Optimal. Leaf size=291 \[ -\frac {35 (i A-5 B) c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a^3 f}+\frac {35 (i A-5 B) c^4 \sqrt {c-i c \tan (e+f x)}}{8 a^3 f}+\frac {35 (i A-5 B) c^3 (c-i c \tan (e+f x))^{3/2}}{48 a^3 f}+\frac {7 (i A-5 B) c^2 (c-i c \tan (e+f x))^{5/2}}{16 a^3 f (1+i \tan (e+f x))}-\frac {(i A-5 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^3 f (1+i \tan (e+f x))^2}+\frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{6 a^3 f (1+i \tan (e+f x))^3} \]
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Rubi [A]
time = 0.20, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3669, 79, 43,
52, 65, 214} \begin {gather*} -\frac {35 c^{9/2} (-5 B+i A) \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a^3 f}+\frac {35 c^4 (-5 B+i A) \sqrt {c-i c \tan (e+f x)}}{8 a^3 f}+\frac {35 c^3 (-5 B+i A) (c-i c \tan (e+f x))^{3/2}}{48 a^3 f}+\frac {7 c^2 (-5 B+i A) (c-i c \tan (e+f x))^{5/2}}{16 a^3 f (1+i \tan (e+f x))}-\frac {c (-5 B+i A) (c-i c \tan (e+f x))^{7/2}}{8 a^3 f (1+i \tan (e+f x))^2}+\frac {(-B+i A) (c-i c \tan (e+f x))^{9/2}}{6 a^3 f (1+i \tan (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 79
Rule 214
Rule 3669
Rubi steps
\begin {align*} \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^3} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {(A+B x) (c-i c x)^{7/2}}{(a+i a x)^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{6 a^3 f (1+i \tan (e+f x))^3}-\frac {((A+5 i B) c) \text {Subst}\left (\int \frac {(c-i c x)^{7/2}}{(a+i a x)^3} \, dx,x,\tan (e+f x)\right )}{4 f}\\ &=-\frac {(i A-5 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^3 f (1+i \tan (e+f x))^2}+\frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac {\left (7 (A+5 i B) c^2\right ) \text {Subst}\left (\int \frac {(c-i c x)^{5/2}}{(a+i a x)^2} \, dx,x,\tan (e+f x)\right )}{16 a f}\\ &=\frac {7 (i A-5 B) c^2 (c-i c \tan (e+f x))^{5/2}}{16 a^3 f (1+i \tan (e+f x))}-\frac {(i A-5 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^3 f (1+i \tan (e+f x))^2}+\frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{6 a^3 f (1+i \tan (e+f x))^3}-\frac {\left (35 (A+5 i B) c^3\right ) \text {Subst}\left (\int \frac {(c-i c x)^{3/2}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{32 a^2 f}\\ &=\frac {35 (i A-5 B) c^3 (c-i c \tan (e+f x))^{3/2}}{48 a^3 f}+\frac {7 (i A-5 B) c^2 (c-i c \tan (e+f x))^{5/2}}{16 a^3 f (1+i \tan (e+f x))}-\frac {(i A-5 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^3 f (1+i \tan (e+f x))^2}+\frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{6 a^3 f (1+i \tan (e+f x))^3}-\frac {\left (35 (A+5 i B) c^4\right ) \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{16 a^2 f}\\ &=\frac {35 (i A-5 B) c^4 \sqrt {c-i c \tan (e+f x)}}{8 a^3 f}+\frac {35 (i A-5 B) c^3 (c-i c \tan (e+f x))^{3/2}}{48 a^3 f}+\frac {7 (i A-5 B) c^2 (c-i c \tan (e+f x))^{5/2}}{16 a^3 f (1+i \tan (e+f x))}-\frac {(i A-5 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^3 f (1+i \tan (e+f x))^2}+\frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{6 a^3 f (1+i \tan (e+f x))^3}-\frac {\left (35 (A+5 i B) c^5\right ) \text {Subst}\left (\int \frac {1}{(a+i a x) \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{8 a^2 f}\\ &=\frac {35 (i A-5 B) c^4 \sqrt {c-i c \tan (e+f x)}}{8 a^3 f}+\frac {35 (i A-5 B) c^3 (c-i c \tan (e+f x))^{3/2}}{48 a^3 f}+\frac {7 (i A-5 B) c^2 (c-i c \tan (e+f x))^{5/2}}{16 a^3 f (1+i \tan (e+f x))}-\frac {(i A-5 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^3 f (1+i \tan (e+f x))^2}+\frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{6 a^3 f (1+i \tan (e+f x))^3}-\frac {\left (35 (i A-5 B) c^4\right ) \text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-i c \tan (e+f x)}\right )}{4 a^2 f}\\ &=-\frac {35 (i A-5 B) c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a^3 f}+\frac {35 (i A-5 B) c^4 \sqrt {c-i c \tan (e+f x)}}{8 a^3 f}+\frac {35 (i A-5 B) c^3 (c-i c \tan (e+f x))^{3/2}}{48 a^3 f}+\frac {7 (i A-5 B) c^2 (c-i c \tan (e+f x))^{5/2}}{16 a^3 f (1+i \tan (e+f x))}-\frac {(i A-5 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^3 f (1+i \tan (e+f x))^2}+\frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{6 a^3 f (1+i \tan (e+f x))^3}\\ \end {align*}
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Mathematica [F]
time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.37, size = 207, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {2 i c^{3} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+7 i B c \sqrt {c -i c \tan \left (f x +e \right )}+A c \sqrt {c -i c \tan \left (f x +e \right )}-8 c^{2} \left (\frac {8 \left (-\frac {81 i B}{512}-\frac {29 A}{512}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}+8 \left (\frac {53}{96} i B c +\frac {17}{96} A c \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}+8 \left (-\frac {63}{128} i B \,c^{2}-\frac {19}{128} A \,c^{2}\right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\left (c +i c \tan \left (f x +e \right )\right )^{3}}+\frac {35 \left (\frac {A}{8}+\frac {5 i B}{8}\right ) \sqrt {2}\, \arctanh \left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{16 \sqrt {c}}\right )\right )}{f \,a^{3}}\) | \(207\) |
