Optimal. Leaf size=62 \[ \frac {2 a (B+i A) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac {2 a B (c-i c \tan (e+f x))^{9/2}}{9 c f} \]
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Rubi [A] time = 0.11, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3588, 43} \[ \frac {2 a (B+i A) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac {2 a B (c-i c \tan (e+f x))^{9/2}}{9 c f} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3588
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int (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 ((A-i B) (c-i c x)^{5/2}+\frac {i B (c-i c x)^{7/2}}{c}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {2 a (i A+B) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac {2 a B (c-i c \tan (e+f x))^{9/2}}{9 c f}\\ \end {align*}
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Mathematica [A] time = 6.59, size = 90, normalized size = 1.45 \[ \frac {2 a c^3 \sec ^3(e+f x) (\cos (f x)-i \sin (f x)) \sqrt {c-i c \tan (e+f x)} (\sin (3 e+2 f x)+i \cos (3 e+2 f x)) (9 A+7 B \tan (e+f x)-2 i B)}{63 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.53, size = 107, normalized size = 1.73 \[ \frac {\sqrt {2} {\left ({\left (144 i \, A + 144 \, B\right )} a c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (144 i \, A - 80 \, B\right )} a c^{3}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{63 \, {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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 = 55, normalized size = 0.89 \[ \frac {2 i a \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}+\frac {\left (-i B c +c A \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}\right )}{f c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 48, normalized size = 0.77 \[ \frac {2 i \, {\left (7 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {9}{2}} B a + 9 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} {\left (A - i \, B\right )} a c\right )}}{63 \, c f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.38, size = 101, normalized size = 1.63 \[ \frac {16\,a\,c^3\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,\left (A\,9{}\mathrm {i}-5\,B+A\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,9{}\mathrm {i}+9\,B\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\right )}{63\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^4} \]
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 \[ i a \left (\int \left (- i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c}\right )\, dx + \int \left (- 2 A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\right )\, dx + \int \left (- 2 A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}\right )\, dx + \int i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}\, dx + \int \left (- i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\right )\, dx + \int i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{5}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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