Optimal. Leaf size=62 \[ \frac {2 a (i A+B) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac {2 a B (c-i c \tan (e+f x))^{7/2}}{7 c f} \]
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Rubi [A]
time = 0.07, 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 = {3669, 45}
\begin {gather*} \frac {2 a (B+i A) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac {2 a B (c-i c \tan (e+f x))^{7/2}}{7 c f} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3669
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))^{5/2} \, dx &=\frac {(a c) \text {Subst}\left (\int (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 ((A-i B) (c-i c x)^{3/2}+\frac {i B (c-i c x)^{5/2}}{c}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {2 a (i A+B) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac {2 a B (c-i c \tan (e+f x))^{7/2}}{7 c f}\\ \end {align*}
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Mathematica [A]
time = 1.43, size = 88, normalized size = 1.42 \begin {gather*} \frac {2 a c^2 \sec ^2(e+f x) (\cos (f x)-i \sin (f x)) (i \cos (2 e+f x)+\sin (2 e+f x)) (7 A-2 i B+5 B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{35 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 55, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {2 i a \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {\left (-i B c +A c \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}\right )}{f c}\) | \(55\) |
default | \(\frac {2 i a \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {\left (-i B c +A c \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}\right )}{f c}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 50, normalized size = 0.81 \begin {gather*} \frac {2 i \, {\left (5 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} B a + 7 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (A - i \, B\right )} a c\right )}}{35 \, 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. 101 vs. \(2 (50) = 100\).
time = 2.71, size = 101, normalized size = 1.63 \begin {gather*} -\frac {8 \, \sqrt {2} {\left (7 \, {\left (-i \, A - B\right )} a c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-7 i \, A + 3 \, B\right )} a c^{2}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{35 \, {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, 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 \left (\int \left (- i A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}\right )\, 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 (- A 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 ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}\right )\, dx + \int \left (- i A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- i B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\right )\, dx + \int \left (- i B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\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 = 15.53, size = 101, normalized size = 1.63 \begin {gather*} \frac {8\,a\,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\,7{}\mathrm {i}-3\,B+A\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,7{}\mathrm {i}+7\,B\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\right )}{35\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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