default | \(\frac {2 i c^{3} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+7 i B c \sqrt {c -i c \tan \left (f x +e \right )}+A c \sqrt {c -i c \tan \left (f x +e \right )}-8 c^{2} \left (\frac {8 \left (-\frac {81 i B}{512}-\frac {29 A}{512}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}+8 \left (\frac {53}{96} i B c +\frac {17}{96} A c \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}+8 \left (-\frac {63}{128} i B \,c^{2}-\frac {19}{128} A \,c^{2}\right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\left (c +i c \tan \left (f x +e \right )\right )^{3}}+\frac {35 \left (\frac {A}{8}+\frac {5 i B}{8}\right ) \sqrt {2}\, \arctanh \left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{16 \sqrt {c}}\right )\right )}{f \,a^{3}}\) | \(207\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 276, normalized size = 0.95 \begin {gather*} \frac {i \, {\left (\frac {105 \, \sqrt {2} {\left (A + 5 i \, B\right )} c^{\frac {11}{2}} \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}} - \frac {4 \, {\left (3 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (29 \, A + 81 i \, B\right )} c^{6} - 16 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (17 \, A + 53 i \, B\right )} c^{7} + 12 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (19 \, A + 63 i \, B\right )} c^{8}\right )}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{3} a^{3} - 6 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} a^{3} c + 12 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} a^{3} c^{2} - 8 \, a^{3} c^{3}} + \frac {32 \, {\left (i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} B c^{4} + 3 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (A + 7 i \, B\right )} c^{5}\right )}}{a^{3}}\right )}}{48 \, 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. 488 vs. \(2 (237) = 474\).
time = 3.78, size = 488, normalized size = 1.68 \begin {gather*} -\frac {105 \, \sqrt {\frac {1}{2}} {\left (a^{3} f e^{\left (8 i \, f x + 8 i \, e\right )} + a^{3} f e^{\left (6 i \, f x + 6 i \, e\right )}\right )} \sqrt {-\frac {{\left (A^{2} + 10 i \, A B - 25 \, B^{2}\right )} c^{9}}{a^{6} f^{2}}} \log \left (-\frac {35 \, {\left ({\left (i \, A - 5 \, B\right )} c^{5} + \sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt {-\frac {{\left (A^{2} + 10 i \, A B - 25 \, B^{2}\right )} c^{9}}{a^{6} f^{2}}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{2 \, a^{3} f}\right ) - 105 \, \sqrt {\frac {1}{2}} {\left (a^{3} f e^{\left (8 i \, f x + 8 i \, e\right )} + a^{3} f e^{\left (6 i \, f x + 6 i \, e\right )}\right )} \sqrt {-\frac {{\left (A^{2} + 10 i \, A B - 25 \, B^{2}\right )} c^{9}}{a^{6} f^{2}}} \log \left (-\frac {35 \, {\left ({\left (i \, A - 5 \, B\right )} c^{5} - \sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt {-\frac {{\left (A^{2} + 10 i \, A B - 25 \, B^{2}\right )} c^{9}}{a^{6} f^{2}}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{2 \, a^{3} f}\right ) + \sqrt {2} {\left (105 \, {\left (-i \, A + 5 \, B\right )} c^{4} e^{\left (8 i \, f x + 8 i \, e\right )} + 140 \, {\left (-i \, A + 5 \, B\right )} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 21 \, {\left (-i \, A + 5 \, B\right )} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, {\left (i \, A - 5 \, B\right )} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 \, {\left (-i \, A + B\right )} c^{4}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{24 \, {\left (a^{3} f e^{\left (8 i \, f x + 8 i \, e\right )} + a^{3} f e^{\left (6 i \, f x + 6 i \, e\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = 9.59, size = 441, normalized size = 1.52 \begin {gather*} \frac {\frac {A\,c^7\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,19{}\mathrm {i}}{a^3\,f}-\frac {A\,c^6\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,68{}\mathrm {i}}{3\,a^3\,f}+\frac {A\,c^5\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,29{}\mathrm {i}}{4\,a^3\,f}}{6\,c\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2-12\,c^2\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )-{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3+8\,c^3}-\frac {63\,B\,c^7\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}-\frac {212\,B\,c^6\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{3}+\frac {81\,B\,c^5\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{4}}{8\,a^3\,c^3\,f-a^3\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3+6\,a^3\,c\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2-12\,a^3\,c^2\,f\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}+\frac {A\,c^4\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,2{}\mathrm {i}}{a^3\,f}-\frac {14\,B\,c^4\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{a^3\,f}-\frac {2\,B\,c^3\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{3\,a^3\,f}-\frac {\sqrt {2}\,A\,{\left (-c\right )}^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,35{}\mathrm {i}}{8\,a^3\,f}-\frac {\sqrt {2}\,B\,c^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,175{}\mathrm {i}}{8\,a^3\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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