Optimal. Leaf size=202 \[ -\frac {i d^2 e^{-2 i e-2 i f x}}{8 a^2 f^3}-\frac {i d^2 e^{-4 i e-4 i f x}}{128 a^2 f^3}+\frac {d e^{-2 i e-2 i f x} (c+d x)}{4 a^2 f^2}+\frac {d e^{-4 i e-4 i f x} (c+d x)}{32 a^2 f^2}+\frac {i e^{-2 i e-2 i f x} (c+d x)^2}{4 a^2 f}+\frac {i e^{-4 i e-4 i f x} (c+d x)^2}{16 a^2 f}+\frac {(c+d x)^3}{12 a^2 d} \]
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Rubi [A]
time = 0.14, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps
used = 8, 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 {d (c+d x) e^{-2 i e-2 i f x}}{4 a^2 f^2}+\frac {d (c+d x) e^{-4 i e-4 i f x}}{32 a^2 f^2}+\frac {i (c+d x)^2 e^{-2 i e-2 i f x}}{4 a^2 f}+\frac {i (c+d x)^2 e^{-4 i e-4 i f x}}{16 a^2 f}+\frac {(c+d x)^3}{12 a^2 d}-\frac {i d^2 e^{-2 i e-2 i f x}}{8 a^2 f^3}-\frac {i d^2 e^{-4 i e-4 i f x}}{128 a^2 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))^2} \, dx &=\int \left (\frac {(c+d x)^2}{4 a^2}+\frac {e^{-2 i e-2 i f x} (c+d x)^2}{2 a^2}+\frac {e^{-4 i e-4 i f x} (c+d x)^2}{4 a^2}\right ) \, dx\\ &=\frac {(c+d x)^3}{12 a^2 d}+\frac {\int e^{-4 i e-4 i f x} (c+d x)^2 \, dx}{4 a^2}+\frac {\int e^{-2 i e-2 i f x} (c+d x)^2 \, dx}{2 a^2}\\ &=\frac {i e^{-2 i e-2 i f x} (c+d x)^2}{4 a^2 f}+\frac {i e^{-4 i e-4 i f x} (c+d x)^2}{16 a^2 f}+\frac {(c+d x)^3}{12 a^2 d}-\frac {(i d) \int e^{-4 i e-4 i f x} (c+d x) \, dx}{8 a^2 f}-\frac {(i d) \int e^{-2 i e-2 i f x} (c+d x) \, dx}{2 a^2 f}\\ &=\frac {d e^{-2 i e-2 i f x} (c+d x)}{4 a^2 f^2}+\frac {d e^{-4 i e-4 i f x} (c+d x)}{32 a^2 f^2}+\frac {i e^{-2 i e-2 i f x} (c+d x)^2}{4 a^2 f}+\frac {i e^{-4 i e-4 i f x} (c+d x)^2}{16 a^2 f}+\frac {(c+d x)^3}{12 a^2 d}-\frac {d^2 \int e^{-4 i e-4 i f x} \, dx}{32 a^2 f^2}-\frac {d^2 \int e^{-2 i e-2 i f x} \, dx}{4 a^2 f^2}\\ &=-\frac {i d^2 e^{-2 i e-2 i f x}}{8 a^2 f^3}-\frac {i d^2 e^{-4 i e-4 i f x}}{128 a^2 f^3}+\frac {d e^{-2 i e-2 i f x} (c+d x)}{4 a^2 f^2}+\frac {d e^{-4 i e-4 i f x} (c+d x)}{32 a^2 f^2}+\frac {i e^{-2 i e-2 i f x} (c+d x)^2}{4 a^2 f}+\frac {i e^{-4 i e-4 i f x} (c+d x)^2}{16 a^2 f}+\frac {(c+d x)^3}{12 a^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.69, size = 282, normalized size = 1.40 \begin {gather*} \frac {\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left ((d+(1+i) c f+(1+i) d f x) ((1+i) c f+d (-i+(1+i) f x)) \cos (2 f x)+\frac {1}{16} (d+(2+2 i) c f+(2+2 i) d f x) ((2+2 i) c f+d (-i+(2+2 i) f x)) \cos (4 f x) (\cos (2 e)-i \sin (2 e))+\frac {2}{3} f^3 x \left (3 c^2+3 c d x+d^2 x^2\right ) (\cos (2 e)+i \sin (2 e))-i (d+(1+i) c f+(1+i) d f x) ((1+i) c f+d (-i+(1+i) f x)) \sin (2 f x)-\frac {1}{16} i (d+(2+2 i) c f+(2+2 i) d f x) ((2+2 i) c f+d (-i+(2+2 i) f x)) (\cos (2 e)-i \sin (2 e)) \sin (4 f x)\right )}{8 f^3 (a+i a \tan (e+f x))^2} \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. 955 vs. \(2 (166 ) = 332\).
