Optimal. Leaf size=223 \[ \frac {(7 i A-3 B) \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{16 \sqrt {2} a c^{5/2} f}-\frac {7 i A-3 B}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac {i A-B}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}-\frac {7 i A-3 B}{24 a c f (c-i c \tan (e+f x))^{3/2}}-\frac {7 i A-3 B}{16 a c^2 f \sqrt {c-i c \tan (e+f x)}} \]
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Rubi [A]
time = 0.20, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {3669, 79, 53,
65, 214} \begin {gather*} \frac {(-3 B+7 i A) \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{16 \sqrt {2} a c^{5/2} f}-\frac {-3 B+7 i A}{16 a c^2 f \sqrt {c-i c \tan (e+f x)}}-\frac {-3 B+7 i A}{24 a c f (c-i c \tan (e+f x))^{3/2}}-\frac {-3 B+7 i A}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac {-B+i A}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 79
Rule 214
Rule 3669
Rubi steps
\begin {align*} \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {A+B x}{(a+i a x)^2 (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i A-B}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}+\frac {((7 A+3 i B) c) \text {Subst}\left (\int \frac {1}{(a+i a x) (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{4 f}\\ &=-\frac {7 i A-3 B}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac {i A-B}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}+\frac {(7 A+3 i B) \text {Subst}\left (\int \frac {1}{(a+i a x) (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=-\frac {7 i A-3 B}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac {i A-B}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}-\frac {7 i A-3 B}{24 a c f (c-i c \tan (e+f x))^{3/2}}+\frac {(7 A+3 i B) \text {Subst}\left (\int \frac {1}{(a+i a x) (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{16 c f}\\ &=-\frac {7 i A-3 B}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac {i A-B}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}-\frac {7 i A-3 B}{24 a c f (c-i c \tan (e+f x))^{3/2}}-\frac {7 i A-3 B}{16 a c^2 f \sqrt {c-i c \tan (e+f x)}}+\frac {(7 A+3 i B) \text {Subst}\left (\int \frac {1}{(a+i a x) \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{32 c^2 f}\\ &=-\frac {7 i A-3 B}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac {i A-B}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}-\frac {7 i A-3 B}{24 a c f (c-i c \tan (e+f x))^{3/2}}-\frac {7 i A-3 B}{16 a c^2 f \sqrt {c-i c \tan (e+f x)}}+\frac {(7 i A-3 B) \text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-i c \tan (e+f x)}\right )}{16 c^3 f}\\ &=\frac {(7 i A-3 B) \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{16 \sqrt {2} a c^{5/2} f}-\frac {7 i A-3 B}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac {i A-B}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}-\frac {7 i A-3 B}{24 a c f (c-i c \tan (e+f x))^{3/2}}-\frac {7 i A-3 B}{16 a c^2 f \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 2.50, size = 213, normalized size = 0.96 \begin {gather*} -\frac {e^{-2 i (e+f x)} \sqrt {\frac {c}{1+e^{2 i (e+f x)}}} \left (\left (1+e^{2 i (e+f x)}\right ) \left (3 B \left (5-8 e^{2 i (e+f x)}+4 e^{4 i (e+f x)}+2 e^{6 i (e+f x)}\right )+i A \left (-15+116 e^{2 i (e+f x)}+32 e^{4 i (e+f x)}+6 e^{6 i (e+f x)}\right )\right )+15 (-7 i A+3 B) e^{2 i (e+f x)} \sqrt {1+e^{2 i (e+f x)}} \tanh ^{-1}\left (\sqrt {1+e^{2 i (e+f x)}}\right )\right )}{240 \sqrt {2} a c^3 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.33, size = 168, normalized size = 0.75
method | result | size |
derivativedivides | \(\frac {2 i c \left (-\frac {A}{12 c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {i B +3 A}{16 c^{3} \sqrt {c -i c \tan \left (f x +e \right )}}-\frac {-i B +A}{20 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {\frac {\left (\frac {A}{4}+\frac {i B}{4}\right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\frac {c}{2}+\frac {i c \tan \left (f x +e \right )}{2}}+\frac {\left (\frac {7 A}{2}+\frac {3 i B}{2}\right ) \sqrt {2}\, \arctanh \left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{2 \sqrt {c}}}{16 c^{3}}\right )}{f a}\) | \(168\) |
