Optimal. Leaf size=311 \[ \frac {21 (B+11 i A) \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{512 \sqrt {2} a^3 c^{5/2} f}-\frac {21 (B+11 i A)}{512 a^3 c^2 f \sqrt {c-i c \tan (e+f x)}}+\frac {-B+i A}{6 a^3 f (1+i \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2}}-\frac {7 (B+11 i A)}{256 a^3 c f (c-i c \tan (e+f x))^{3/2}}-\frac {21 (B+11 i A)}{640 a^3 f (c-i c \tan (e+f x))^{5/2}}+\frac {3 (B+11 i A)}{64 a^3 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}+\frac {B+11 i A}{48 a^3 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2}} \]
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Rubi [A] time = 0.35, antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {3588, 78, 51, 63, 208} \[ -\frac {21 (B+11 i A)}{512 a^3 c^2 f \sqrt {c-i c \tan (e+f x)}}+\frac {21 (B+11 i A) \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{512 \sqrt {2} a^3 c^{5/2} f}+\frac {-B+i A}{6 a^3 f (1+i \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2}}-\frac {7 (B+11 i A)}{256 a^3 c f (c-i c \tan (e+f x))^{3/2}}-\frac {21 (B+11 i A)}{640 a^3 f (c-i c \tan (e+f x))^{5/2}}+\frac {3 (B+11 i A)}{64 a^3 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}+\frac {B+11 i A}{48 a^3 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rule 3588
Rubi steps
\begin {align*} \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {A+B x}{(a+i a x)^4 (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i A-B}{6 a^3 f (1+i \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2}}+\frac {((11 A-i B) c) \operatorname {Subst}\left (\int \frac {1}{(a+i a x)^3 (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{12 f}\\ &=\frac {i A-B}{6 a^3 f (1+i \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2}}+\frac {11 i A+B}{48 a^3 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2}}+\frac {(3 (11 A-i B) c) \operatorname {Subst}\left (\int \frac {1}{(a+i a x)^2 (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{32 a f}\\ &=\frac {i A-B}{6 a^3 f (1+i \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2}}+\frac {11 i A+B}{48 a^3 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2}}+\frac {3 (11 i A+B)}{64 a^3 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}+\frac {(21 (11 A-i B) c) \operatorname {Subst}\left (\int \frac {1}{(a+i a x) (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{128 a^2 f}\\ &=-\frac {21 (11 i A+B)}{640 a^3 f (c-i c \tan (e+f x))^{5/2}}+\frac {i A-B}{6 a^3 f (1+i \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2}}+\frac {11 i A+B}{48 a^3 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2}}+\frac {3 (11 i A+B)}{64 a^3 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}+\frac {(21 (11 A-i B)) \operatorname {Subst}\left (\int \frac {1}{(a+i a x) (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{256 a^2 f}\\ &=-\frac {21 (11 i A+B)}{640 a^3 f (c-i c \tan (e+f x))^{5/2}}+\frac {i A-B}{6 a^3 f (1+i \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2}}+\frac {11 i A+B}{48 a^3 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2}}+\frac {3 (11 i A+B)}{64 a^3 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}-\frac {7 (11 i A+B)}{256 a^3 c f (c-i c \tan (e+f x))^{3/2}}+\frac {(21 (11 A-i B)) \operatorname {Subst}\left (\int \frac {1}{(a+i a x) (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{512 a^2 c f}\\ &=-\frac {21 (11 i A+B)}{640 a^3 f (c-i c \tan (e+f x))^{5/2}}+\frac {i A-B}{6 a^3 f (1+i \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2}}+\frac {11 i A+B}{48 a^3 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2}}+\frac {3 (11 i A+B)}{64 a^3 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}-\frac {7 (11 i A+B)}{256 a^3 c f (c-i c \tan (e+f x))^{3/2}}-\frac {21 (11 i A+B)}{512 a^3 c^2 f \sqrt {c-i c \tan (e+f x)}}+\frac {(21 (11 A-i B)) \operatorname {Subst}\left (\int \frac {1}{(a+i a x) \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{1024 a^2 c^2 f}\\ &=-\frac {21 (11 i A+B)}{640 a^3 f (c-i c \tan (e+f x))^{5/2}}+\frac {i A-B}{6 a^3 f (1+i \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2}}+\frac {11 i A+B}{48 a^3 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2}}+\frac {3 (11 i A+B)}{64 a^3 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}-\frac {7 (11 i A+B)}{256 a^3 c f (c-i c \tan (e+f x))^{3/2}}-\frac {21 (11 i A+B)}{512 a^3 c^2 f \sqrt {c-i c \tan (e+f x)}}+\frac {(21 (11 i A+B)) \operatorname {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-i c \tan (e+f x)}\right )}{512 a^2 c^3 f}\\ &=\frac {21 (11 i A+B) \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{512 \sqrt {2} a^3 c^{5/2} f}-\frac {21 (11 i A+B)}{640 a^3 f (c-i c \tan (e+f x))^{5/2}}+\frac {i A-B}{6 a^3 f (1+i \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2}}+\frac {11 i A+B}{48 a^3 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2}}+\frac {3 (11 i A+B)}{64 a^3 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}-\frac {7 (11 i A+B)}{256 a^3 c f (c-i c \tan (e+f x))^{3/2}}-\frac {21 (11 i A+B)}{512 a^3 c^2 f \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 12.38, size = 256, normalized size = 0.82 \[ \frac {e^{-6 i (e+f x)} \sqrt {c-i c \tan (e+f x)} \left (315 (B+11 i A) e^{6 i (e+f x)} \sqrt {1+e^{2 i (e+f x)}} \tanh ^{-1}\left (\sqrt {1+e^{2 i (e+f x)}}\right )-i \left (1+e^{2 i (e+f x)}\right ) \left (A \left (-310 e^{2 i (e+f x)}-1335 e^{4 i (e+f x)}+2768 e^{6 i (e+f x)}+416 e^{8 i (e+f x)}+48 e^{10 i (e+f x)}-40\right )-i B \left (190 e^{2 i (e+f x)}+315 e^{4 i (e+f x)}+688 e^{6 i (e+f x)}+256 e^{8 i (e+f x)}+48 e^{10 i (e+f x)}+40\right )\right )\right )}{15360 a^3 c^3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 3.06, size = 461, normalized size = 1.48 \[ \frac {{\left (15 \, \sqrt {\frac {1}{2}} a^{3} c^{3} f \sqrt {-\frac {53361 \, A^{2} - 9702 i \, A B - 441 \, B^{2}}{a^{6} c^{5} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{3} c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} c^{2} f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {53361 \, A^{2} - 