Optimal. Leaf size=144 \[ \frac {2 a^3 (5 B+i A) (c-i c \tan (e+f x))^{11/2}}{11 c^2 f}-\frac {8 a^3 (2 B+i A) (c-i c \tan (e+f x))^{9/2}}{9 c f}+\frac {8 a^3 (B+i A) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac {2 a^3 B (c-i c \tan (e+f x))^{13/2}}{13 c^3 f} \]
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Rubi [A] time = 0.21, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {3588, 77} \[ \frac {2 a^3 (5 B+i A) (c-i c \tan (e+f x))^{11/2}}{11 c^2 f}-\frac {8 a^3 (2 B+i A) (c-i c \tan (e+f x))^{9/2}}{9 c f}+\frac {8 a^3 (B+i A) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac {2 a^3 B (c-i c \tan (e+f x))^{13/2}}{13 c^3 f} \]
Antiderivative was successfully verified.
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Rule 77
Rule 3588
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int (a+i a x)^2 (A+B x) (c-i c x)^{5/2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left (4 a^2 (A-i B) (c-i c x)^{5/2}-\frac {4 a^2 (A-2 i B) (c-i c x)^{7/2}}{c}+\frac {a^2 (A-5 i B) (c-i c x)^{9/2}}{c^2}+\frac {i a^2 B (c-i c x)^{11/2}}{c^3}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {8 a^3 (i A+B) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac {8 a^3 (i A+2 B) (c-i c \tan (e+f x))^{9/2}}{9 c f}+\frac {2 a^3 (i A+5 B) (c-i c \tan (e+f x))^{11/2}}{11 c^2 f}-\frac {2 a^3 B (c-i c \tan (e+f x))^{13/2}}{13 c^3 f}\\ \end {align*}
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Mathematica [A] time = 13.45, size = 127, normalized size = 0.88 \[ -\frac {2 a^3 c^3 (\cos (3 e)-i \sin (3 e)) \sec ^5(e+f x) \sqrt {c-i c \tan (e+f x)} (7 (169 A-86 i B) \tan (e+f x)+\cos (2 (e+f x)) (7 (169 A-185 i B) \tan (e+f x)-1391 i A-1279 B)-572 i A+737 B)}{9009 f (\cos (f x)+i \sin (f x))^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.47, size = 181, normalized size = 1.26 \[ \frac {\sqrt {2} {\left ({\left (82368 i \, A + 82368 \, B\right )} a^{3} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (118976 i \, A - 9152 \, B\right )} a^{3} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (43264 i \, A - 3328 \, B\right )} a^{3} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (6656 i \, A - 512 \, B\right )} a^{3} c^{3}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{9009 \, {\left (f e^{\left (12 i \, f x + 12 i \, e\right )} + 6 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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 = 0.40, size = 121, normalized size = 0.84 \[ \frac {2 i a^{3} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {13}{2}}}{13}+\frac {\left (-5 i B c +c A \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}+\frac {\left (-4 \left (-i B c +c A \right ) c +4 i B \,c^{2}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}+\frac {4 \left (-i B c +c A \right ) c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}\right )}{f \,c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 104, normalized size = 0.72 \[ \frac {2 i \, {\left (693 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {13}{2}} B a^{3} + 819 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {11}{2}} {\left (A - 5 i \, B\right )} a^{3} c - 4004 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {9}{2}} {\left (A - 2 i \, B\right )} a^{3} c^{2} + 5148 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} {\left (A - i \, B\right )} a^{3} c^{3}\right )}}{9009 \, c^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.07, size = 349, normalized size = 2.42 \[ -\frac {\left (\frac {a^3\,c^3\,\left (A-B\,1{}\mathrm {i}\right )\,64{}\mathrm {i}}{9\,f}+\frac {a^3\,c^3\,\left (A-B\,3{}\mathrm {i}\right )\,64{}\mathrm {i}}{9\,f}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^4}-\frac {\left (\frac {a^3\,c^3\,\left (A-B\,1{}\mathrm {i}\right )\,64{}\mathrm {i}}{13\,f}-\frac {a^3\,c^3\,\left (A+B\,1{}\mathrm {i}\right )\,64{}\mathrm {i}}{13\,f}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^6}+\frac {\left (\frac {256\,B\,a^3\,c^3}{11\,f}+\frac {a^3\,c^3\,\left (A-B\,1{}\mathrm {i}\right )\,64{}\mathrm {i}}{11\,f}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^5}+\frac {a^3\,c^3\,\left (A-B\,1{}\mathrm {i}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{7\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3} \]
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 \[ - i a^{3} \left (\int i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c}\, dx + \int 3 i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\, dx + \int 3 i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}\, dx + \int i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{6}{\left (e + f x \right )}\, dx + \int i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\, dx + \int 3 i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\, dx + \int 3 i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{5}{\left (e + f x \right )}\, dx + \int i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{7}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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