Optimal. Leaf size=144 \[ \frac {8 a^3 (i A+B) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac {8 a^3 (i A+2 B) (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac {2 a^3 (i A+5 B) (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac {2 a^3 B (c-i c \tan (e+f x))^{11/2}}{11 c^3 f} \]
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Rubi [A]
time = 0.15, 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 = {3669, 78}
\begin {gather*} \frac {2 a^3 (5 B+i A) (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac {8 a^3 (2 B+i A) (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac {8 a^3 (B+i A) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac {2 a^3 B (c-i c \tan (e+f x))^{11/2}}{11 c^3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 3669
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))^{5/2} \, dx &=\frac {(a c) \text {Subst}\left (\int (a+i a x)^2 (A+B x) (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \text {Subst}\left (\int \left (4 a^2 (A-i B) (c-i c x)^{3/2}-\frac {4 a^2 (A-2 i B) (c-i c x)^{5/2}}{c}+\frac {a^2 (A-5 i B) (c-i c x)^{7/2}}{c^2}+\frac {i a^2 B (c-i c x)^{9/2}}{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))^{5/2}}{5 f}-\frac {8 a^3 (i A+2 B) (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac {2 a^3 (i A+5 B) (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac {2 a^3 B (c-i c \tan (e+f x))^{11/2}}{11 c^3 f}\\ \end {align*}
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Mathematica [A]
time = 3.76, size = 139, normalized size = 0.97 \begin {gather*} -\frac {2 a^3 c^2 \sec ^4(e+f x) (\cos (2 e-f x)-i \sin (2 e-f x)) \sqrt {c-i c \tan (e+f x)} (9 (-44 i A+31 B)+5 (121 A-74 i B) \tan (e+f x)+\cos (2 (e+f x)) (-781 i A-701 B+(605 A-685 i B) \tan (e+f x)))}{3465 f (\cos (f x)+i \sin (f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 121, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {2 i a^{3} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}+\frac {\left (-5 i B c +A c \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}+\frac {\left (-4 \left (-i B c +A c \right ) c +4 i B \,c^{2}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {4 \left (-i B c +A c \right ) c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}\right )}{f \,c^{3}}\) | \(121\) |
default | \(\frac {2 i a^{3} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}+\frac {\left (-5 i B c +A c \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}+\frac {\left (-4 \left (-i B c +A c \right ) c +4 i B \,c^{2}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {4 \left (-i B c +A c \right ) c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}\right )}{f \,c^{3}}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 108, normalized size = 0.75 \begin {gather*} \frac {2 i \, {\left (315 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {11}{2}} B a^{3} + 385 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {9}{2}} {\left (A - 5 i \, B\right )} a^{3} c - 1980 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} {\left (A - 2 i \, B\right )} a^{3} c^{2} + 2772 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (A - i \, B\right )} a^{3} c^{3}\right )}}{3465 \, c^{3} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.23, size = 182, normalized size = 1.26 \begin {gather*} -\frac {32 \, \sqrt {2} {\left (693 \, {\left (-i \, A - B\right )} a^{3} c^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 99 \, {\left (-11 i \, A - B\right )} a^{3} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 44 \, {\left (-11 i \, A - B\right )} a^{3} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 \, {\left (-11 i \, A - B\right )} a^{3} c^{2}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{3465 \, {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \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*} - i a^{3} \left (\int i A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}\, dx + \int \left (- A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\right )\, dx + \int \left (- 2 A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{5}{\left (e + f x \right )}\right )\, dx + \int \left (- B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}\right )\, dx + \int \left (- B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{6}{\left (e + f x \right )}\right )\, dx + \int 2 i A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\, dx + \int i A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}\, dx + \int i B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\, dx + \int 2 i B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\, dx + \int i B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{5}{\left (e + f x \right )}\, dx\right ) \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 = 14.48, size = 349, normalized size = 2.42 \begin {gather*} -\frac {\left (\frac {a^3\,c^2\,\left (A-B\,1{}\mathrm {i}\right )\,32{}\mathrm {i}}{7\,f}+\frac {a^3\,c^2\,\left (A-B\,3{}\mathrm {i}\right )\,32{}\mathrm {i}}{7\,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 )}^3}-\frac {\left (\frac {a^3\,c^2\,\left (A-B\,1{}\mathrm {i}\right )\,32{}\mathrm {i}}{11\,f}-\frac {a^3\,c^2\,\left (A+B\,1{}\mathrm {i}\right )\,32{}\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 {\left (\frac {128\,B\,a^3\,c^2}{9\,f}+\frac {a^3\,c^2\,\left (A-B\,1{}\mathrm {i}\right )\,32{}\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 {a^3\,c^2\,\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}}\,32{}\mathrm {i}}{5\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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