Optimal. Leaf size=270 \[ -\frac {3 i d^2 (c+d x) e^{2 i e+2 i f x}}{8 a^2 f^3}+\frac {3 i d^2 (c+d x) e^{4 i e+4 i f x}}{128 a^2 f^3}-\frac {3 d (c+d x)^2 e^{2 i e+2 i f x}}{8 a^2 f^2}+\frac {3 d (c+d x)^2 e^{4 i e+4 i f x}}{64 a^2 f^2}+\frac {i (c+d x)^3 e^{2 i e+2 i f x}}{4 a^2 f}-\frac {i (c+d x)^3 e^{4 i e+4 i f x}}{16 a^2 f}+\frac {(c+d x)^4}{16 a^2 d}+\frac {3 d^3 e^{2 i e+2 i f x}}{16 a^2 f^4}-\frac {3 d^3 e^{4 i e+4 i f x}}{512 a^2 f^4} \]
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Rubi [A] time = 0.28, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3729, 2176, 2194} \[ -\frac {3 i d^2 (c+d x) e^{2 i e+2 i f x}}{8 a^2 f^3}+\frac {3 i d^2 (c+d x) e^{4 i e+4 i f x}}{128 a^2 f^3}-\frac {3 d (c+d x)^2 e^{2 i e+2 i f x}}{8 a^2 f^2}+\frac {3 d (c+d x)^2 e^{4 i e+4 i f x}}{64 a^2 f^2}+\frac {i (c+d x)^3 e^{2 i e+2 i f x}}{4 a^2 f}-\frac {i (c+d x)^3 e^{4 i e+4 i f x}}{16 a^2 f}+\frac {(c+d x)^4}{16 a^2 d}+\frac {3 d^3 e^{2 i e+2 i f x}}{16 a^2 f^4}-\frac {3 d^3 e^{4 i e+4 i f x}}{512 a^2 f^4} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rule 3729
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{(a+i a \cot (e+f x))^2} \, dx &=\int \left (\frac {(c+d x)^3}{4 a^2}-\frac {e^{2 i e+2 i f x} (c+d x)^3}{2 a^2}+\frac {e^{4 i e+4 i f x} (c+d x)^3}{4 a^2}\right ) \, dx\\ &=\frac {(c+d x)^4}{16 a^2 d}+\frac {\int e^{4 i e+4 i f x} (c+d x)^3 \, dx}{4 a^2}-\frac {\int e^{2 i e+2 i f x} (c+d x)^3 \, dx}{2 a^2}\\ &=\frac {i e^{2 i e+2 i f x} (c+d x)^3}{4 a^2 f}-\frac {i e^{4 i e+4 i f x} (c+d x)^3}{16 a^2 f}+\frac {(c+d x)^4}{16 a^2 d}+\frac {(3 i d) \int e^{4 i e+4 i f x} (c+d x)^2 \, dx}{16 a^2 f}-\frac {(3 i d) \int e^{2 i e+2 i f x} (c+d x)^2 \, dx}{4 a^2 f}\\ &=-\frac {3 d e^{2 i e+2 i f x} (c+d x)^2}{8 a^2 f^2}+\frac {3 d e^{4 i e+4 i f x} (c+d x)^2}{64 a^2 f^2}+\frac {i e^{2 i e+2 i f x} (c+d x)^3}{4 a^2 f}-\frac {i e^{4 i e+4 i f x} (c+d x)^3}{16 a^2 f}+\frac {(c+d x)^4}{16 a^2 d}-\frac {\left (3 d^2\right ) \int e^{4 i e+4 i f x} (c+d x) \, dx}{32 a^2 f^2}+\frac {\left (3 d^2\right ) \int e^{2 i e+2 i f x} (c+d x) \, dx}{4 a^2 f^2}\\ &=-\frac {3 i d^2 e^{2 i e+2 i f x} (c+d x)}{8 a^2 f^3}+\frac {3 i d^2 e^{4 i e+4 i f x} (c+d x)}{128 a^2 f^3}-\frac {3 d e^{2 i e+2 i f x} (c+d x)^2}{8 a^2 f^2}+\frac {3 d e^{4 i e+4 i f x} (c+d x)^2}{64 a^2 f^2}+\frac {i e^{2 i e+2 i f x} (c+d x)^3}{4 a^2 f}-\frac {i e^{4 i e+4 i f x} (c+d x)^3}{16 a^2 f}+\frac {(c+d x)^4}{16 a^2 d}-\frac {\left (3 i d^3\right ) \int e^{4 i e+4 i f x} \, dx}{128 a^2 f^3}+\frac {\left (3 i d^3\right ) \int e^{2 i e+2 i f x} \, dx}{8 a^2 f^3}\\ &=\frac {3 d^3 e^{2 i e+2 i f x}}{16 a^2 f^4}-\frac {3 d^3 e^{4 i e+4 i f x}}{512 a^2 f^4}-\frac {3 i d^2 e^{2 i e+2 i f x} (c+d x)}{8 a^2 f^3}+\frac {3 i d^2 e^{4 i e+4 i f x} (c+d x)}{128 a^2 f^3}-\frac {3 d e^{2 i e+2 i f x} (c+d x)^2}{8 a^2 f^2}+\frac {3 d e^{4 i e+4 i f x} (c+d x)^2}{64 a^2 f^2}+\frac {i e^{2 i e+2 i f x} (c+d x)^3}{4 a^2 f}-\frac {i e^{4 i e+4 i f x} (c+d x)^3}{16 a^2 f}+\frac {(c+d x)^4}{16 a^2 d}\\ \end {align*}