default | \(\frac {2 i c \left (-\frac {A}{12 c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {i B +3 A}{16 c^{3} \sqrt {c -i c \tan \left (f x +e \right )}}-\frac {-i B +A}{20 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {\frac {\left (\frac {A}{4}+\frac {i B}{4}\right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\frac {c}{2}+\frac {i c \tan \left (f x +e \right )}{2}}+\frac {\left (\frac {7 A}{2}+\frac {3 i B}{2}\right ) \sqrt {2}\, \arctanh \left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{2 \sqrt {c}}}{16 c^{3}}\right )}{f a}\) | \(168\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 202, normalized size = 0.91 \begin {gather*} -\frac {i \, {\left (\frac {4 \, {\left (15 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{3} {\left (7 \, A + 3 i \, B\right )} - 20 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} {\left (7 \, A + 3 i \, B\right )} c - 8 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} {\left (7 \, A + 3 i \, B\right )} c^{2} - 48 \, {\left (A - i \, B\right )} c^{3}\right )}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} a c - 2 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} a c^{2}} + \frac {15 \, \sqrt {2} {\left (7 \, A + 3 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 c^{\frac {3}{2}}}\right )}}{960 \, c f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 432 vs. \(2 (178) = 356\).
time = 4.19, size = 432, normalized size = 1.94 \begin {gather*} \frac {{\left (15 \, \sqrt {\frac {1}{2}} a c^{3} f \sqrt {-\frac {49 \, A^{2} + 42 i \, A B - 9 \, B^{2}}{a^{2} c^{5} f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a c^{2} f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {49 \, A^{2} + 42 i \, A B - 9 \, B^{2}}{a^{2} c^{5} f^{2}}} + 7 i \, A - 3 \, B\right )} e^{\left (-i \, f x - i \, e\right )}}{8 \, a c^{2} f}\right ) - 15 \, \sqrt {\frac {1}{2}} a c^{3} f \sqrt {-\frac {49 \, A^{2} + 42 i \, A B - 9 \, B^{2}}{a^{2} c^{5} f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a c^{2} f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {49 \, A^{2} + 42 i \, A B - 9 \, B^{2}}{a^{2} c^{5} f^{2}}} - 7 i \, A + 3 \, B\right )} e^{\left (-i \, f x - i \, e\right )}}{8 \, a c^{2} f}\right ) - \sqrt {2} {\left (6 \, {\left (i \, A + B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + 2 \, {\left (19 i \, A + 9 \, B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + 4 \, {\left (37 i \, A - 3 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - {\left (-101 i \, A + 9 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - 15 i \, A + 15 \, B\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{480 \, a c^{3} f} \end {gather*}
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 \begin {gather*} - \frac {i \left (\int \frac {A}{- c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )} - 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 ^{3}{\left (e + f x \right )} - 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} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.83, size = 308, normalized size = 1.38 \begin {gather*} \frac {\frac {B\,c}{5}-\frac {B\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}{10}-\frac {B\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2}{4\,c}+\frac {3\,B\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3}{16\,c^2}}{a\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2}-2\,a\,c\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}-\frac {\frac {A\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,7{}\mathrm {i}}{30\,a\,f}+\frac {A\,c\,1{}\mathrm {i}}{5\,a\,f}+\frac {A\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2\,7{}\mathrm {i}}{12\,a\,c\,f}-\frac {A\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3\,7{}\mathrm {i}}{16\,a\,c^2\,f}}{2\,c\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}-{\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 )\,7{}\mathrm {i}}{32\,a\,{\left (-c\right )}^{5/2}\,f}-\frac {3\,\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 )}{32\,a\,c^{5/2}\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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