Optimal. Leaf size=148 \[ -\frac {12 i d^2 (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right )}{a f^3}+\frac {6 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a f^2}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f}-\frac {i (c+d x)^3}{a f}+\frac {12 d^3 \text {Li}_3\left (i e^{i (e+f x)}\right )}{a f^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.31, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3318, 4184, 3717, 2190, 2531, 2282, 6589} \[ -\frac {12 i d^2 (c+d x) \text {PolyLog}\left (2,i e^{i (e+f x)}\right )}{a f^3}+\frac {12 d^3 \text {PolyLog}\left (3,i e^{i (e+f x)}\right )}{a f^4}+\frac {6 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a f^2}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f}-\frac {i (c+d x)^3}{a f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2282
Rule 2531
Rule 3318
Rule 3717
Rule 4184
Rule 6589
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{a+a \sin (e+f x)} \, dx &=\frac {\int (c+d x)^3 \csc ^2\left (\frac {1}{2} \left (e+\frac {\pi }{2}\right )+\frac {f x}{2}\right ) \, dx}{2 a}\\ &=-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {(3 d) \int (c+d x)^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{a f}\\ &=-\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {(6 d) \int \frac {e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)^2}{1-i e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a f}\\ &=-\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {6 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a f^2}-\frac {\left (12 d^2\right ) \int (c+d x) \log \left (1-i e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a f^2}\\ &=-\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {6 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right )}{a f^3}+\frac {\left (12 i d^3\right ) \int \text {Li}_2\left (i e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a f^3}\\ &=-\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {6 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right )}{a f^3}+\frac {\left (12 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a f^4}\\ &=-\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {6 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right )}{a f^3}+\frac {12 d^3 \text {Li}_3\left (i e^{i (e+f x)}\right )}{a f^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.19, size = 126, normalized size = 0.85 \[ \frac {-12 i d^2 f (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right )+f^2 (c+d x)^2 \left (f (c+d x) \tan \left (\frac {1}{4} (2 e+2 f x-\pi )\right )-i f (c+d x)+6 d \log \left (1-i e^{i (e+f x)}\right )\right )+12 d^3 \text {Li}_3\left (i e^{i (e+f x)}\right )}{a f^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.89, size = 915, normalized size = 6.18 \[ -\frac {d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + c^{3} f^{3} + {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + c^{3} f^{3}\right )} \cos \left (f x + e\right ) - {\left (-6 i \, d^{3} f x - 6 i \, c d^{2} f + {\left (-6 i \, d^{3} f x - 6 i \, c d^{2} f\right )} \cos \left (f x + e\right ) + {\left (-6 i \, d^{3} f x - 6 i \, c d^{2} f\right )} \sin \left (f x + e\right )\right )} {\rm Li}_2\left (i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) - {\left (6 i \, d^{3} f x + 6 i \, c d^{2} f + {\left (6 i \, d^{3} f x + 6 i \, c d^{2} f\right )} \cos \left (f x + e\right ) + {\left (6 i \, d^{3} f x + 6 i \, c d^{2} f\right )} \sin \left (f x + e\right )\right )} {\rm Li}_2\left (-i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) - 3 \, {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2} + {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \cos \left (f x + e\right ) + {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \sin \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + i\right ) - 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} \cos \left (f x + e\right ) + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} \sin \left (f x + e\right )\right )} \log \left (i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} \cos \left (f x + e\right ) + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} \sin \left (f x + e\right )\right )} \log \left (-i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2} + {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \cos \left (f x + e\right ) + {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \sin \left (f x + e\right )\right )} \log \left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + i\right ) - 6 \, {\left (d^{3} \cos \left (f x + e\right ) + d^{3} \sin \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) - 6 \, {\left (d^{3} \cos \left (f x + e\right ) + d^{3} \sin \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, -i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) - {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + c^{3} f^{3}\right )} \sin \left (f x + e\right )}{a f^{4} \cos \left (f x + e\right ) + a f^{4} \sin \left (f x + e\right ) + a f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{3}}{a \sin \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.20, size = 484, normalized size = 3.27 \[ -\frac {2 \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}+\frac {6 d^{3} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{a \,f^{4}}+\frac {12 d^{3} \polylog \left (3, i {\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{4}}+\frac {12 c \,d^{2} e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{3}}-\frac {6 \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right ) c^{2} d}{a \,f^{2}}-\frac {12 c \,d^{2} e \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{a \,f^{3}}+\frac {6 d^{3} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) x^{2}}{a \,f^{2}}-\frac {6 d^{3} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) e^{2}}{a \,f^{4}}+\frac {6 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c^{2} d}{a \,f^{2}}+\frac {4 i d^{3} e^{3}}{a \,f^{4}}-\frac {12 i c \,d^{2} \polylog \left (2, i {\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{3}}-\frac {6 d^{3} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{4}}-\frac {6 i c \,d^{2} e^{2}}{a \,f^{3}}-\frac {12 i d^{3} \polylog \left (2, i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{a \,f^{3}}+\frac {6 i d^{3} e^{2} x}{a \,f^{3}}-\frac {6 i c \,d^{2} x^{2}}{a f}-\frac {2 i d^{3} x^{3}}{a f}-\frac {12 i c \,d^{2} e x}{a \,f^{2}}+\frac {12 c \,d^{2} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{a \,f^{2}}+\frac {12 c \,d^{2} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) e}{a \,f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.70, size = 974, normalized size = 6.58 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^3}{a+a\,\sin \left (e+f\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {c^{3}}{\sin {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{3} x^{3}}{\sin {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c d^{2} x^{2}}{\sin {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c^{2} d x}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________