Optimal. Leaf size=162 \[ \frac {(a-b) \text {ArcTan}\left (1-\sqrt {2} \sqrt {-\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a-b) \text {ArcTan}\left (1+\sqrt {2} \sqrt {-\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {-\tan (c+d x)}-\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {-\tan (c+d x)}-\tan (c+d x)\right )}{2 \sqrt {2} d} \]
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Rubi [A]
time = 0.07, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3615, 1182,
1176, 631, 210, 1179, 642} \begin {gather*} \frac {(a-b) \text {ArcTan}\left (1-\sqrt {2} \sqrt {-\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a-b) \text {ArcTan}\left (\sqrt {2} \sqrt {-\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {(a+b) \log \left (-\tan (c+d x)-\sqrt {2} \sqrt {-\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {(a+b) \log \left (-\tan (c+d x)+\sqrt {2} \sqrt {-\tan (c+d x)}+1\right )}{2 \sqrt {2} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3615
Rubi steps
\begin {align*} \int \frac {a+b \tan (c+d x)}{\sqrt {-\tan (c+d x)}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {-a+b x^2}{1+x^4} \, dx,x,\sqrt {-\tan (c+d x)}\right )}{d}\\ &=-\frac {(a-b) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {-\tan (c+d x)}\right )}{d}-\frac {(a+b) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {-\tan (c+d x)}\right )}{d}\\ &=-\frac {(a-b) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {-\tan (c+d x)}\right )}{2 d}-\frac {(a-b) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {-\tan (c+d x)}\right )}{2 d}+\frac {(a+b) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {-\tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {(a+b) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {-\tan (c+d x)}\right )}{2 \sqrt {2} d}\\ &=\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {-\tan (c+d x)}-\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {-\tan (c+d x)}-\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a-b) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {-\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {-\tan (c+d x)}\right )}{\sqrt {2} d}\\ &=\frac {(a-b) \tan ^{-1}\left (1-\sqrt {2} \sqrt {-\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a-b) \tan ^{-1}\left (1+\sqrt {2} \sqrt {-\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {-\tan (c+d x)}-\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {-\tan (c+d x)}-\tan (c+d x)\right )}{2 \sqrt {2} d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.16, size = 82, normalized size = 0.51 \begin {gather*} \frac {\sqrt [4]{-1} \left ((a-i b) \text {ArcTan}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+(a+i b) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right ) \tan ^{\frac {3}{2}}(c+d x)}{d (-\tan (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 202, normalized size = 1.25
method | result | size |
derivativedivides | \(\frac {-\frac {a \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}-\tan \left (d x +c \right )}{1-\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}-\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )\right )}{4}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}-\tan \left (d x +c \right )}{1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}-\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )\right )}{4}}{d}\) | \(202\) |
default | \(\frac {-\frac {a \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}-\tan \left (d x +c \right )}{1-\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}-\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )\right )}{4}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}-\tan \left (d x +c \right )}{1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}-\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )\right )}{4}}{d}\) | \(202\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 136, normalized size = 0.84 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {-\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {-\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a + b\right )} \log \left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a + b\right )} \log \left (-\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \tan \left (d x + c\right ) + 1\right )}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2694 vs.
\(2 (134) = 268\).
time = 0.99, size = 2694, normalized size = 16.63 \begin {gather*} \text {Too large to display} \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 {a + b \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.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.59, size = 94, normalized size = 0.58 \begin {gather*} \frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {-\mathrm {tan}\left (c+d\,x\right )}\right )}{d}-\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {-\mathrm {tan}\left (c+d\,x\right )}\right )}{d}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {-\mathrm {tan}\left (c+d\,x\right )}\right )\,1{}\mathrm {i}}{d}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {-\mathrm {tan}\left (c+d\,x\right )}\right )\,1{}\mathrm {i}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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