Optimal. Leaf size=323 \[ \frac {(4+4 i) a^{5/2} (B+i A) \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {8 a^2 (655 A-649 i B) \sqrt {a+i a \tan (c+d x)}}{3465 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {4 a^2 (253 B+250 i A) \sqrt {a+i a \tan (c+d x)}}{1155 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 a^2 (212 A-209 i B) \sqrt {a+i a \tan (c+d x)}}{693 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 a^2 (11 B+14 i A) \sqrt {a+i a \tan (c+d x)}}{99 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {8 a^2 (2167 B+2155 i A) \sqrt {a+i a \tan (c+d x)}}{3465 d \sqrt {\tan (c+d x)}}-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{11 d \tan ^{\frac {11}{2}}(c+d x)} \]
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Rubi [A] time = 1.16, antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {3593, 3598, 12, 3544, 205} \[ -\frac {8 a^2 (655 A-649 i B) \sqrt {a+i a \tan (c+d x)}}{3465 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {4 a^2 (253 B+250 i A) \sqrt {a+i a \tan (c+d x)}}{1155 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 a^2 (212 A-209 i B) \sqrt {a+i a \tan (c+d x)}}{693 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 a^2 (11 B+14 i A) \sqrt {a+i a \tan (c+d x)}}{99 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {8 a^2 (2167 B+2155 i A) \sqrt {a+i a \tan (c+d x)}}{3465 d \sqrt {\tan (c+d x)}}+\frac {(4+4 i) a^{5/2} (B+i A) \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{11 d \tan ^{\frac {11}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 3544
Rule 3593
Rule 3598
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac {13}{2}}(c+d x)} \, dx &=-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{11 d \tan ^{\frac {11}{2}}(c+d x)}+\frac {2}{11} \int \frac {(a+i a \tan (c+d x))^{3/2} \left (\frac {1}{2} a (14 i A+11 B)-\frac {1}{2} a (8 A-11 i B) \tan (c+d x)\right )}{\tan ^{\frac {11}{2}}(c+d x)} \, dx\\ &=-\frac {2 a^2 (14 i A+11 B) \sqrt {a+i a \tan (c+d x)}}{99 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{11 d \tan ^{\frac {11}{2}}(c+d x)}+\frac {4}{99} \int \frac {\sqrt {a+i a \tan (c+d x)} \left (-\frac {1}{4} a^2 (212 A-209 i B)-\frac {1}{4} a^2 (184 i A+187 B) \tan (c+d x)\right )}{\tan ^{\frac {9}{2}}(c+d x)} \, dx\\ &=-\frac {2 a^2 (14 i A+11 B) \sqrt {a+i a \tan (c+d x)}}{99 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {2 a^2 (212 A-209 i B) \sqrt {a+i a \tan (c+d x)}}{693 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{11 d \tan ^{\frac {11}{2}}(c+d x)}+\frac {8 \int \frac {\sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{4} a^3 (250 i A+253 B)+\frac {3}{4} a^3 (212 A-209 i B) \tan (c+d x)\right )}{\tan ^{\frac {7}{2}}(c+d x)} \, dx}{693 a}\\ &=-\frac {2 a^2 (14 i A+11 B) \sqrt {a+i a \tan (c+d x)}}{99 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {2 a^2 (212 A-209 i B) \sqrt {a+i a \tan (c+d x)}}{693 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {4 a^2 (250 i A+253 B) \sqrt {a+i a \tan (c+d x)}}{1155 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{11 d \tan ^{\frac {11}{2}}(c+d x)}+\frac {16 \int \frac {\sqrt {a+i a \tan (c+d x)} \left (\frac {3}{4} a^4 (655 A-649 i B)+\frac {3}{2} a^4 (250 i A+253 B) \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx}{3465 a^2}\\ &=-\frac {2 a^2 (14 i A+11 B) \sqrt {a+i a \tan (c+d x)}}{99 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {2 a^2 (212 A-209 i B) \sqrt {a+i a \tan (c+d x)}}{693 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {4 a^2 (250 i A+253 B) \sqrt {a+i a \tan (c+d x)}}{1155 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {8 a^2 (655 A-649 i B) \sqrt {a+i a \tan (c+d x)}}{3465 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{11 d \tan ^{\frac {11}{2}}(c+d x)}+\frac {32 \int \frac {\sqrt {a+i a \tan (c+d x)} \left (\frac {3}{8} a^5 (2155 i A+2167 B)-\frac {3}{4} a^5 (655 A-649 i B) \tan (c+d x)\right )}{\tan ^{\frac {3}{2}}(c+d x)} \, dx}{10395 a^3}\\ &=-\frac {2 a^2 (14 i A+11 B) \sqrt {a+i a \tan (c+d x)}}{99 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {2 a^2 (212 A-209 i B) \sqrt {a+i a \tan (c+d x)}}{693 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {4 a^2 (250 i A+253 B) \sqrt {a+i a \tan (c+d x)}}{1155 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {8 a^2 (655 A-649 i B) \sqrt {a+i a \tan (c+d x)}}{3465 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {8 a^2 (2155 i A+2167 B) \sqrt {a+i a \tan (c+d x)}}{3465 d \sqrt {\tan (c+d x)}}-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{11 d \tan ^{\frac {11}{2}}(c+d x)}+\frac {64 \int -\frac {10395 a^6 (A-i B) \sqrt {a+i a \tan (c+d x)}}{16 \sqrt {\tan (c+d x)}} \, dx}{10395 a^4}\\ &=-\frac {2 a^2 (14 i A+11 B) \sqrt {a+i a \tan (c+d x)}}{99 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {2 a^2 (212 A-209 i B) \sqrt {a+i a \tan (c+d x)}}{693 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {4 a^2 (250 i A+253 B) \sqrt {a+i a \tan (c+d x)}}{1155 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {8 a^2 (655 A-649 i B) \sqrt {a+i a \tan (c+d x)}}{3465 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {8 a^2 (2155 i A+2167 B) \sqrt {a+i a \tan (c+d x)}}{3465 d \sqrt {\tan (c+d x)}}-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{11 d \tan ^{\frac {11}{2}}(c+d x)}-\left (4 a^2 (A-i B)\right ) \int \frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx\\ &=-\frac {2 a^2 (14 i A+11 B) \sqrt {a+i a \tan (c+d x)}}{99 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {2 a^2 (212 A-209 i B) \sqrt {a+i a \tan (c+d x)}}{693 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {4 a^2 (250 i A+253 B) \sqrt {a+i a \tan (c+d x)}}{1155 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {8 a^2 (655 A-649 i B) \sqrt {a+i a \tan (c+d x)}}{3465 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {8 a^2 (2155 i A+2167 B) \sqrt {a+i a \tan (c+d x)}}{3465 d \sqrt {\tan (c+d x)}}-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{11 d \tan ^{\frac {11}{2}}(c+d x)}+\frac {\left (8 a^4 (i A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{-i a-2 a^2 x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}\\ &=\frac {(4+4 i) a^{5/2} (i A+B) \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {2 a^2 (14 i A+11 B) \sqrt {a+i a \tan (c+d x)}}{99 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {2 a^2 (212 A-209 i B) \sqrt {a+i a \tan (c+d x)}}{693 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {4 a^2 (250 i A+253 B) \sqrt {a+i a \tan (c+d x)}}{1155 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {8 a^2 (655 A-649 i B) \sqrt {a+i a \tan (c+d x)}}{3465 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {8 a^2 (2155 i A+2167 B) \sqrt {a+i a \tan (c+d x)}}{3465 d \sqrt {\tan (c+d x)}}-\frac {2 a A (a+i a \tan (c+d x))^{3/2}}{11 d \tan ^{\frac {11}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 20.24, size = 328, normalized size = 1.02 \[ \frac {4 \sqrt {2} a^2 (B+i A) e^{-i (c+d x)} \sqrt {-1+e^{2 i (c+d x)}} \sqrt {\frac {a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left (\frac {e^{i (c+d x)}}{\sqrt {-1+e^{2 i (c+d x)}}}\right )}{d \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}}}-\frac {a^2 \csc ^3(c+d x) \sec ^2(c+d x) \sqrt {a+i a \tan (c+d x)} (66 (95 A-47 i B) \cos (c+d x)+(-5225 A+6743 i B) \cos (3 (c+d x))+84810 i A \sin (c+d x)-42185 i A \sin (3 (c+d x))+10925 i A \sin (5 (c+d x))+3995 A \cos (5 (c+d x))+84414 B \sin (c+d x)-43703 B \sin (3 (c+d x))+10571 B \sin (5 (c+d x))-3641 i B \cos (5 (c+d x)))}{27720 d \tan ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.27, size = 755, normalized size = 2.34 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\tan \left (d x + c\right )^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 976, normalized size = 3.02 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{{\mathrm {tan}\left (c+d\,x\right )}^{13/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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