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Mathematica [A] time = 1.61, size = 362, normalized size = 1.34 \[ \frac {(\cos (2 (e+f x))+i \sin (2 (e+f x))) \left (\left (32 c^3 f^3 (4 f x-i)+24 c^2 d f^2 \left (8 f^2 x^2-4 i f x+1\right )+4 c d^2 f \left (32 f^3 x^3-24 i f^2 x^2+12 f x+3 i\right )+d^3 \left (32 f^4 x^4-32 i f^3 x^3+24 f^2 x^2+12 i f x-3\right )\right ) \cos (2 (e+f x))-i \left (\left (32 c^3 f^3 (4 f x+i)+24 c^2 d f^2 \left (8 f^2 x^2+4 i f x-1\right )+4 c d^2 f \left (32 f^3 x^3+24 i f^2 x^2-12 f x-3 i\right )+d^3 \left (32 f^4 x^4+32 i f^3 x^3-24 f^2 x^2-12 i f x+3\right )\right ) \sin (2 (e+f x))-32 \left (4 c^3 f^3+6 c^2 d f^2 (2 f x+i)+6 c d^2 f \left (2 f^2 x^2+2 i f x-1\right )+d^3 \left (4 f^3 x^3+6 i f^2 x^2-6 f x-3 i\right )\right )\right )\right )}{512 a^2 f^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.90, size = 253, normalized size = 0.94 \[ \frac {32 \, d^{3} f^{4} x^{4} + 128 \, c d^{2} f^{4} x^{3} + 192 \, c^{2} d f^{4} x^{2} + 128 \, c^{3} f^{4} x + {\left (-32 i \, d^{3} f^{3} x^{3} - 32 i \, c^{3} f^{3} + 24 \, c^{2} d f^{2} + 12 i \, c d^{2} f - 3 \, d^{3} + {\left (-96 i \, c d^{2} f^{3} + 24 \, d^{3} f^{2}\right )} x^{2} + {\left (-96 i \, c^{2} d f^{3} + 48 \, c d^{2} f^{2} + 12 i \, d^{3} f\right )} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (128 i \, d^{3} f^{3} x^{3} + 128 i \, c^{3} f^{3} - 192 \, c^{2} d f^{2} - 192 i \, c d^{2} f + 96 \, d^{3} + {\left (384 i \, c d^{2} f^{3} - 192 \, d^{3} f^{2}\right )} x^{2} + {\left (384 i \, c^{2} d f^{3} - 384 \, c d^{2} f^{2} - 192 i \, d^{3} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{512 \, a^{2} f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.51, size = 433, normalized size = 1.60 \[ \frac {32 \, d^{3} f^{4} x^{4} + 128 \, c d^{2} f^{4} x^{3} - 32 i \, d^{3} f^{3} x^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 128 i \, d^{3} f^{3} x^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 192 \, c^{2} d f^{4} x^{2} - 96 i \, c d^{2} f^{3} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 384 i \, c d^{2} f^{3} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 128 \, c^{3} f^{4} x - 96 i \, c^{2} d f^{3} x e^{\left (4 i \, f x + 4 i \, e\right )} + 24 \, d^{3} f^{2} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 384 i \, c^{2} d f^{3} x e^{\left (2 i \, f x + 2 i \, e\right )} - 192 \, d^{3} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 32 i \, c^{3} f^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 48 \, c d^{2} f^{2} x e^{\left (4 i \, f x + 4 i \, e\right )} + 128 i \, c^{3} f^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 384 \, c d^{2} f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 24 \, c^{2} d f^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 12 i \, d^{3} f x e^{\left (4 i \, f x + 4 i \, e\right )} - 192 \, c^{2} d f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 192 i \, d^{3} f x e^{\left (2 i \, f x + 2 i \, e\right )} + 12 i \, c d^{2} f e^{\left (4 i \, f x + 4 