Integrand size = 36, antiderivative size = 129 \[ \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {4 \sqrt [4]{-1} a^2 (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}+\frac {4 a^2 (i A+B) \sqrt {\tan (c+d x)}}{d}-\frac {2 a^2 (5 A-7 i B) \tan ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 i B \tan ^{\frac {3}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{5 d} \]
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Time = 0.43 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {3675, 3673, 3609, 3614, 211} \[ \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {4 \sqrt [4]{-1} a^2 (B+i A) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}-\frac {2 a^2 (5 A-7 i B) \tan ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {4 a^2 (B+i A) \sqrt {\tan (c+d x)}}{d}+\frac {2 i B \tan ^{\frac {3}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{5 d} \]
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Rule 211
Rule 3609
Rule 3614
Rule 3673
Rule 3675
Rubi steps \begin{align*} \text {integral}& = \frac {2 i B \tan ^{\frac {3}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{5 d}+\frac {2}{5} \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x)) \left (\frac {1}{2} a (5 A-3 i B)+\frac {1}{2} a (5 i A+7 B) \tan (c+d x)\right ) \, dx \\ & = -\frac {2 a^2 (5 A-7 i B) \tan ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 i B \tan ^{\frac {3}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{5 d}+\frac {2}{5} \int \sqrt {\tan (c+d x)} \left (5 a^2 (A-i B)+5 a^2 (i A+B) \tan (c+d x)\right ) \, dx \\ & = \frac {4 a^2 (i A+B) \sqrt {\tan (c+d x)}}{d}-\frac {2 a^2 (5 A-7 i B) \tan ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 i B \tan ^{\frac {3}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{5 d}+\frac {2}{5} \int \frac {-5 a^2 (i A+B)+5 a^2 (A-i B) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx \\ & = \frac {4 a^2 (i A+B) \sqrt {\tan (c+d x)}}{d}-\frac {2 a^2 (5 A-7 i B) \tan ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 i B \tan ^{\frac {3}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{5 d}+\frac {\left (20 a^4 (i A+B)^2\right ) \text {Subst}\left (\int \frac {1}{-5 a^2 (i A+B)-5 a^2 (A-i B) x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = \frac {4 \sqrt [4]{-1} a^2 (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}+\frac {4 a^2 (i A+B) \sqrt {\tan (c+d x)}}{d}-\frac {2 a^2 (5 A-7 i B) \tan ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 i B \tan ^{\frac {3}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{5 d} \\ \end{align*}
Time = 1.36 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {2 a^2 \left ((15+15 i) \sqrt {2} (A-i B) \text {arctanh}\left (\frac {(1+i) \sqrt {\tan (c+d x)}}{\sqrt {2}}\right )+\sqrt {\tan (c+d x)} \left (-30 i A-30 B+5 (A-2 i B) \tan (c+d x)+3 B \tan ^2(c+d x)\right )\right )}{15 d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (108 ) = 216\).
Time = 0.03 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.95
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {2 B \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {4 i B \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {2 A \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+4 i A \left (\sqrt {\tan }\left (d x +c \right )\right )+4 B \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\left (-2 i A -2 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-2 i B +2 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\) | \(252\) |
default | \(\frac {a^{2} \left (-\frac {2 B \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {4 i B \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {2 A \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+4 i A \left (\sqrt {\tan }\left (d x +c \right )\right )+4 B \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\left (-2 i A -2 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-2 i B +2 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\) | \(252\) |
parts | \(\frac {\left (2 i A \,a^{2}+B \,a^{2}\right ) \left (2 \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}+\frac {\left (2 i B \,a^{2}-A \,a^{2}\right ) \left (\frac {2 \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}+\frac {A \,a^{2} \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4 d}-\frac {B \,a^{2} \left (\frac {2 \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}-2 \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\) | \(436\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (103) = 206\).
Time = 0.27 (sec) , antiderivative size = 441, normalized size of antiderivative = 3.42 \[ \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {15 \, \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (-\frac {2 \, {\left ({\left (A - i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{2}}\right ) - 15 \, \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (-\frac {2 \, {\left ({\left (A - i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{2}}\right ) + 2 \, {\left ({\left (-35 i \, A - 43 \, B\right )} a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, {\left (-10 i \, A - 9 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-25 i \, A - 23 \, B\right )} a^{2}\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{15 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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\[ \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=- a^{2} \left (\int \left (- A \sqrt {\tan {\left (c + d x \right )}}\right )\, dx + \int A \tan ^{\frac {5}{2}}{\left (c + d x \right )}\, dx + \int \left (- B \tan ^{\frac {3}{2}}{\left (c + d x \right )}\right )\, dx + \int B \tan ^{\frac {7}{2}}{\left (c + d x \right )}\, dx + \int \left (- 2 i A \tan ^{\frac {3}{2}}{\left (c + d x \right )}\right )\, dx + \int \left (- 2 i B \tan ^{\frac {5}{2}}{\left (c + d x \right )}\right )\, dx\right ) \]
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Time = 0.29 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.50 \[ \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {12 \, B a^{2} \tan \left (d x + c\right )^{\frac {5}{2}} + 20 \, {\left (A - 2 i \, B\right )} a^{2} \tan \left (d x + c\right )^{\frac {3}{2}} + 120 \, {\left (-i \, A - B\right )} a^{2} \sqrt {\tan \left (d x + c\right )} + 15 \, {\left (2 \, \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )} a^{2}}{30 \, d} \]
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Time = 0.75 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.98 \[ \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {\left (2 i - 2\right ) \, \sqrt {2} {\left (A a^{2} - i \, B a^{2}\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{d} - \frac {2 \, {\left (3 \, B a^{2} d^{4} \tan \left (d x + c\right )^{\frac {5}{2}} + 5 \, A a^{2} d^{4} \tan \left (d x + c\right )^{\frac {3}{2}} - 10 i \, B a^{2} d^{4} \tan \left (d x + c\right )^{\frac {3}{2}} - 30 i \, A a^{2} d^{4} \sqrt {\tan \left (d x + c\right )} - 30 \, B a^{2} d^{4} \sqrt {\tan \left (d x + c\right )}\right )}}{15 \, d^{5}} \]
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Time = 9.11 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.98 \[ \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {A\,a^2\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,4{}\mathrm {i}}{d}-\frac {2\,A\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}}{3\,d}+\frac {4\,B\,a^2\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}}{d}+\frac {B\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}\,4{}\mathrm {i}}{3\,d}-\frac {2\,B\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^{5/2}}{5\,d}+\frac {\sqrt {2}\,A\,a^2\,\ln \left (4\,A\,a^2\,d+\sqrt {2}\,A\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\left (-2-2{}\mathrm {i}\right )\right )\,\left (1+1{}\mathrm {i}\right )}{d}-\frac {\sqrt {4{}\mathrm {i}}\,A\,a^2\,\ln \left (4\,A\,a^2\,d+2\,\sqrt {4{}\mathrm {i}}\,A\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{d}+\frac {\sqrt {2}\,B\,a^2\,\ln \left (-B\,a^2\,d\,4{}\mathrm {i}+\sqrt {2}\,B\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\left (-2+2{}\mathrm {i}\right )\right )\,\left (1-\mathrm {i}\right )}{d}-\frac {\sqrt {-4{}\mathrm {i}}\,B\,a^2\,\ln \left (-B\,a^2\,d\,4{}\mathrm {i}+2\,\sqrt {-4{}\mathrm {i}}\,B\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{d} \]
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