Optimal. Leaf size=89 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}}\right )}{\sqrt {2-3 i} d}+\frac {\text {ArcTan}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, 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 = {3656, 926, 95,
211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}}\right )}{\sqrt {2-3 i} d}+\frac {\text {ArcTan}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 95
Rule 211
Rule 926
Rule 3656
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-2 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {3-2 x} \sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {i}{2 \sqrt {3-2 x} (i-x) \sqrt {x}}+\frac {i}{2 \sqrt {3-2 x} \sqrt {x} (i+x)}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {i \text {Subst}\left (\int \frac {1}{\sqrt {3-2 x} (i-x) \sqrt {x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {i \text {Subst}\left (\int \frac {1}{\sqrt {3-2 x} \sqrt {x} (i+x)} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {i \text {Subst}\left (\int \frac {1}{i-(3-2 i) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}}\right )}{d}+\frac {i \text {Subst}\left (\int \frac {1}{i+(3+2 i) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}}\right )}{d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}}\right )}{\sqrt {2-3 i} d}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 101, normalized size = 1.13 \begin {gather*} \frac {-\sqrt {2+3 i} \text {ArcTan}\left (\frac {\sqrt {\frac {2}{13}+\frac {3 i}{13}} \sqrt {3-2 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )+\sqrt {-2+3 i} \tanh ^{-1}\left (\frac {\sqrt {-\frac {2}{13}+\frac {3 i}{13}} \sqrt {3-2 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )}{\sqrt {13} d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(434\) vs.
\(2(73)=146\).
time = 0.82, size = 435, normalized size = 4.89
method | result | size |
derivativedivides | \(\frac {\sqrt {3-2 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right ) \left (4 \sqrt {-4+2 \sqrt {13}}\, \sqrt {13}\, \sqrt {2 \sqrt {13}+4}\, \arctanh \left (\frac {\left (2+\sqrt {13}\right ) \left (\sqrt {13}+2+3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{351 \sqrt {-4+2 \sqrt {13}}\, \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}\right )-17 \sqrt {-4+2 \sqrt {13}}\, \sqrt {2 \sqrt {13}+4}\, \arctanh \left (\frac {\left (2+\sqrt {13}\right ) \left (\sqrt {13}+2+3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{351 \sqrt {-4+2 \sqrt {13}}\, \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}\right )-18 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right ) \sqrt {13}+36 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+4}\, \left (-3+2 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right )}\) | \(435\) |
default | \(\frac {\sqrt {3-2 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right ) \left (4 \sqrt {-4+2 \sqrt {13}}\, \sqrt {13}\, \sqrt {2 \sqrt {13}+4}\, \arctanh \left (\frac {\left (2+\sqrt {13}\right ) \left (\sqrt {13}+2+3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{351 \sqrt {-4+2 \sqrt {13}}\, \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}\right )-17 \sqrt {-4+2 \sqrt {13}}\, \sqrt {2 \sqrt {13}+4}\, \arctanh \left (\frac {\left (2+\sqrt {13}\right ) \left (\sqrt {13}+2+3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{351 \sqrt {-4+2 \sqrt {13}}\, \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}\right )-18 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right ) \sqrt {13}+36 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+4}\, \left (-3+2 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right )}\) | \(435\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {3 - 2 \tan {\left (c + d x \right )}} \sqrt {\tan {\left (c + d x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1061 vs.
\(2 (65) = 130\).
time = 0.69, size = 1061, normalized size = 11.92 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 6.36, size = 201, normalized size = 2.26 \begin {gather*} -\mathrm {atan}\left (\frac {\sqrt {3}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,\left (6-4{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,\sqrt {3-2\,\mathrm {tan}\left (c+d\,x\right )}\,\left (-6+4{}\mathrm {i}\right )}{2\,\mathrm {tan}\left (c+d\,x\right )+\sqrt {3}\,\sqrt {3-2\,\mathrm {tan}\left (c+d\,x\right )}-3}\right )\,\sqrt {\frac {-\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {\sqrt {3}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,\left (6+4{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,\sqrt {3-2\,\mathrm {tan}\left (c+d\,x\right )}\,\left (-6-4{}\mathrm {i}\right )}{2\,\mathrm {tan}\left (c+d\,x\right )+\sqrt {9-6\,\mathrm {tan}\left (c+d\,x\right )}-3}\right )\,\sqrt {\frac {-\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________