Integrand size = 36, antiderivative size = 134 \[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {8 \sqrt [4]{-1} a^3 (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}-\frac {16 a^3 B \sqrt {\tan (c+d x)}}{3 d}-\frac {2 a A (a+i a \tan (c+d x))^2}{d \sqrt {\tan (c+d x)}}+\frac {2 (3 i A-B) \sqrt {\tan (c+d x)} \left (a^3+i a^3 \tan (c+d x)\right )}{3 d} \]
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Time = 0.54 (sec) , antiderivative size = 134, 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 = {3674, 3675, 3673, 3614, 211} \[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {8 \sqrt [4]{-1} a^3 (B+i A) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}+\frac {2 (-B+3 i A) \sqrt {\tan (c+d x)} \left (a^3+i a^3 \tan (c+d x)\right )}{3 d}-\frac {16 a^3 B \sqrt {\tan (c+d x)}}{3 d}-\frac {2 a A (a+i a \tan (c+d x))^2}{d \sqrt {\tan (c+d x)}} \]
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Rule 211
Rule 3614
Rule 3673
Rule 3674
Rule 3675
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a A (a+i a \tan (c+d x))^2}{d \sqrt {\tan (c+d x)}}+2 \int \frac {(a+i a \tan (c+d x))^2 \left (\frac {1}{2} a (5 i A+B)+\frac {1}{2} a (3 A+i B) \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx \\ & = -\frac {2 a A (a+i a \tan (c+d x))^2}{d \sqrt {\tan (c+d x)}}+\frac {2 (3 i A-B) \sqrt {\tan (c+d x)} \left (a^3+i a^3 \tan (c+d x)\right )}{3 d}+\frac {4}{3} \int \frac {(a+i a \tan (c+d x)) \left (a^2 (3 i A+B)+2 i a^2 B \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx \\ & = -\frac {16 a^3 B \sqrt {\tan (c+d x)}}{3 d}-\frac {2 a A (a+i a \tan (c+d x))^2}{d \sqrt {\tan (c+d x)}}+\frac {2 (3 i A-B) \sqrt {\tan (c+d x)} \left (a^3+i a^3 \tan (c+d x)\right )}{3 d}+\frac {4}{3} \int \frac {3 a^3 (i A+B)-3 a^3 (A-i B) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx \\ & = -\frac {16 a^3 B \sqrt {\tan (c+d x)}}{3 d}-\frac {2 a A (a+i a \tan (c+d x))^2}{d \sqrt {\tan (c+d x)}}+\frac {2 (3 i A-B) \sqrt {\tan (c+d x)} \left (a^3+i a^3 \tan (c+d x)\right )}{3 d}+\frac {\left (24 a^6 (i A+B)^2\right ) \text {Subst}\left (\int \frac {1}{3 a^3 (i A+B)+3 a^3 (A-i B) x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = -\frac {8 \sqrt [4]{-1} a^3 (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}-\frac {16 a^3 B \sqrt {\tan (c+d x)}}{3 d}-\frac {2 a A (a+i a \tan (c+d x))^2}{d \sqrt {\tan (c+d x)}}+\frac {2 (3 i A-B) \sqrt {\tan (c+d x)} \left (a^3+i a^3 \tan (c+d x)\right )}{3 d} \\ \end{align*}
Time = 1.89 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.70 \[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 i a^3 \left (-3 i A+12 \sqrt [4]{-1} (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)}+3 (A-3 i B) \tan (c+d x)+B \tan ^2(c+d x)\right )}{3 d \sqrt {\tan (c+d x)}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (114 ) = 228\).
Time = 0.04 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.80
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {2 i B \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-2 i A \left (\sqrt {\tan }\left (d x +c \right )\right )-6 B \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {2 A}{\sqrt {\tan \left (d x +c \right )}}+\frac {\left (4 i A +4 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (4 i B -4 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\) | \(241\) |
default | \(\frac {a^{3} \left (-\frac {2 i B \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-2 i A \left (\sqrt {\tan }\left (d x +c \right )\right )-6 B \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {2 A}{\sqrt {\tan \left (d x +c \right )}}+\frac {\left (4 i A +4 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (4 i B -4 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\) | \(241\) |
parts | \(\frac {\left (-i A \,a^{3}-3 B \,a^{3}\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 (3 i A \,a^{3}+B \,a^{3}\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 d}+\frac {\left (3 i B \,a^{3}-3 A \,a^{3}\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 d}+\frac {A \,a^{3} \left (-\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}-\frac {2}{\sqrt {\tan \left (d x +c \right )}}\right )}{d}-\frac {i B \,a^{3} \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}\) | \(529\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (108) = 216\).
