Integrand size = 32, antiderivative size = 121 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=-\frac {i \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{\sqrt {2} \sqrt {a} f}+\frac {i \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}} \]
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Time = 0.24 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {3627, 3625, 214} \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\frac {i \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}-\frac {i \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{\sqrt {2} \sqrt {a} f} \]
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Rule 214
Rule 3625
Rule 3627
Rubi steps \begin{align*} \text {integral}& = \frac {i \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}+\frac {(c-i d) \int \frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 a} \\ & = \frac {i \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}-\frac {(a (i c+d)) \text {Subst}\left (\int \frac {1}{a c-i a d-2 a^2 x^2} \, dx,x,\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}}\right )}{f} \\ & = -\frac {i \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{\sqrt {2} \sqrt {a} f}+\frac {i \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}} \\ \end{align*}
Time = 1.15 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\frac {i \left (\frac {\sqrt {2} \sqrt {-a (c-i d)} \arctan \left (\frac {\sqrt {-a (c-i d)} \sqrt {a+i a \tan (e+f x)}}{\sqrt {2} a \sqrt {c+d \tan (e+f x)}}\right )}{a}+\frac {2 \sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}}\right )}{2 f} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 876 vs. \(2 (95 ) = 190\).
Time = 3.48 (sec) , antiderivative size = 877, normalized size of antiderivative = 7.25
method | result | size |
derivativedivides | \(-\frac {\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \left (-i \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, d \left (\tan ^{2}\left (f x +e \right )\right )+2 i \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, c \tan \left (f x +e \right )-\ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, c \left (\tan ^{2}\left (f x +e \right )\right )+i \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, d -2 \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, d \tan \left (f x +e \right )+\ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, c -4 i c \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \tan \left (f x +e \right )-4 i \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, d +4 \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, d \tan \left (f x +e \right )-4 c \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\right )}{4 f a \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \left (i c -d \right ) \left (-\tan \left (f x +e \right )+i\right )^{2}}\) | \(877\) |
default | \(-\frac {\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \left (-i \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, d \left (\tan ^{2}\left (f x +e \right )\right )+2 i \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, c \tan \left (f x +e \right )-\ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, c \left (\tan ^{2}\left (f x +e \right )\right )+i \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, d -2 \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, d \tan \left (f x +e \right )+\ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, c -4 i c \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \tan \left (f x +e \right )-4 i \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, d +4 \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, d \tan \left (f x +e \right )-4 c \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\right )}{4 f a \sqrt {a \left (1+i \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )}\, \left (i c -d \right ) \left (-\tan \left (f x +e \right )+i\right )^{2}}\) | \(877\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (89) = 178\).
Time = 0.26 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.91 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\frac {{\left (\sqrt {2} a f \sqrt {-\frac {c - i \, d}{a f^{2}}} e^{\left (i \, f x + i \, e\right )} \log \left (i \, \sqrt {2} a f \sqrt {-\frac {c - i \, d}{a f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}\right ) - \sqrt {2} a f \sqrt {-\frac {c - i \, d}{a f^{2}}} e^{\left (i \, f x + i \, e\right )} \log \left (-i \, \sqrt {2} a f \sqrt {-\frac {c - i \, d}{a f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}\right ) - 2 \, \sqrt {2} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (-i \, e^{\left (2 i \, f x + 2 i \, e\right )} - i\right )}\right )} e^{\left (-i \, f x - i \, e\right )}}{4 \, a f} \]
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\[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\int \frac {\sqrt {c + d \tan {\left (e + f x \right )}}}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )}}\, dx \]
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Timed out. \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\text {Exception raised: TypeError} \]
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Time = 21.84 (sec) , antiderivative size = 1724, normalized size of antiderivative = 14.25 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\text {Too large to display} \]
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