctanh \left (\frac {\left (\sqrt {13}+3\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{52 \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \sqrt {-6+2 \sqrt {13}}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {2 \sqrt {13}+6}\, \sqrt {-6+2 \sqrt {13}}-4 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right ) \sqrt {13}+12 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \left (-2+3 \tan \left (d x +c \right )\right ) \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )} \]
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 \[ \int \frac {1}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2} \sqrt {\tan \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.33, size = 205, normalized size = 2.30 \[ \mathrm {atan}\left (\frac {\sqrt {2}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (4+6{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}\,\left (-4-6{}\mathrm {i}\right )}{3\,\mathrm {tan}\left (c+d\,x\right )+\sqrt {2}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}-2}\right )\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {\sqrt {2}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (4-6{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}\,\left (-4+6{}\mathrm {i}\right )}{3\,\mathrm {tan}\left (c+d\,x\right )+\sqrt {2}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}-2}\right )\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i} \]
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 \[ \int \frac {1}{\sqrt {2 - 3 \tan {\left (c + d x \right )}} \sqrt {\tan {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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