Optimal. Leaf size=94 \[ \frac {8 i a^3 (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac {8 i a^3 (c-i c \tan (e+f x))^{5/2}}{5 c f}+\frac {2 i a^3 (c-i c \tan (e+f x))^{7/2}}{7 c^2 f} \]
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Rubi [A]
time = 0.12, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3603, 3568, 45}
\begin {gather*} \frac {2 i a^3 (c-i c \tan (e+f x))^{7/2}}{7 c^2 f}-\frac {8 i a^3 (c-i c \tan (e+f x))^{5/2}}{5 c f}+\frac {8 i a^3 (c-i c \tan (e+f x))^{3/2}}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rule 3603
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^{3/2} \, dx &=\left (a^3 c^3\right ) \int \frac {\sec ^6(e+f x)}{(c-i c \tan (e+f x))^{3/2}} \, dx\\ &=\frac {\left (i a^3\right ) \text {Subst}\left (\int (c-x)^2 \sqrt {c+x} \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac {\left (i a^3\right ) \text {Subst}\left (\int \left (4 c^2 \sqrt {c+x}-4 c (c+x)^{3/2}+(c+x)^{5/2}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac {8 i a^3 (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac {8 i a^3 (c-i c \tan (e+f x))^{5/2}}{5 c f}+\frac {2 i a^3 (c-i c \tan (e+f x))^{7/2}}{7 c^2 f}\\ \end {align*}
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Mathematica [A]
time = 1.37, size = 94, normalized size = 1.00 \begin {gather*} \frac {2 a^3 c \sec ^3(e+f x) (i \cos (e-2 f x)+\sin (e-2 f x)) (28+43 \cos (2 (e+f x))+27 i \sin (2 (e+f x))) \sqrt {c-i c \tan (e+f x)}}{105 f (\cos (f x)+i \sin (f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 66, normalized size = 0.70
method | result | size |
derivativedivides | \(\frac {2 i a^{3} \left (\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}-\frac {4 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {4 c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}\right )}{f \,c^{2}}\) | \(66\) |
default | \(\frac {2 i a^{3} \left (\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}-\frac {4 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {4 c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}\right )}{f \,c^{2}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 70, normalized size = 0.74 \begin {gather*} \frac {2 i \, {\left (15 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} a^{3} - 84 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} a^{3} c + 140 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} a^{3} c^{2}\right )}}{105 \, c^{2} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.15, size = 104, normalized size = 1.11 \begin {gather*} -\frac {16 \, \sqrt {2} {\left (-35 i \, a^{3} c e^{\left (4 i \, f x + 4 i \, e\right )} - 28 i \, a^{3} c e^{\left (2 i \, f x + 2 i \, e\right )} - 8 i \, a^{3} c\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{105 \, {\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^{3} \left (\int i c \sqrt {- i c \tan {\left (e + f x \right )} + c}\, dx + \int \left (- 2 c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\right )\, dx + \int \left (- 2 c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\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 = 8.62, size = 95, normalized size = 1.01 \begin {gather*} \frac {16\,a^3\,c\,\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}}\,28{}\mathrm {i}+{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,35{}\mathrm {i}+8{}\mathrm {i}\right )}{105\,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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