Optimal. Leaf size=294 \[ -\frac {3 d (c+d x) e^{2 i e+2 i f x}}{16 a^3 f^2}+\frac {3 d (c+d x) e^{4 i e+4 i f x}}{64 a^3 f^2}-\frac {d (c+d x) e^{6 i e+6 i f x}}{144 a^3 f^2}+\frac {3 i (c+d x)^2 e^{2 i e+2 i f x}}{16 a^3 f}-\frac {3 i (c+d x)^2 e^{4 i e+4 i f x}}{32 a^3 f}+\frac {i (c+d x)^2 e^{6 i e+6 i f x}}{48 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}-\frac {3 i d^2 e^{2 i e+2 i f x}}{32 a^3 f^3}+\frac {3 i d^2 e^{4 i e+4 i f x}}{256 a^3 f^3}-\frac {i d^2 e^{6 i e+6 i f x}}{864 a^3 f^3} \]
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Rubi [A] time = 0.27, 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 = {3729, 2176, 2194} \[ -\frac {3 d (c+d x) e^{2 i e+2 i f x}}{16 a^3 f^2}+\frac {3 d (c+d x) e^{4 i e+4 i f x}}{64 a^3 f^2}-\frac {d (c+d x) e^{6 i e+6 i f x}}{144 a^3 f^2}+\frac {3 i (c+d x)^2 e^{2 i e+2 i f x}}{16 a^3 f}-\frac {3 i (c+d x)^2 e^{4 i e+4 i f x}}{32 a^3 f}+\frac {i (c+d x)^2 e^{6 i e+6 i f x}}{48 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}-\frac {3 i d^2 e^{2 i e+2 i f x}}{32 a^3 f^3}+\frac {3 i d^2 e^{4 i e+4 i f x}}{256 a^3 f^3}-\frac {i d^2 e^{6 i e+6 i f x}}{864 a^3 f^3} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rule 3729
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{(a+i a \cot (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 = 0.81, size = 369, normalized size = 1.26 \[ \frac {288 f^3 x \left (3 c^2+3 c d x+d^2 x^2\right )+648 (\cos (2 e)+i \sin (2 e)) \cos (2 f x) ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d ((1+i) f x+i))-81 (\cos (4 e)+i \sin (4 e)) \cos (4 f x) ((2+2 i) c f+d (-1+(2+2 i) f x)) ((2+2 i) c f+d ((2+2 i) f x+i))+8 (\cos (6 e)+i \sin (6 e)) \cos (6 f x) ((3+3 i) c f+d (-1+(3+3 i) f x)) ((3+3 i) c f+d ((3+3 i) f x+i))+648 i (\cos (2 e)+i \sin (2 e)) \sin (2 f x) ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d ((1+i) f x+i))-81 (\cos (4 e)+i \sin (4 e)) \sin (4 f x) (-(2+2 i) c f+(-2-2 i) d f x+d) ((2-2 i) c f+(2-2 i) d f x+d)+8 i (\cos (6 e)+i \sin (6 e)) \sin (6 f x) ((3+3 i) c f+d (-1+(3+3 i) f x)) ((3+3 i) c f+d ((3+3 i) f x+i))}{6912 a^3 f^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 207, normalized size = 0.70 \[ \frac {288 \, d^{2} f^{3} x^{3} + 864 \, c d f^{3} x^{2} + 864 \, c^{2} f^{3} x + {\left (144 i \, d^{2} f^{2} x^{2} + 144 i \, c^{2} f^{2} - 48 \, c d f - 8 i \, d^{2} + {\left (288 i \, c d f^{2} - 48 \, d^{2} f\right )} x\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-648 i \, d^{2} f^{2} x^{2} - 648 i \, c^{2} f^{2} + 324 \, c d f + 81 i \, d^{2} + {\left (-1296 i \, c d f^{2} + 324 \, d^{2} f\right )} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (1296 i \, d^{2} f^{2} x^{2} + 1296 i \, c^{2} f^{2} - 1296 \, c d f - 648 i \, d^{2} + {\left (2592 i \, c d f^{2} - 1296 \, d^{2} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{6912 \, a^{3} f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 351, normalized size = 1.19 \[ \frac {288 \, d^{2} f^{3} x^{3} + 864 \, c d f^{3} x^{2} + 144 i \, d^{2} f^{2} x^{2} e^{\left (6 i \, f x + 6 i \, e\right )} - 648 i \, d^{2} f^{2} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 1296 i \, d^{2} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 864 \, c^{2} f^{3} x + 288 i \, c d f^{2} x e^{\left (6 i \, f x + 6 i \, e\right )} - 1296 i \, c d f^{2} x e^{\left (4 i \, f x + 4 i \, e\right )} + 2592 i \, c d f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 144 i \, c^{2} f^{2} e^{\left (6 i \, f x + 6 i \, e\right )} - 48 \, d^{2} f x e^{\left (6 i \, f x + 6 i \, e\right )} - 648 i \, c^{2} f^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 324 \, d^{2} f x e^{\left (4 i \, f x + 4 i \, e\right )} + 1296 i \, c^{2} f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 1296 \, d^{2} f x e^{\left (2 i \, f x + 2 i \, e\right )} - 48 \, c d f e^{\left (6 i \, f x + 6 i \, e\right )} + 324 \, c d f e^{\left (4 i \, f x + 4 i \, e\right )} - 1296 \, c d f e^{\left (2 i \, f x + 2 i \, e\right )} - 8 i \, d^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 81 i \, d^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - 648 i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )}}{6912 \, a^{3} f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.48, size = 1843, normalized size = 6.27 \[ \text {Expression too large to display} \]
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 = 1.00, size = 263, normalized size = 0.89 \[ {\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (\frac {\left (6\,c^2\,f^2+c\,d\,f\,6{}\mathrm {i}-3\,d^2\right )\,1{}\mathrm {i}}{32\,a^3\,f^3}+\frac {d^2\,x^2\,3{}\mathrm {i}}{16\,a^3\,f}+\frac {d\,x\,\left (2\,c\,f+d\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{16\,a^3\,f^2}\right )-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (\frac {\left (24\,c^2\,f^2+c\,d\,f\,12{}\mathrm {i}-3\,d^2\right )\,1{}\mathrm {i}}{256\,a^3\,f^3}+\frac {d^2\,x^2\,3{}\mathrm {i}}{32\,a^3\,f}+\frac {d\,x\,\left (4\,c\,f+d\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{64\,a^3\,f^2}\right )+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (\frac {\left (18\,c^2\,f^2+c\,d\,f\,6{}\mathrm {i}-d^2\right )\,1{}\mathrm {i}}{864\,a^3\,f^3}+\frac {d^2\,x^2\,1{}\mathrm {i}}{48\,a^3\,f}+\frac {d\,x\,\left (6\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{144\,a^3\,f^2}\right )+\frac {c^2\,x}{8\,a^3}+\frac {d^2\,x^3}{24\,a^3}+\frac {c\,d\,x^2}{8\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.64, size = 578, normalized size = 1.97 \[ \begin {cases} - \frac {\left (- 1327104 i a^{6} c^{2} f^{8} e^{2 i e} - 2654208 i a^{6} c d f^{8} x e^{2 i e} + 1327104 a^{6} c d f^{7} e^{2 i e} - 1327104 i a^{6} d^{2} f^{8} x^{2} e^{2 i e} + 1327104 a^{6} d^{2} f^{7} x e^{2 i e} + 663552 i a^{6} d^{2} f^{6} e^{2 i e}\right ) e^{2 i f x} + \left (663552 i a^{6} c^{2} f^{8} e^{4 i e} + 1327104 i a^{6} c d f^{8} x e^{4 i e} - 331776 a^{6} c d f^{7} e^{4 i e} + 663552 i a^{6} d^{2} f^{8} x^{2} e^{4 i e} - 331776 a^{6} d^{2} f^{7} x e^{4 i e} - 82944 i a^{6} d^{2} f^{6} e^{4 i e}\right ) e^{4 i f x} + \left (- 147456 i a^{6} c^{2} f^{8} e^{6 i e} - 294912 i a^{6} c d f^{8} x e^{6 i e} + 49152 a^{6} c d f^{7} e^{6 i e} - 147456 i a^{6} d^{2} f^{8} x^{2} e^{6 i e} + 49152 a^{6} d^{2} f^{7} x e^{6 i e} + 8192 i a^{6} d^{2} f^{6} e^{6 i e}\right ) e^{6 i f x}}{7077888 a^{9} f^{9}} & \text {for}\: 7077888 a^{9} f^{9} \neq 0 \\\frac {x^{3} \left (- d^{2} e^{6 i e} + 3 d^{2} e^{4 i e} - 3 d^{2} e^{2 i e}\right )}{24 a^{3}} + \frac {x^{2} \left (- c d e^{6 i e} + 3 c d e^{4 i e} - 3 c d e^{2 i e}\right )}{8 a^{3}} + \frac {x \left (- c^{2} e^{6 i e} + 3 c^{2} e^{4 i e} - 3 c^{2} e^{2 i e}\right )}{8 a^{3}} & \text {otherwise} \end {cases} + \frac {c^{2} x}{8 a^{3}} + \frac {c d x^{2}}{8 a^{3}} + \frac {d^{2} x^{3}}{24 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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