Optimal. Leaf size=294 \[ -\frac {3 i d^2 e^{-2 i e-2 i f x}}{32 a^3 f^3}-\frac {3 i d^2 e^{-4 i e-4 i f x}}{256 a^3 f^3}-\frac {i d^2 e^{-6 i e-6 i f x}}{864 a^3 f^3}+\frac {3 d e^{-2 i e-2 i f x} (c+d x)}{16 a^3 f^2}+\frac {3 d e^{-4 i e-4 i f x} (c+d x)}{64 a^3 f^2}+\frac {d e^{-6 i e-6 i f x} (c+d x)}{144 a^3 f^2}+\frac {3 i e^{-2 i e-2 i f x} (c+d x)^2}{16 a^3 f}+\frac {3 i e^{-4 i e-4 i f x} (c+d x)^2}{32 a^3 f}+\frac {i e^{-6 i e-6 i f x} (c+d x)^2}{48 a^3 f}+\frac {(c+d x)^3}{24 a^3 d} \]
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Rubi [A]
time = 0.19, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3810, 2207,
2225} \begin {gather*} \frac {3 d (c+d x) e^{-2 i e-2 i f x}}{16 a^3 f^2}+\frac {3 d (c+d x) e^{-4 i e-4 i f x}}{64 a^3 f^2}+\frac {d (c+d x) e^{-6 i e-6 i f x}}{144 a^3 f^2}+\frac {3 i (c+d x)^2 e^{-2 i e-2 i f x}}{16 a^3 f}+\frac {3 i (c+d x)^2 e^{-4 i e-4 i f x}}{32 a^3 f}+\frac {i (c+d x)^2 e^{-6 i e-6 i f x}}{48 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}-\frac {3 i d^2 e^{-2 i e-2 i f x}}{32 a^3 f^3}-\frac {3 i d^2 e^{-4 i e-4 i f x}}{256 a^3 f^3}-\frac {i d^2 e^{-6 i e-6 i f x}}{864 a^3 f^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rule 3810
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{(a+i a \tan (e+f x))^3} \, dx &=\int \left (\frac {(c+d x)^2}{8 a^3}+\frac {3 e^{-2 i e-2 i f x} (c+d x)^2}{8 a^3}+\frac {3 e^{-4 i e-4 i f x} (c+d x)^2}{8 a^3}+\frac {e^{-6 i e-6 i f x} (c+d x)^2}{8 a^3}\right ) \, dx\\ &=\frac {(c+d x)^3}{24 a^3 d}+\frac {\int e^{-6 i e-6 i f x} (c+d x)^2 \, dx}{8 a^3}+\frac {3 \int e^{-2 i e-2 i f x} (c+d x)^2 \, dx}{8 a^3}+\frac {3 \int e^{-4 i e-4 i f x} (c+d x)^2 \, dx}{8 a^3}\\ &=\frac {3 i e^{-2 i e-2 i f x} (c+d x)^2}{16 a^3 f}+\frac {3 i e^{-4 i e-4 i f x} (c+d x)^2}{32 a^3 f}+\frac {i e^{-6 i e-6 i f x} (c+d x)^2}{48 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}-\frac {(i d) \int e^{-6 i e-6 i f x} (c+d x) \, dx}{24 a^3 f}-\frac {(3 i d) \int e^{-4 i e-4 i f x} (c+d x) \, dx}{16 a^3 f}-\frac {(3 i d) \int e^{-2 i e-2 i f x} (c+d x) \, dx}{8 a^3 f}\\ &=\frac {3 d e^{-2 i e-2 i f x} (c+d x)}{16 a^3 f^2}+\frac {3 d e^{-4 i e-4 i f x} (c+d x)}{64 a^3 f^2}+\frac {d e^{-6 i e-6 i f x} (c+d x)}{144 a^3 f^2}+\frac {3 i e^{-2 i e-2 i f x} (c+d x)^2}{16 a^3 f}+\frac {3 i e^{-4 i e-4 i f x} (c+d x)^2}{32 a^3 f}+\frac {i e^{-6 i e-6 i f x} (c+d x)^2}{48 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}-\frac {d^2 \int e^{-6 i e-6 i f x} \, dx}{144 a^3 f^2}-\frac {\left (3 d^2\right ) \int e^{-4 i e-4 i f x} \, dx}{64 a^3 f^2}-\frac {\left (3 d^2\right ) \int e^{-2 i e-2 i f x} \, dx}{16 a^3 f^2}\\ &=-\frac {3 i d^2 e^{-2 i e-2 i f x}}{32 a^3 f^3}-\frac {3 i d^2 e^{-4 i e-4 i f x}}{256 a^3 f^3}-\frac {i d^2 e^{-6 i e-6 i f x}}{864 a^3 f^3}+\frac {3 d e^{-2 i e-2 i f x} (c+d x)}{16 a^3 f^2}+\frac {3 d e^{-4 i e-4 i f x} (c+d x)}{64 a^3 f^2}+\frac {d e^{-6 i e-6 i f x} (c+d x)}{144 a^3 f^2}+\frac {3 i e^{-2 i e-2 i f x} (c+d x)^2}{16 a^3 f}+\frac {3 i e^{-4 i e-4 i f x} (c+d x)^2}{32 a^3 f}+\frac {i e^{-6 i e-6 i f x} (c+d x)^2}{48 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}\\ \end {align*}
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Mathematica [A]
