Integrand size = 34, antiderivative size = 137 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {2 \sqrt {2} a^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {2 a (A-i B) \sqrt {a+i a \tan (c+d x)}}{d}+\frac {2 A (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {2 i B (a+i a \tan (c+d x))^{5/2}}{5 a d} \]
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Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3673, 3608, 3559, 3561, 212} \[ \int \tan (c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {2 \sqrt {2} a^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {2 a (A-i B) \sqrt {a+i a \tan (c+d x)}}{d}+\frac {2 A (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {2 i B (a+i a \tan (c+d x))^{5/2}}{5 a d} \]
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Rule 212
Rule 3559
Rule 3561
Rule 3608
Rule 3673
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i B (a+i a \tan (c+d x))^{5/2}}{5 a d}+\int (a+i a \tan (c+d x))^{3/2} (-B+A \tan (c+d x)) \, dx \\ & = \frac {2 A (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {2 i B (a+i a \tan (c+d x))^{5/2}}{5 a d}-(i A+B) \int (a+i a \tan (c+d x))^{3/2} \, dx \\ & = \frac {2 a (A-i B) \sqrt {a+i a \tan (c+d x)}}{d}+\frac {2 A (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {2 i B (a+i a \tan (c+d x))^{5/2}}{5 a d}-(2 a (i A+B)) \int \sqrt {a+i a \tan (c+d x)} \, dx \\ & = \frac {2 a (A-i B) \sqrt {a+i a \tan (c+d x)}}{d}+\frac {2 A (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {2 i B (a+i a \tan (c+d x))^{5/2}}{5 a d}-\frac {\left (4 a^2 (A-i B)\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{d} \\ & = -\frac {2 \sqrt {2} a^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {2 a (A-i B) \sqrt {a+i a \tan (c+d x)}}{d}+\frac {2 A (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {2 i B (a+i a \tan (c+d x))^{5/2}}{5 a d} \\ \end{align*}
Time = 1.05 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.83 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {-30 \sqrt {2} a^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )+2 a \sqrt {a+i a \tan (c+d x)} \left (20 A-18 i B+(5 i A+6 B) \tan (c+d x)+3 i B \tan ^2(c+d x)\right )}{15 d} \]
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Time = 0.13 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {-\frac {2 i B \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 A a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 i a^{2} B \sqrt {a +i a \tan \left (d x +c \right )}+2 A \,a^{2} \sqrt {a +i a \tan \left (d x +c \right )}-2 a^{\frac {5}{2}} \left (-i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{a d}\) | \(123\) |
default | \(\frac {-\frac {2 i B \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 A a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 i a^{2} B \sqrt {a +i a \tan \left (d x +c \right )}+2 A \,a^{2} \sqrt {a +i a \tan \left (d x +c \right )}-2 a^{\frac {5}{2}} \left (-i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{a d}\) | \(123\) |
parts | \(\frac {A \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 a \sqrt {a +i a \tan \left (d x +c \right )}-2 a^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )\right )}{d}+\frac {2 i B \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-a^{2} \sqrt {a +i a \tan \left (d x +c \right )}+a^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )\right )}{d a}\) | \(148\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (104) = 208\).
Time = 0.27 (sec) , antiderivative size = 415, normalized size of antiderivative = 3.03 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {15 \, \sqrt {2} \sqrt {\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {4 \, {\left ({\left (-i \, A - B\right )} a^{2} e^{\left (i \, d x + i \, c\right )} - \sqrt {\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{3}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (-i \, A - B\right )} a}\right ) - 15 \, \sqrt {2} \sqrt {\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{3}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {4 \, {\left ({\left (-i \, A - B\right )} a^{2} e^{\left (i \, d x + i \, c\right )} - \sqrt {\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{3}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (-i \, A - B\right )} a}\right ) - 2 \, \sqrt {2} {\left ({\left (25 \, A - 27 i \, B\right )} a e^{\left (5 i \, d x + 5 i \, c\right )} + 10 \, {\left (4 \, A - 3 i \, B\right )} a e^{\left (3 i \, d x + 3 i \, c\right )} + 15 \, {\left (A - i \, B\right )} a e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{15 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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\[ \int \tan (c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}} \left (A + B \tan {\left (c + d x \right )}\right ) \tan {\left (c + d x \right )}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.95 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {15 \, \sqrt {2} {\left (A - i \, B\right )} a^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) - 6 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} B a + 10 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} A a^{2} + 30 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (A - i \, B\right )} a^{3}}{15 \, a^{2} d} \]
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\[ \int \tan (c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \tan \left (d x + c\right ) \,d x } \]
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Time = 7.95 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.19 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {2\,A\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{3\,d}+\frac {2\,A\,a\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{d}-\frac {B\,a\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,2{}\mathrm {i}}{d}-\frac {B\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,2{}\mathrm {i}}{5\,a\,d}-\frac {\sqrt {2}\,B\,{\left (-a\right )}^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-a}}\right )\,2{}\mathrm {i}}{d}-\frac {2\,\sqrt {2}\,A\,a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {a}}\right )}{d} \]
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