9702 i \, A B - 441 \, B^{2}}{a^{6} c^{5} f^{2}}} + 231 i \, A + 21 \, B\right )} e^{\left (-i \, f x - i \, e\right )}}{256 \, a^{3} c^{2} f}\right ) - 15 \, \sqrt {\frac {1}{2}} a^{3} c^{3} f \sqrt {-\frac {53361 \, A^{2} - 9702 i \, A B - 441 \, B^{2}}{a^{6} c^{5} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (-\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{3} c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} c^{2} f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {53361 \, A^{2} - 9702 i \, A B - 441 \, B^{2}}{a^{6} c^{5} f^{2}}} - 231 i \, A - 21 \, B\right )} e^{\left (-i \, f x - i \, e\right )}}{256 \, a^{3} c^{2} f}\right ) + \sqrt {2} {\left ({\left (-48 i \, A - 48 \, B\right )} e^{\left (12 i \, f x + 12 i \, e\right )} + {\left (-464 i \, A - 304 \, B\right )} e^{\left (10 i \, f x + 10 i \, e\right )} + {\left (-3184 i \, A - 944 \, B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (-1433 i \, A - 1003 \, B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (1645 i \, A - 505 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (350 i \, A - 230 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 40 i \, A - 40 \, B\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{15360 \, a^{3} c^{3} f} \]
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 \[ \int \frac {B \tan \left (f x + e\right ) + A}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 233, normalized size = 0.75 \[ \frac {2 i c^{3} \left (-\frac {\frac {\left (\frac {11 i B}{32}+\frac {71 A}{32}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}+\left (-\frac {59}{6} c A -\frac {11}{6} i B c \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}+\left (\frac {21}{8} i B \,c^{2}+\frac {89}{8} A \,c^{2}\right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\left (-c -i c \tan \left (f x +e \right )\right )^{3}}-\frac {21 \left (-i B +11 A \right ) \sqrt {2}\, \arctanh \left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{64 \sqrt {c}}}{32 c^{5}}-\frac {-i B +5 A}{32 c^{5} \sqrt {c -i c \tan \left (f x +e \right )}}-\frac {-i B +2 A}{48 c^{4} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {-i B +A}{80 c^{3} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}\right )}{f \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 291, normalized size = 0.94 \[ -\frac {i \, {\left (\frac {4 \, {\left (315 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{5} {\left (11 \, A - i \, B\right )} - 1680 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{4} {\left (11 \, A - i \, B\right )} c + 2772 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{3} {\left (11 \, A - i \, B\right )} c^{2} - 1152 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} {\left (11 \, A - i \, B\right )} c^{3} - 256 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} {\left (11 \, A - i \, B\right )} c^{4} - 1536 \, {\left (A - i \, B\right )} c^{5}\right )}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {11}{2}} a^{3} c - 6 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {9}{2}} a^{3} c^{2} + 12 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} a^{3} c^{3} - 8 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} a^{3} c^{4}} + \frac {315 \, \sqrt {2} {\left (11 \, A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{\sqrt {2} \sqrt {c} + \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}{a^{3} c^{\frac {3}{2}}}\right )}}{30720 \, c f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.80, size = 490, normalized size = 1.58 \[ -\frac {-\frac {A\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3\,2541{}\mathrm {i}}{640\,a^3\,f}+\frac {A\,c^3\,1{}\mathrm {i}}{5\,a^3\,f}+\frac {A\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^4\,77{}\mathrm {i}}{32\,a^3\,c\,f}-\frac {A\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^5\,231{}\mathrm {i}}{512\,a^3\,c^2\,f}+\frac {A\,c\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2\,33{}\mathrm {i}}{20\,a^3\,f}+\frac {A\,c^2\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,11{}\mathrm {i}}{30\,a^3\,f}}{6\,c\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{9/2}-{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{11/2}+8\,c^3\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}-12\,c^2\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2}}+\frac {\frac {B\,c^3}{5}-\frac {231\,B\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3}{640}+\frac {3\,B\,c\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2}{20}+\frac {B\,c^2\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}{30}+\frac {7\,B\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^4}{32\,c}-\frac {21\,B\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^5}{512\,c^2}}{a^3\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{11/2}-6\,a^3\,c\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{9/2}-8\,a^3\,c^3\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}+12\,a^3\,c^2\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2}}-\frac {\sqrt {2}\,A\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,231{}\mathrm {i}}{1024\,a^3\,{\left (-c\right )}^{5/2}\,f}+\frac {21\,\sqrt {2}\,B\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {c}}\right )}{1024\,a^3\,c^{5/2}\,f} \]
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 \[ \frac {i \left (\int \frac {A}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{5}{\left (e + f x \right )} + i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )} - 2 c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )} + 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \frac {B \tan {\left (e + f x \right )}}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{5}{\left (e + f x \right )} + i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )} - 2 c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )} + 2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )} - c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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