Optimal. Leaf size=113 \[ \frac {3 d^3 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i d^2 (c+d x) \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {i (c+d x)^3}{b} \]
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Rubi [A] time = 0.21, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4184, 3717, 2190, 2531, 2282, 6589} \[ -\frac {3 i d^2 (c+d x) \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^3 \text {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {i (c+d x)^3}{b} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 4184
Rule 6589
Rubi steps
\begin {align*} \int (c+d x)^3 \csc ^2(a+b x) \, dx &=-\frac {(c+d x)^3 \cot (a+b x)}{b}+\frac {(3 d) \int (c+d x)^2 \cot (a+b x) \, dx}{b}\\ &=-\frac {i (c+d x)^3}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {(6 i d) \int \frac {e^{2 i (a+b x)} (c+d x)^2}{1-e^{2 i (a+b x)}} \, dx}{b}\\ &=-\frac {i (c+d x)^3}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {\left (6 d^2\right ) \int (c+d x) \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {i (c+d x)^3}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i d^2 (c+d x) \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {\left (3 i d^3\right ) \int \text {Li}_2\left (e^{2 i (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac {i (c+d x)^3}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i d^2 (c+d x) \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {\left (3 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^4}\\ &=-\frac {i (c+d x)^3}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i d^2 (c+d x) \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^3 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^4}\\ \end {align*}
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Mathematica [B] time = 6.96, size = 478, normalized size = 4.23 \[ \frac {3 c^2 d \csc (a) (\sin (a) \log (\sin (a) \cos (b x)+\cos (a) \sin (b x))-b x \cos (a))}{b^2 \left (\sin ^2(a)+\cos ^2(a)\right )}-\frac {3 c d^2 \csc (a) \sec (a) \left (b^2 x^2 e^{i \tan ^{-1}(\tan (a))}+\frac {\tan (a) \left (i \text {Li}_2\left (e^{2 i \left (b x+\tan ^{-1}(\tan (a))\right )}\right )+i b x \left (2 \tan ^{-1}(\tan (a))-\pi \right )-2 \left (\tan ^{-1}(\tan (a))+b x\right ) \log \left (1-e^{2 i \left (\tan ^{-1}(\tan (a))+b x\right )}\right )+2 \tan ^{-1}(\tan (a)) \log \left (\sin \left (\tan ^{-1}(\tan (a))+b x\right )\right )-\pi \log \left (1+e^{-2 i b x}\right )+\pi \log (\cos (b x))\right )}{\sqrt {\tan ^2(a)+1}}\right )}{b^3 \sqrt {\sec ^2(a) \left (\sin ^2(a)+\cos ^2(a)\right )}}-\frac {e^{i a} d^3 \csc (a) \left (2 e^{-2 i a} b^3 x^3+3 i \left (1-e^{-2 i a}\right ) b^2 x^2 \log \left (1-e^{-i (a+b x)}\right )+3 i \left (1-e^{-2 i a}\right ) b^2 x^2 \log \left (1+e^{-i (a+b x)}\right )-6 e^{-2 i a} \left (-1+e^{2 i a}\right ) \left (b x \text {Li}_2\left (-e^{-i (a+b x)}\right )-i \text {Li}_3\left (-e^{-i (a+b x)}\right )\right )-6 e^{-2 i a} \left (-1+e^{2 i a}\right ) \left (b x \text {Li}_2\left (e^{-i (a+b x)}\right )-i \text {Li}_3\left (e^{-i (a+b x)}\right )\right )\right )}{2 b^4}+\frac {\csc (a) \csc (a+b x) \left (c^3 \sin (b x)+3 c^2 d x \sin (b x)+3 c d^2 x^2 \sin (b x)+d^3 x^3 \sin (b x)\right )}{b} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.90, size = 672, normalized size = 5.95 \[ \frac {6 \, d^{3} {\rm polylog}\left (3, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + 6 \, d^{3} {\rm polylog}\left (3, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + 6 \, d^{3} {\rm polylog}\left (3, -\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + 6 \, d^{3} {\rm polylog}\left (3, -\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + {\left (-6 i \, b d^{3} x - 6 i \, b c d^{2}\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + {\left (6 i \, b d^{3} x + 6 i \, b c d^{2}\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + {\left (6 i \, b d^{3} x + 6 i \, b c d^{2}\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + {\left (-6 i \, b d^{3} x - 6 i \, b c d^{2}\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) \sin \left (b x + a\right ) + 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) - \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) \sin \left (b x + a\right ) + 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + 2 \, a b c d^{2} - a^{2} d^{3}\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + 2 \, a b c d^{2} - a^{2} d^{3}\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 2 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cos \left (b x + a\right )}{2 \, b^{4} \sin \left (b x + a\right )} \]
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 \[ \int {\left (d x + c\right )}^{3} \csc \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 541, normalized size = 4.79 \[ -\frac {6 d^{3} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {3 d^{3} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{4}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{2}}-\frac {6 d \,c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{2}}+\frac {3 d^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{b^{2}}+\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}-\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a^{2}}{b^{4}}+\frac {4 i d^{3} a^{3}}{b^{4}}-\frac {2 i d^{3} x^{3}}{b}-\frac {2 i \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}+\frac {6 d^{3} \polylog \left (3, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {6 d^{3} \polylog \left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{3}}+\frac {12 d^{2} c a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {6 d^{2} c a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}+\frac {6 d^{2} c \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b^{2}}+\frac {6 i d^{3} a^{2} x}{b^{3}}-\frac {6 i d^{2} c \,x^{2}}{b}-\frac {6 i d^{2} c \,a^{2}}{b^{3}}-\frac {6 i d^{3} \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}-\frac {6 i d^{3} \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}-\frac {6 i d^{2} c \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {6 i d^{2} c \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {12 i d^{2} c a x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.72, size = 1650, normalized size = 14.60 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^3}{{\sin \left (a+b\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right )^{3} \csc ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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