time = 1.62, size = 405, normalized size = 1.38 \begin {gather*} \frac {i \sec ^3(e+f x) \left (81 \left (24 i c^2 f^2+4 c d f (5+12 i f x)+d^2 \left (-9 i+20 f x+24 i f^2 x^2\right )\right ) \cos (e+f x)+8 \left (18 c^2 f^2 (i+6 f x)+6 c d f \left (1+6 i f x+18 f^2 x^2\right )+d^2 \left (-i+6 f x+18 i f^2 x^2+36 f^3 x^3\right )\right ) \cos (3 (e+f x))+567 d^2 \sin (e+f x)+972 i c d f \sin (e+f x)-648 c^2 f^2 \sin (e+f x)+972 i d^2 f x \sin (e+f x)-1296 c d f^2 x \sin (e+f x)-648 d^2 f^2 x^2 \sin (e+f x)-8 d^2 \sin (3 (e+f x))-48 i c d f \sin (3 (e+f x))+144 c^2 f^2 \sin (3 (e+f x))-48 i d^2 f x \sin (3 (e+f x))+288 c d f^2 x \sin (3 (e+f x))+864 i c^2 f^3 x \sin (3 (e+f x))+144 d^2 f^2 x^2 \sin (3 (e+f x))+864 i c d f^3 x^2 \sin (3 (e+f x))+288 i d^2 f^3 x^3 \sin (3 (e+f x))\right )}{6912 a^3 f^3 (-i+\tan (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1512 vs. \(2 (241 ) = 482\).
time = 0.54, size = 1513, normalized size = 5.15
method | result | size |
risch | \(\frac {d^{2} x^{3}}{24 a^{3}}+\frac {d c \,x^{2}}{8 a^{3}}+\frac {c^{2} x}{8 a^{3}}+\frac {c^{3}}{24 a^{3} d}+\frac {3 i \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x -2 i d^{2} f x +2 c^{2} f^{2}-2 i c d f -d^{2}\right ) {\mathrm e}^{-2 i \left (f x +e \right )}}{32 a^{3} f^{3}}+\frac {3 i \left (8 d^{2} x^{2} f^{2}+16 c d \,f^{2} x -4 i d^{2} f x +8 c^{2} f^{2}-4 i c d f -d^{2}\right ) {\mathrm e}^{-4 i \left (f x +e \right )}}{256 a^{3} f^{3}}+\frac {i \left (18 d^{2} x^{2} f^{2}+36 c d \,f^{2} x -6 i d^{2} f x +18 c^{2} f^{2}-6 i c d f -d^{2}\right ) {\mathrm e}^{-6 i \left (f x +e \right )}}{864 a^{3} f^{3}}\) | \(238\) |
default | \(\text {Expression too large to display}\) | \(1513\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 225, normalized size = 0.77 \begin {gather*} \frac {{\left (144 i \, d^{2} f^{2} x^{2} + 144 i \, c^{2} f^{2} + 48 \, c d f - 8 i \, d^{2} - 48 \, {\left (-6 i \, c d f^{2} - d^{2} f\right )} x + 288 \, {\left (d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2} + 3 \, c^{2} f^{3} x\right )} e^{\left (6 i \, f x + 6 i \, e\right )} - 648 \, {\left (-2 i \, d^{2} f^{2} x^{2} - 2 i \, c^{2} f^{2} - 2 \, c d f + i \, d^{2} + 2 \, {\left (-2 i \, c d f^{2} - d^{2} f\right )} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 81 \, {\left (-8 i \, d^{2} f^{2} x^{2} - 8 i \, c^{2} f^{2} - 4 \, c d f + i \, d^{2} + 4 \, {\left (-4 i \, c d f^{2} - d^{2} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{6912 \, a^{3} f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.39, size = 588, normalized size = 2.00 \begin {gather*} \begin {cases} \frac {\left (\left (147456 i a^{6} c^{2} f^{8} e^{6 i e} + 294912 i a^{6} c d f^{8} x e^{6 i e} + 49152 a^{6} c d f^{7} e^{6 i e} + 147456 i a^{6} d^{2} f^{8} x^{2} e^{6 i e} + 49152 a^{6} d^{2} f^{7} x e^{6 i e} - 8192 i a^{6} d^{2} f^{6} e^{6 i e}\right ) e^{- 6 i f x} + \left (663552 i a^{6} c^{2} f^{8} e^{8 i e} + 1327104 i a^{6} c d f^{8} x e^{8 i e} + 331776 a^{6} c d f^{7} e^{8 i e} + 663552 i a^{6} d^{2} f^{8} x^{2} e^{8 i e} + 331776 a^{6} d^{2} f^{7} x e^{8 i e} - 82944 i a^{6} d^{2} f^{6} e^{8 i e}\right ) e^{- 4 i f x} + \left (1327104 i a^{6} c^{2} f^{8} e^{10 i e} + 2654208 i a^{6} c d