Optimal. Leaf size=246 \[ \frac {4 \sqrt {2} a^{5/2} (B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {2 a^2 (3 A-4 i B) \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}+\frac {2 a^2 (46 B+45 i A) \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {8 a^2 (46 B+45 i A) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {8 a (59 B+60 i A) (a+i a \tan (c+d x))^{3/2}}{315 d}+\frac {2 i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d} \]
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Rubi [A] time = 0.75, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3594, 3597, 3592, 3527, 3480, 206} \[ -\frac {2 a^2 (3 A-4 i B) \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}+\frac {2 a^2 (46 B+45 i A) \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {8 a^2 (46 B+45 i A) \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {4 \sqrt {2} a^{5/2} (B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {8 a (59 B+60 i A) (a+i a \tan (c+d x))^{3/2}}{315 d}+\frac {2 i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3480
Rule 3527
Rule 3592
Rule 3594
Rule 3597
Rubi steps
\begin {align*} \int \tan ^2(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx &=\frac {2 i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\frac {2}{9} \int \tan ^2(c+d x) (a+i a \tan (c+d x))^{3/2} \left (\frac {3}{2} a (3 A-2 i B)+\frac {3}{2} a (3 i A+4 B) \tan (c+d x)\right ) \, dx\\ &=-\frac {2 a^2 (3 A-4 i B) \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}+\frac {2 i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\frac {4}{63} \int \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (\frac {3}{4} a^2 (39 A-38 i B)+\frac {3}{4} a^2 (45 i A+46 B) \tan (c+d x)\right ) \, dx\\ &=\frac {2 a^2 (45 i A+46 B) \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (3 A-4 i B) \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}+\frac {2 i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\frac {8 \int \tan (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{2} a^3 (45 i A+46 B)+\frac {3}{2} a^3 (60 A-59 i B) \tan (c+d x)\right ) \, dx}{315 a}\\ &=\frac {2 a^2 (45 i A+46 B) \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (3 A-4 i B) \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}-\frac {8 a (60 i A+59 B) (a+i a \tan (c+d x))^{3/2}}{315 d}+\frac {2 i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\frac {8 \int \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{2} a^3 (60 A-59 i B)-\frac {3}{2} a^3 (45 i A+46 B) \tan (c+d x)\right ) \, dx}{315 a}\\ &=-\frac {8 a^2 (45 i A+46 B) \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {2 a^2 (45 i A+46 B) \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (3 A-4 i B) \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}-\frac {8 a (60 i A+59 B) (a+i a \tan (c+d x))^{3/2}}{315 d}+\frac {2 i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}-\left (4 a^2 (A-i B)\right ) \int \sqrt {a+i a \tan (c+d x)} \, dx\\ &=-\frac {8 a^2 (45 i A+46 B) \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {2 a^2 (45 i A+46 B) \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (3 A-4 i B) \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}-\frac {8 a (60 i A+59 B) (a+i a \tan (c+d x))^{3/2}}{315 d}+\frac {2 i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}+\frac {\left (8 a^3 (i A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{d}\\ &=\frac {4 \sqrt {2} a^{5/2} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {8 a^2 (45 i A+46 B) \sqrt {a+i a \tan (c+d x)}}{105 d}+\frac {2 a^2 (45 i A+46 B) \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{105 d}-\frac {2 a^2 (3 A-4 i B) \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{21 d}-\frac {8 a (60 i A+59 B) (a+i a \tan (c+d x))^{3/2}}{315 d}+\frac {2 i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{9 d}\\ \end {align*}