i \, e\right )} - 192 i \, c d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - 3 \, d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 96 \, d^{3} e^{\left (2 i \, f x + 2 i \, e\right )}}{512 \, a^{2} f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.16, size = 2261, normalized size = 8.37 \[ \text {Expression too large to display} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 294, normalized size = 1.09 \[ {\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (-\frac {\left (-4\,c^3\,f^3-c^2\,d\,f^2\,6{}\mathrm {i}+6\,c\,d^2\,f+d^3\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,a^2\,f^4}+\frac {d^3\,x^3\,1{}\mathrm {i}}{4\,a^2\,f}+\frac {d\,x\,\left (2\,c^2\,f^2+c\,d\,f\,2{}\mathrm {i}-d^2\right )\,3{}\mathrm {i}}{8\,a^2\,f^3}+\frac {d^2\,x^2\,\left (2\,c\,f+d\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8\,a^2\,f^2}\right )-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (-\frac {\left (-32\,c^3\,f^3-c^2\,d\,f^2\,24{}\mathrm {i}+12\,c\,d^2\,f+d^3\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{512\,a^2\,f^4}+\frac {d^3\,x^3\,1{}\mathrm {i}}{16\,a^2\,f}+\frac {d\,x\,\left (8\,c^2\,f^2+c\,d\,f\,4{}\mathrm {i}-d^2\right )\,3{}\mathrm {i}}{128\,a^2\,f^3}+\frac {d^2\,x^2\,\left (4\,c\,f+d\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{64\,a^2\,f^2}\right )+\frac {c^3\,x}{4\,a^2}+\frac {d^3\,x^4}{16\,a^2}+\frac {3\,c^2\,d\,x^2}{8\,a^2}+\frac {c\,d^2\,x^3}{4\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.64, size = 653, normalized size = 2.42 \[ \begin {cases} \frac {\left (2048 i a^{2} c^{3} f^{7} e^{2 i e} + 6144 i a^{2} c^{2} d f^{7} x e^{2 i e} - 3072 a^{2} c^{2} d f^{6} e^{2 i e} + 6144 i a^{2} c d^{2} f^{7} x^{2} e^{2 i e} - 6144 a^{2} c d^{2} f^{6} x e^{2 i e} - 3072 i a^{2} c d^{2} f^{5} e^{2 i e} + 2048 i a^{2} d^{3} f^{7} x^{3} e^{2 i e} - 3072 a^{2} d^{3} f^{6} x^{2} e^{2 i e} - 3072 i a^{2} d^{3} f^{5} x e^{2 i e} + 1536 a^{2} d^{3} f^{4} e^{2 i e}\right ) e^{2 i f x} + \left (- 512 i a^{2} c^{3} f^{7} e^{4 i e} - 1536 i a^{2} c^{2} d f^{7} x e^{4 i e} + 384 a^{2} c^{2} d f^{6} e^{4 i e} - 1536 i a^{2} c d^{2} f^{7} x^{2} e^{4 i e} + 768 a^{2} c d^{2} f^{6} x e^{4 i e} + 192 i a^{2} c d^{2} f^{5} e^{4 i e} - 512 i a^{2} d^{3} f^{7} x^{3} e^{4 i e} + 384 a^{2} d^{3} f^{6} x^{2} e^{4 i e} + 192 i a^{2} d^{3} f^{5} x e^{4 i e} - 48 a^{2} d^{3} f^{4} e^{4 i e}\right ) e^{4 i f x}}{8192 a^{4} f^{8}} & \text {for}\: 8192 a^{4} f^{8} \neq 0 \\\frac {x^{4} \left (d^{3} e^{4 i e} - 2 d^{3} e^{2 i e}\right )}{16 a^{2}} + \frac {x^{3} \left (c d^{2} e^{4 i e} - 2 c d^{2} e^{2 i e}\right )}{4 a^{2}} + \frac {x^{2} \left (3 c^{2} d e^{4 i e} - 6 c^{2} d e^{2 i e}\right )}{8 a^{2}} + \frac {x \left (c^{3} e^{4 i e} - 2 c^{3} e^{2 i e}\right )}{4 a^{2}} & \text {otherwise} \end {cases} + \frac {c^{3} x}{4 a^{2}} + \frac {3 c^{2} d x^{2}}{8 a^{2}} + \frac {c d^{2} x^{3}}{4 a^{2}} + \frac {d^{3} x^{4}}{16 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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