Optimal. Leaf size=105 \[ \frac {4 a^2 (i A+B) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac {2 a^2 (i A+3 B) (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac {2 a^2 B (c-i c \tan (e+f x))^{9/2}}{9 c^2 f} \]
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Rubi [A]
time = 0.12, antiderivative size = 105, 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^2 (3 B+i A) (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac {4 a^2 (B+i A) (c-i c \tan (e+f x))^{5/2}}{5 f}+\frac {2 a^2 B (c-i c \tan (e+f x))^{9/2}}{9 c^2 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))^2 (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) (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 (2 a (A-i B) (c-i c x)^{3/2}-\frac {a (A-3 i B) (c-i c x)^{5/2}}{c}-\frac {i a B (c-i c x)^{7/2}}{c^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {4 a^2 (i A+B) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac {2 a^2 (i A+3 B) (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac {2 a^2 B (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}\\ \end {align*}
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Mathematica [A]
time = 2.06, size = 112, normalized size = 1.07 \begin {gather*} \frac {a^2 c^2 \sec ^4(e+f x) (i \cos (2 e)+\sin (2 e)) (81 A+9 i B+(81 A-61 i B) \cos (2 (e+f x))+5 (9 i A+13 B) \sin (2 (e+f x))) \sqrt {c-i c \tan (e+f x)}}{315 f (\cos (f x)+i \sin (f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 83, normalized size = 0.79
method | result | size |
derivativedivides | \(-\frac {2 i a^{2} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}+\frac {\left (-3 i B c +A c \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}-\frac {2 \left (-i B c +A c \right ) c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}\right )}{f \,c^{2}}\) | \(83\) |
default | \(-\frac {2 i a^{2} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}+\frac {\left (-3 i B c +A c \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}-\frac {2 \left (-i B c +A c \right ) c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}\right )}{f \,c^{2}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 81, normalized size = 0.77 \begin {gather*} -\frac {2 i \, {\left (35 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {9}{2}} B a^{2} + 45 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} {\left (A - 3 i \, B\right )} a^{2} c - 126 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (A - i \, B\right )} a^{2} c^{2}\right )}}{315 \, c^{2} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 5.44, size = 140, normalized size = 1.33 \begin {gather*} -\frac {16 \, \sqrt {2} {\left (63 \, {\left (-i \, A - B\right )} a^{2} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 9 \, {\left (-9 i \, A + B\right )} a^{2} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 \, {\left (-9 i \, A + B\right )} a^{2} c^{2}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{315 \, {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, 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*} - a^{2} \left (\int \left (- A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}\right )\, dx + \int \left (- 2 A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- A 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 {\left (e + f x \right )}\right )\, dx + \int \left (- 2 B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{5}{\left (e + f x \right )}\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 = 15.34, size = 132, normalized size = 1.26 \begin {gather*} \frac {16\,a^2\,c^2\,\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 (A\,18{}\mathrm {i}-2\,B+A\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,81{}\mathrm {i}+A\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,63{}\mathrm {i}-9\,B\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+63\,B\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\right )}{315\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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