Optimal. Leaf size=255 \[ -\frac {18 i d^3 \text {Li}_4\left (-e^{i (a+b x)}\right )}{b^4}+\frac {18 i d^3 \text {Li}_4\left (e^{i (a+b x)}\right )}{b^4}+\frac {24 d^3 \sin (a+b x)}{b^4}-\frac {18 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {18 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac {24 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {9 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac {9 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac {12 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac {4 (c+d x)^3 \cos (a+b x)}{b}-\frac {6 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b} \]
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Rubi [A] time = 0.35, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {4431, 4408, 3296, 2637, 4183, 2531, 6609, 2282, 6589} \[ -\frac {18 d^2 (c+d x) \text {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {18 d^2 (c+d x) \text {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {9 i d (c+d x)^2 \text {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {9 i d (c+d x)^2 \text {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {18 i d^3 \text {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {18 i d^3 \text {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}-\frac {24 d^2 (c+d x) \cos (a+b x)}{b^3}-\frac {12 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac {24 d^3 \sin (a+b x)}{b^4}+\frac {4 (c+d x)^3 \cos (a+b x)}{b}-\frac {6 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 2637
Rule 3296
Rule 4183
Rule 4408
Rule 4431
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int (c+d x)^3 \csc ^2(a+b x) \sin (3 a+3 b x) \, dx &=\int \left (3 (c+d x)^3 \cos (a+b x) \cot (a+b x)-(c+d x)^3 \sin (a+b x)\right ) \, dx\\ &=3 \int (c+d x)^3 \cos (a+b x) \cot (a+b x) \, dx-\int (c+d x)^3 \sin (a+b x) \, dx\\ &=\frac {(c+d x)^3 \cos (a+b x)}{b}+3 \int (c+d x)^3 \csc (a+b x) \, dx-3 \int (c+d x)^3 \sin (a+b x) \, dx-\frac {(3 d) \int (c+d x)^2 \cos (a+b x) \, dx}{b}\\ &=-\frac {6 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {4 (c+d x)^3 \cos (a+b x)}{b}-\frac {3 d (c+d x)^2 \sin (a+b x)}{b^2}-\frac {(9 d) \int (c+d x)^2 \cos (a+b x) \, dx}{b}-\frac {(9 d) \int (c+d x)^2 \log \left (1-e^{i (a+b x)}\right ) \, dx}{b}+\frac {(9 d) \int (c+d x)^2 \log \left (1+e^{i (a+b x)}\right ) \, dx}{b}+\frac {\left (6 d^2\right ) \int (c+d x) \sin (a+b x) \, dx}{b^2}\\ &=-\frac {6 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {4 (c+d x)^3 \cos (a+b x)}{b}+\frac {9 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac {9 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac {12 d (c+d x)^2 \sin (a+b x)}{b^2}-\frac {\left (18 i d^2\right ) \int (c+d x) \text {Li}_2\left (-e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (18 i d^2\right ) \int (c+d x) \text {Li}_2\left (e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (18 d^2\right ) \int (c+d x) \sin (a+b x) \, dx}{b^2}+\frac {\left (6 d^3\right ) \int \cos (a+b x) \, dx}{b^3}\\ &=-\frac {6 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {24 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {4 (c+d x)^3 \cos (a+b x)}{b}+\frac {9 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac {9 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac {18 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {18 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}+\frac {6 d^3 \sin (a+b x)}{b^4}-\frac {12 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac {\left (18 d^3\right ) \int \cos (a+b x) \, dx}{b^3}+\frac {\left (18 d^3\right ) \int \text {Li}_3\left (-e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (18 d^3\right ) \int \text {Li}_3\left (e^{i (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac {6 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {24 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {4 (c+d x)^3 \cos (a+b x)}{b}+\frac {9 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac {9 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac {18 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {18 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}+\frac {24 d^3 \sin (a+b x)}{b^4}-\frac {12 d (c+d x)^2 \sin (a+b x)}{b^2}-\frac {\left (18 i d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}+\frac {\left (18 i d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}\\ &=-\frac {6 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {24 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {4 (c+d x)^3 \cos (a+b x)}{b}+\frac {9 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac {9 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac {18 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {18 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac {18 i d^3 \text {Li}_4\left (-e^{i (a+b x)}\right )}{b^4}+\frac {18 i d^3 \text {Li}_4\left (e^{i (a+b x)}\right )}{b^4}+\frac {24 d^3 \sin (a+b x)}{b^4}-\frac {12 d (c+d x)^2 \sin (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 1.54, size = 459, normalized size = 1.80 \[ \frac {4 \cos (b x) \left (b^3 c^3 \cos (a)+3 b^3 c^2 d x \cos (a)+3 b^3 c d^2 x^2 \cos (a)+b^3 d^3 x^3 \cos (a)-3 b^2 c^2 d \sin (a)-6 b^2 c d^2 x \sin (a)-3 b^2 d^3 x^2 \sin (a)-6 b c d^2 \cos (a)-6 b d^3 x \cos (a)+6 d^3 \sin (a)\right )}{b^4}-\frac {4 \sin (b x) \left (b^3 c^3 \sin (a)+3 b^3 c^2 d x \sin (a)+3 b^3 c d^2 x^2 \sin (a)+b^3 d^3 x^3 \sin (a)+3 b^2 c^2 d \cos (a)+6 b^2 c d^2 x \cos (a)+3 b^2 d^3 x^2 \cos (a)-6 b c d^2 \sin (a)-6 b d^3 x \sin (a)-6 d^3 \cos (a)\right )}{b^4}+\frac {3 \left (-2 b^3 (c+d x)^3 \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x))+3 i d \left (b^2 (c+d x)^2 \text {Li}_2(-\cos (a+b x)-i \sin (a+b x))+2 i b d (c+d x) \text {Li}_3(-\cos (a+b x)-i \sin (a+b x))-2 d^2 \text {Li}_4(-\cos (a+b x)-i \sin (a+b x))\right )-3 i d \left (b^2 (c+d x)^2 \text {Li}_2(\cos (a+b x)+i \sin (a+b x))+2 i b d (c+d x) \text {Li}_3(\cos (a+b x)+i \sin (a+b x))-2 d^2 \text {Li}_4(\cos (a+b x)+i \sin (a+b x))\right )\right )}{b^4} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.56, size = 925, normalized size = 3.63 \[ \text {result too large to display} \]
