Integrand size = 19, antiderivative size = 116 \[ \int (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=-\frac {a \left (a^2-3 b^2\right ) \cos (c+d x)}{d}+\frac {3 a^2 b \cos ^2(c+d x)}{2 d}+\frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \]
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Time = 0.12 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4482, 2800, 908} \[ \int (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {a \left (a^2-3 b^2\right ) \cos (c+d x)}{d}-\frac {b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac {3 a^2 b \cos ^2(c+d x)}{2 d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \]
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Rule 908
Rule 2800
Rule 4482
Rubi steps \begin{align*} \text {integral}& = \int (b+a \cos (c+d x))^3 \tan ^3(c+d x) \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {(b+x)^3 \left (a^2-x^2\right )}{x^3} \, dx,x,a \cos (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \left (a^2 \left (1-\frac {3 b^2}{a^2}\right )+\frac {a^2 b^3}{x^3}+\frac {3 a^2 b^2}{x^2}+\frac {3 a^2 b-b^3}{x}-3 b x-x^2\right ) \, dx,x,a \cos (c+d x)\right )}{d} \\ & = -\frac {a \left (a^2-3 b^2\right ) \cos (c+d x)}{d}+\frac {3 a^2 b \cos ^2(c+d x)}{2 d}+\frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.88 \[ \int (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\frac {-9 a \left (a^2-4 b^2\right ) \cos (c+d x)+9 a^2 b \cos (2 (c+d x))+a^3 \cos (3 (c+d x))-36 a^2 b \log (\cos (c+d x))+12 b^3 \log (\cos (c+d x))+36 a b^2 \sec (c+d x)+6 b^3 \sec ^2(c+d x)}{12 d} \]
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Time = 5.01 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {-\frac {a^{3} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}+3 a^{2} b \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+b^{3} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(116\) |
default | \(\frac {-\frac {a^{3} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}+3 a^{2} b \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+b^{3} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(116\) |
parts | \(-\frac {a^{3} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3 d}+\frac {b^{3} \left (\frac {\tan \left (d x +c \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}\right )}{d}+\frac {3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )}{d}+\frac {3 a^{2} b \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(130\) |
risch | \(-i x \,b^{3}-\frac {2 i b^{3} c}{d}+\frac {a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}+\frac {3 b \,{\mathrm e}^{2 i \left (d x +c \right )} a^{2}}{8 d}-\frac {3 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {3 a \,{\mathrm e}^{i \left (d x +c \right )} b^{2}}{2 d}-\frac {3 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {3 a \,{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{2 d}+\frac {3 b \,{\mathrm e}^{-2 i \left (d x +c \right )} a^{2}}{8 d}+\frac {a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}+3 i a^{2} b x +\frac {6 i b \,a^{2} c}{d}+\frac {2 b^{2} \left (3 a \,{\mathrm e}^{3 i \left (d x +c \right )}+b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 a \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{d}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(275\) |
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Time = 0.28 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.06 \[ \int (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\frac {4 \, a^{3} \cos \left (d x + c\right )^{5} + 18 \, a^{2} b \cos \left (d x + c\right )^{4} - 9 \, a^{2} b \cos \left (d x + c\right )^{2} + 36 \, a b^{2} \cos \left (d x + c\right ) - 12 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 12 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) + 6 \, b^{3}}{12 \, d \cos \left (d x + c\right )^{2}} \]
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\[ \int (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right )^{3}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.97 \[ \int (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\frac {{\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} a^{3}}{3 \, d} - \frac {3 \, {\left (\sin \left (d x + c\right )^{2} + \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )} a^{2} b}{2 \, d} - \frac {b^{3} {\left (\frac {1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{2 \, d} + \frac {3 \, a b^{2} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )}}{d} \]
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Exception generated. \[ \int (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\text {Exception raised: TypeError} \]
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Time = 26.13 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.89 \[ \int (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-\frac {4\,a^3}{3}-6\,a^2\,b+12\,a\,b^2+2\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (4\,a^3-6\,a^2\,b+12\,a\,b^2-6\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {20\,a^3}{3}+6\,a^2\,b-12\,a\,b^2+6\,b^3\right )+12\,a\,b^2-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (6\,a^2\,b-2\,b^3\right )-\frac {4\,a^3}{3}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}^2\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3}-\frac {2\,b^3\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )-6\,a^2\,b\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d} \]
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