time = 0.49, size = 956, normalized size = 4.73
method | result | size |
risch | \(\frac {d^{2} x^{3}}{12 a^{2}}+\frac {d c \,x^{2}}{4 a^{2}}+\frac {c^{2} x}{4 a^{2}}+\frac {c^{3}}{12 a^{2} d}+\frac {i \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x -2 i d^{2} f x +2 c^{2} f^{2}-2 i c d f -d^{2}\right ) {\mathrm e}^{-2 i \left (f x +e \right )}}{8 f^{3} a^{2}}+\frac {i \left (8 d^{2} x^{2} f^{2}+16 c d \,f^{2} x -4 i d^{2} f x +8 c^{2} f^{2}-4 i c d f -d^{2}\right ) {\mathrm e}^{-4 i \left (f x +e \right )}}{128 f^{3} a^{2}}\) | \(173\) |
default | \(\text {Expression too large to display}\) | \(956\) |
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 = 0.38, size = 166, normalized size = 0.82 \begin {gather*} \frac {{\left (24 i \, d^{2} f^{2} x^{2} + 24 i \, c^{2} f^{2} + 12 \, c d f - 3 i \, d^{2} - 12 \, {\left (-4 i \, c d f^{2} - d^{2} f\right )} x + 32 \, {\left (d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2} + 3 \, c^{2} f^{3} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 48 \, {\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 (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{384 \, a^{2} f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 418, normalized size = 2.07 \begin {gather*} \begin {cases} \frac {\left (\left (64 i a^{2} c^{2} f^{5} e^{2 i e} + 128 i a^{2} c d f^{5} x e^{2 i e} + 32 a^{2} c d f^{4} e^{2 i e} + 64 i a^{2} d^{2} f^{5} x^{2} e^{2 i e} + 32 a^{2} d^{2} f^{4} x e^{2 i e} - 8 i a^{2} d^{2} f^{3} e^{2 i e}\right ) e^{- 4 i f x} + \left (256 i a^{2} c^{2} f^{5} e^{4 i e} + 512 i a^{2} c d f^{5} x e^{4 i e} + 256 a^{2} c d f^{4} e^{4 i e} + 256 i a^{2} d^{2} f^{5} x^{2} e^{4 i e} + 256 a^{2} d^{2} f^{4} x e^{4 i e} - 128 i a^{2} d^{2} f^{3} e^{4 i e}\right ) e^{- 2 i f x}\right ) e^{- 6 i e}}{1024 a^{4} f^{6}} & \text {for}\: a^{4} f^{6} e^{6 i e} \neq 0 \\\frac {x^{3} \cdot \left (2 d^{2} e^{2 i e} + d^{2}\right ) e^{- 4 i e}}{12 a^{2}} + \frac {x^{2} \cdot \left (2 c d e^{2 i e} + c d\right ) e^{- 4 i e}}{4 a^{2}} + \frac {x \left (2 c^{2} e^{2 i e} + c^{2}\right ) e^{- 4 i e}}{4 a^{2}} & \text {otherwise} \end {cases} + \frac {c^{2} x}{4 a^{2}} + \frac {c d x^{2}}{4 a^{2}} + \frac {d^{2} x^{3}}{12 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 227, normalized size = 1.12 \begin {gather*} \frac {{\left (32 \, d^{2} f^{3} x^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 96 \, c d f^{3} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 96 \, c^{2} f^{3} x e^{\left (4 i \, f x + 4 i \, e\right )} + 96 i \, d^{2} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 24 i \, d^{2} f^{2} x^{2} + 192 i \, c d f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 48 i \, c d f^{2} x + 96 i \, c^{2} f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 96 \, d^{2} f x e^{\left (2 i \, f x + 2 i \, e\right )} + 24 i \, c^{2} f^{2} + 12 \, d^{2} f x + 96 \, c d f e^{\left (2 i \, f x + 2 i \, e\right )} + 12 \, c d f - 48 i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 3 i \, d^{2}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{384 \, a^{2} f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.99, size = 183, normalized size = 0.91 \begin {gather*} \frac {c^2\,x}{4\,a^2}-{\mathrm {e}}^{-e\,4{}\mathrm {i}-f\,x\,4{}\mathrm {i}}\,\left (\frac {\left (-8\,c^2\,f^2+c\,d\,f\,4{}\mathrm {i}+d^2\right )\,1{}\mathrm {i}}{128\,a^2\,f^3}-\frac {d^2\,x^2\,1{}\mathrm {i}}{16\,a^2\,f}+\frac {d\,x\,\left (-4\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,a^2\,f^2}\right )-{\mathrm {e}}^{-e\,2{}\mathrm {i}-f\,x\,2{}\mathrm {i}}\,\left (\frac {\left (-2\,c^2\,f^2+c\,d\,f\,2{}\mathrm {i}+d^2\right )\,1{}\mathrm {i}}{8\,a^2\,f^3}-\frac {d^2\,x^2\,1{}\mathrm {i}}{4\,a^2\,f}+\frac {d\,x\,\left (-2\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,a^2\,f^2}\right )+\frac {d^2\,x^3}{12\,a^2}+\frac {c\,d\,x^2}{4\,a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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