Optimal. Leaf size=89 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {-3 \tan (c+d x)-2}}\right )}{\sqrt {3-2 i} d}+\frac {\tan ^{-1}\left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {-3 \tan (c+d x)-2}}\right )}{\sqrt {3+2 i} d} \]
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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} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {-3 \tan (c+d x)-2}}\right )}{\sqrt {3-2 i} d}+\frac {\tan ^{-1}\left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {-3 \tan (c+d x)-2}}\right )}{\sqrt {3+2 i} d} \]
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 = 0.15, size = 101, normalized size = 1.13 \[ \frac {\sqrt {-3+2 i} \tanh ^{-1}\left (\frac {\sqrt {-\frac {3}{13}+\frac {2 i}{13}} \sqrt {-3 \tan (c+d x)-2}}{\sqrt {\tan (c+d x)}}\right )-\sqrt {3+2 i} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{13}+\frac {2 i}{13}} \sqrt {-3 \tan (c+d x)-2}}{\sqrt {\tan (c+d x)}}\right )}{\sqrt {13} d} \]
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 = 0.30, size = 435, normalized size = 4.89 \[ -\frac {\sqrt {-2-3 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \left (3 \sqrt {-6+2 \sqrt {13}}\, \sqrt {13}\, \sqrt {2 \sqrt {13}+6}\, \arctanh \left (\frac {\left (\sqrt {13}+3\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{52 \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \sqrt {-6+2 \sqrt {13}}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}\right )-11 \sqrt {-6+2 \sqrt {13}}\, \sqrt {2 \sqrt {13}+6}\, \arctanh \left (\frac {\left (\sqrt {13}+3\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{52 \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \sqrt {-6+2 \sqrt {13}}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}\right )-4 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right ) \sqrt {13}+12 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \left (2+3 \tan \left (d x +c \right )\right ) \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )} \]
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 = 6.08, size = 129, normalized size = 1.45 \[ -2\,\mathrm {atanh}\left (\frac {4\,d\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}+6\,d\,\mathrm {tan}\left (c+d\,x\right )\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-3\,\mathrm {tan}\left (c+d\,x\right )-2}}\right )\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}-2\,\mathrm {atanh}\left (\frac {4\,d\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}+6\,d\,\mathrm {tan}\left (c+d\,x\right )\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-3\,\mathrm {tan}\left (c+d\,x\right )-2}}\right )\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}} \]
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 \[ \int \frac {1}{\sqrt {- 3 \tan {\left (c + d x \right )} - 2} \sqrt {\tan {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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