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 \[ \int {\left (d x + c\right )}^{3} \csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 849, normalized size = 3.33 \[ -\frac {18 i c \,d^{2} \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {18 i c \,d^{2} \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{3}}{b}+\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a^{3}}{b^{4}}-\frac {3 d^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{3}}{b}+\frac {18 i d^{3} \polylog \left (4, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {18 c \,d^{2} \polylog \left (3, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {18 c \,d^{2} \polylog \left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {18 d^{3} \polylog \left (3, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}-\frac {18 d^{3} \polylog \left (3, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}-\frac {18 i d^{3} \polylog \left (4, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {2 \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 i b^{2} d^{3} x^{2}+3 b^{3} c^{2} d x +6 i b^{2} c \,d^{2} x +b^{3} c^{3}+3 i b^{2} c^{2} d -6 b \,d^{3} x -6 c \,d^{2} b -6 i d^{3}\right ) {\mathrm e}^{i \left (b x +a \right )}}{b^{4}}+\frac {2 \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}-3 i b^{2} d^{3} x^{2}+3 b^{3} c^{2} d x -6 i b^{2} c \,d^{2} x +b^{3} c^{3}-3 i b^{2} c^{2} d -6 b \,d^{3} x -6 c \,d^{2} b +6 i d^{3}\right ) {\mathrm e}^{-i \left (b x +a \right )}}{b^{4}}-\frac {6 c^{3} \arctanh \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b}-\frac {9 c^{2} d \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) a}{b^{2}}+\frac {9 c \,d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{3}}-\frac {18 c \,d^{2} a^{2} \arctanh \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {18 c^{2} d a \arctanh \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {9 i d^{3} \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}-\frac {9 i c^{2} d \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {9 i c^{2} d \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {9 i d^{3} \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}-\frac {9 c^{2} d \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b}+\frac {9 c^{2} d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b}+\frac {9 c^{2} d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{2}}-\frac {9 c \,d^{2} a^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {9 c \,d^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b}-\frac {9 c \,d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{b}-\frac {3 d^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) a^{3}}{b^{4}}+\frac {6 d^{3} a^{3} \arctanh \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 602, normalized size = 2.36 \[ \frac {c^{3} {\left (8 \, \cos \left (b x + a\right ) - 3 \, \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \relax (a) + \cos \relax (a)^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \relax (a) + \sin \relax (a)^{2}\right ) + 3 \, \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \relax (a) + \cos \relax (a)^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \relax (a) + \sin \relax (a)^{2}\right )\right )}}{2 \, b} - \frac {36 i \, d^{3} {\rm Li}_{4}(-e^{\left (i \, b x + i \, a\right )}) - 36 i \, d^{3} {\rm Li}_{4}(e^{\left (i \, b x + i \, a\right )}) + {\left (6 i \, b^{3} d^{3} x^{3} + 18 i \, b^{3} c d^{2} x^{2} + 18 i \, b^{3} c^{2} d x\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) + {\left (6 i \, b^{3} d^{3} x^{3} + 18 i \, b^{3} c d^{2} x^{2} + 18 i \, b^{3} c^{2} d x\right )} \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) - 8 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} - 6 \, b c d^{2} + 3 \, {\left (b^{3} c^{2} d - 2 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right ) + {\left (-18 i \, b^{2} d^{3} x^{2} - 36 i \, b^{2} c d^{2} x - 18 i \, b^{2} c^{2} d\right )} {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) + {\left (18 i \, b^{2} d^{3} x^{2} + 36 i \, b^{2} c d^{2} x + 18 i \, b^{2} c^{2} d\right )} {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) + 3 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) - 3 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) + 36 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm Li}_{3}(-e^{\left (i \, b x + i \, a\right )}) - 36 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm Li}_{3}(e^{\left (i \, b x + i \, a\right )}) + 24 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - 2 \, d^{3}\right )} \sin \left (b x + a\right )}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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