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Mathematica [A] time = 6.01, size = 284, normalized size = 1.15 \[ \frac {(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \left (4 \sqrt {2} (B+i A) e^{-3 i (c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \sinh ^{-1}\left (e^{i (c+d x)}\right )-\frac {i (\cos (2 c)-i \sin (2 c)) \sec ^{\frac {9}{2}}(c+d x) (12 (260 A-251 i B) \cos (2 (c+d x))+(915 A-961 i B) \cos (4 (c+d x))+390 i A \sin (2 (c+d x))+285 i A \sin (4 (c+d x))+2205 A+282 B \sin (2 (c+d x))+331 B \sin (4 (c+d x))-2331 i B)}{1260 (\cos (d x)+i \sin (d x))^2}\right )}{d \sec ^{\frac {7}{2}}(c+d x) (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 535, normalized size = 2.17 \[ \frac {315 \, \sqrt {-\frac {{\left (128 \, A^{2} - 256 i \, A B - 128 \, B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left ({\left (16 i \, A + 16 \, B\right )} a^{3} e^{\left (i \, d x + i \, c\right )} + \sqrt {2} \sqrt {-\frac {{\left (128 \, A^{2} - 256 i \, A B - 128 \, B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + 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 (4 i \, A + 4 \, B\right )} a^{2}}\right ) - 315 \, \sqrt {-\frac {{\left (128 \, A^{2} - 256 i \, A B - 128 \, B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left ({\left (16 i \, A + 16 \, B\right )} a^{3} e^{\left (i \, d x + i \, c\right )} - \sqrt {2} \sqrt {-\frac {{\left (128 \, A^{2} - 256 i \, A B - 128 \, B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + 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 (4 i \, A + 4 \, B\right )} a^{2}}\right ) + \sqrt {2} {\left ({\left (-9600 i \, A - 10336 \, B\right )} a^{2} e^{\left (9 i \, d x + 9 i \, c\right )} + {\left (-28080 i \, A - 26352 \, B\right )} a^{2} e^{\left (7 i \, d x + 7 i \, c\right )} + {\left (-35280 i \, A - 37296 \, B\right )} a^{2} e^{\left (5 i \, d x + 5 i \, c\right )} + {\left (-21840 i \, A - 21840 \, B\right )} a^{2} e^{\left (3 i \, d x + 3 i \, c\right )} + {\left (-5040 i \, A - 5040 \, B\right )} a^{2} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{1260 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 206, normalized size = 0.84 \[ -\frac {2 i \left (-\frac {i B \left (a +i a \tan \left (d x +c \right )\right )^{\frac {9}{2}}}{9}+\frac {i B \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}} a}{7}+\frac {A \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}} a}{7}-\frac {i a^{2} B \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {i B \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}} a^{3}}{3}+\frac {A \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}} a^{3}}{3}-2 i B \,a^{4} \sqrt {a +i a \tan \left (d x +c \right )}+2 A \,a^{4} \sqrt {a +i a \tan \left (d x +c \right )}-2 a^{\frac {9}{2}} \left (-i B +A \right ) \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )\right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 176, normalized size = 0.72 \[ -\frac {2 i \, {\left (315 \, \sqrt {2} {\left (A - i \, B\right )} a^{\frac {11}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) - 35 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} B a + 45 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} {\left (A + i \, B\right )} a^{2} - 63 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} B a^{3} + 105 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} {\left (A - i \, B\right )} a^{4} + 630 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (A - i \, B\right )} a^{5}\right )}}{315 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.57, size = 258, normalized size = 1.05 \[ -\frac {2\,B\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{5\,d}-\frac {A\,a\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,2{}\mathrm {i}}{3\,d}-\frac {2\,B\,a\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{3\,d}-\frac {A\,a^2\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,4{}\mathrm {i}}{d}-\frac {A\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2}\,2{}\mathrm {i}}{7\,a\,d}-\frac {4\,B\,a^2\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{d}+\frac {2\,B\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2}}{7\,a\,d}-\frac {2\,B\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{9/2}}{9\,a^2\,d}+\frac {\sqrt {2}\,A\,{\left (-a\right )}^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-a}}\right )\,4{}\mathrm {i}}{d}-\frac {\sqrt {2}\,B\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,\sqrt {a}}\right )\,4{}\mathrm {i}}{d} \]
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 \[ \int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}} \left (A + B \tan {\left (c + d x \right )}\right ) \tan ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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