f^{8} x e^{10 i e} + 1327104 a^{6} c d f^{7} e^{10 i e} + 1327104 i a^{6} d^{2} f^{8} x^{2} e^{10 i e} + 1327104 a^{6} d^{2} f^{7} x e^{10 i e} - 663552 i a^{6} d^{2} f^{6} e^{10 i e}\right ) e^{- 2 i f x}\right ) e^{- 12 i e}}{7077888 a^{9} f^{9}} & \text {for}\: a^{9} f^{9} e^{12 i e} \neq 0 \\\frac {x^{3} \cdot \left (3 d^{2} e^{4 i e} + 3 d^{2} e^{2 i e} + d^{2}\right ) e^{- 6 i e}}{24 a^{3}} + \frac {x^{2} \cdot \left (3 c d e^{4 i e} + 3 c d e^{2 i e} + c d\right ) e^{- 6 i e}}{8 a^{3}} + \frac {x \left (3 c^{2} e^{4 i e} + 3 c^{2} e^{2 i e} + c^{2}\right ) e^{- 6 i e}}{8 a^{3}} & \text {otherwise} \end {cases} + \frac {c^{2} x}{8 a^{3}} + \frac {c d x^{2}}{8 a^{3}} + \frac {d^{2} x^{3}}{24 a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 331, normalized size = 1.13 \begin {gather*} \frac {{\left (288 \, d^{2} f^{3} x^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 864 \, c d f^{3} x^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 864 \, c^{2} f^{3} x e^{\left (6 i \, f x + 6 i \, e\right )} + 1296 i \, d^{2} f^{2} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 648 i \, d^{2} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 144 i \, d^{2} f^{2} x^{2} + 2592 i \, c d f^{2} x e^{\left (4 i \, f x + 4 i \, e\right )} + 1296 i \, c d f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 288 i \, c d f^{2} x + 1296 i \, c^{2} f^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 1296 \, d^{2} f x e^{\left (4 i \, f x + 4 i \, e\right )} + 648 i \, c^{2} f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 324 \, d^{2} f x e^{\left (2 i \, f x + 2 i \, e\right )} + 144 i \, c^{2} f^{2} + 48 \, d^{2} f x + 1296 \, c d f e^{\left (4 i \, f x + 4 i \, e\right )} + 324 \, c d f e^{\left (2 i \, f x + 2 i \, e\right )} + 48 \, c d f - 648 i \, d^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - 81 i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 8 i \, d^{2}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{6912 \, a^{3} f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.37, size = 263, normalized size = 0.89 \begin {gather*} \frac {c^2\,x}{8\,a^3}-{\mathrm {e}}^{-e\,2{}\mathrm {i}-f\,x\,2{}\mathrm {i}}\,\left (\frac {\left (-6\,c^2\,f^2+c\,d\,f\,6{}\mathrm {i}+3\,d^2\right )\,1{}\mathrm {i}}{32\,a^3\,f^3}-\frac {d^2\,x^2\,3{}\mathrm {i}}{16\,a^3\,f}+\frac {d\,x\,\left (-2\,c\,f+d\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{16\,a^3\,f^2}\right )-{\mathrm {e}}^{-e\,4{}\mathrm {i}-f\,x\,4{}\mathrm {i}}\,\left (\frac {\left (-24\,c^2\,f^2+c\,d\,f\,12{}\mathrm {i}+3\,d^2\right )\,1{}\mathrm {i}}{256\,a^3\,f^3}-\frac {d^2\,x^2\,3{}\mathrm {i}}{32\,a^3\,f}+\frac {d\,x\,\left (-4\,c\,f+d\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{64\,a^3\,f^2}\right )-{\mathrm {e}}^{-e\,6{}\mathrm {i}-f\,x\,6{}\mathrm {i}}\,\left (\frac {\left (-18\,c^2\,f^2+c\,d\,f\,6{}\mathrm {i}+d^2\right )\,1{}\mathrm {i}}{864\,a^3\,f^3}-\frac {d^2\,x^2\,1{}\mathrm {i}}{48\,a^3\,f}+\frac {d\,x\,\left (-6\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{144\,a^3\,f^2}\right )+\frac {d^2\,x^3}{24\,a^3}+\frac {c\,d\,x^2}{8\,a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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