Time = 0.26 (sec) , antiderivative size = 405, normalized size of antiderivative = 3.02 \[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \, {\left (3 \, \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - d\right )} \log \left (-\frac {2 \, {\left ({\left (A - i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{3}}\right ) - 3 \, \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - d\right )} \log \left (-\frac {2 \, {\left ({\left (A - i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {-\frac {{\left (-i \, A^{2} - 2 \, A B + i \, B^{2}\right )} a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{3}}\right ) - 2 \, {\left ({\left (3 i \, A + 5 \, B\right )} a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (3 i \, A - B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 4 \, B a^{3}\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )}}{3 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - d\right )}} \]
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\[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=- i a^{3} \left (\int \left (- \frac {3 A}{\sqrt {\tan {\left (c + d x \right )}}}\right )\, dx + \int A \tan ^{\frac {3}{2}}{\left (c + d x \right )}\, dx + \int \left (- 3 B \sqrt {\tan {\left (c + d x \right )}}\right )\, dx + \int B \tan ^{\frac {5}{2}}{\left (c + d x \right )}\, dx + \int \frac {i A}{\tan ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \left (- 3 i A \sqrt {\tan {\left (c + d x \right )}}\right )\, dx + \int \frac {i B}{\sqrt {\tan {\left (c + d x \right )}}}\, dx + \int \left (- 3 i B \tan ^{\frac {3}{2}}{\left (c + d x \right )}\right )\, dx\right ) \]
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Time = 0.44 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.42 \[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 i \, B a^{3} \tan \left (d x + c\right )^{\frac {3}{2}} + 6 \, {\left (i \, A + 3 \, B\right )} a^{3} \sqrt {\tan \left (d x + c\right )} - 3 \, {\left (2 \, \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )} a^{3} + \frac {6 \, A a^{3}}{\sqrt {\tan \left (d x + c\right )}}}{3 \, d} \]
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Time = 1.06 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.81 \[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 \, A a^{3}}{d \sqrt {\tan \left (d x + c\right )}} + \frac {\left (4 i - 4\right ) \, \sqrt {2} {\left (A a^{3} - i \, B a^{3}\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{d} - \frac {2 \, {\left (i \, B a^{3} d^{2} \tan \left (d x + c\right )^{\frac {3}{2}} + 3 i \, A a^{3} d^{2} \sqrt {\tan \left (d x + c\right )} + 9 \, B a^{3} d^{2} \sqrt {\tan \left (d x + c\right )}\right )}}{3 \, d^{3}} \]
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Time = 8.63 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.78 \[ \int \frac {(a+i a \tan (c+d x))^3 (A+B \tan (c+d x))}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2\,A\,a^3}{d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}}-\frac {A\,a^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,2{}\mathrm {i}}{d}-\frac {6\,B\,a^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}}{d}-\frac {B\,a^3\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}\,2{}\mathrm {i}}{3\,d}+\frac {\sqrt {2}\,A\,a^3\,\ln \left (-8\,A\,a^3\,d+\sqrt {2}\,A\,a^3\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\left (-4-4{}\mathrm {i}\right )\right )\,\left (2+2{}\mathrm {i}\right )}{d}-\frac {\sqrt {16{}\mathrm {i}}\,A\,a^3\,\ln \left (-8\,A\,a^3\,d+2\,\sqrt {16{}\mathrm {i}}\,A\,a^3\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{d}+\frac {\sqrt {2}\,B\,a^3\,\ln \left (B\,a^3\,d\,8{}\mathrm {i}+\sqrt {2}\,B\,a^3\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\left (-4+4{}\mathrm {i}\right )\right )\,\left (2-2{}\mathrm {i}\right )}{d}-\frac {\sqrt {-16{}\mathrm {i}}\,B\,a^3\,\ln \left (B\,a^3\,d\,8{}\mathrm {i}+2\,\sqrt {-16{}\mathrm {i}}\,B\,a^3\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{d} \]
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