Integrand size = 21, antiderivative size = 168 \[ \int \frac {\sin ^3(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {a^3 b \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2} d}+\frac {a b^2 \cos (c+d x)}{\left (a^2+b^2\right )^2 d}-\frac {a \cos (c+d x)}{\left (a^2+b^2\right ) d}+\frac {a \cos ^3(c+d x)}{3 \left (a^2+b^2\right ) d}+\frac {a^2 b \sin (c+d x)}{\left (a^2+b^2\right )^2 d}+\frac {b \sin ^3(c+d x)}{3 \left (a^2+b^2\right ) d} \]
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Time = 0.40 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3599, 3188, 2644, 30, 2713, 3178, 3153, 212, 2718} \[ \int \frac {\sin ^3(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {b \sin ^3(c+d x)}{3 d \left (a^2+b^2\right )}+\frac {a^2 b \sin (c+d x)}{d \left (a^2+b^2\right )^2}+\frac {a \cos ^3(c+d x)}{3 d \left (a^2+b^2\right )}-\frac {a \cos (c+d x)}{d \left (a^2+b^2\right )}+\frac {a b^2 \cos (c+d x)}{d \left (a^2+b^2\right )^2}+\frac {a^3 b \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{d \left (a^2+b^2\right )^{5/2}} \]
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Rule 30
Rule 212
Rule 2644
Rule 2713
Rule 2718
Rule 3153
Rule 3178
Rule 3188
Rule 3599
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx \\ & = \frac {a \int \sin ^3(c+d x) \, dx}{a^2+b^2}+\frac {b \int \cos (c+d x) \sin ^2(c+d x) \, dx}{a^2+b^2}-\frac {(a b) \int \frac {\sin ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{a^2+b^2} \\ & = \frac {a^2 b \sin (c+d x)}{\left (a^2+b^2\right )^2 d}-\frac {\left (a^3 b\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (a b^2\right ) \int \sin (c+d x) \, dx}{\left (a^2+b^2\right )^2}-\frac {a \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{\left (a^2+b^2\right ) d}+\frac {b \text {Subst}\left (\int x^2 \, dx,x,\sin (c+d x)\right )}{\left (a^2+b^2\right ) d} \\ & = \frac {a b^2 \cos (c+d x)}{\left (a^2+b^2\right )^2 d}-\frac {a \cos (c+d x)}{\left (a^2+b^2\right ) d}+\frac {a \cos ^3(c+d x)}{3 \left (a^2+b^2\right ) d}+\frac {a^2 b \sin (c+d x)}{\left (a^2+b^2\right )^2 d}+\frac {b \sin ^3(c+d x)}{3 \left (a^2+b^2\right ) d}+\frac {\left (a^3 b\right ) \text {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{\left (a^2+b^2\right )^2 d} \\ & = \frac {a^3 b \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2} d}+\frac {a b^2 \cos (c+d x)}{\left (a^2+b^2\right )^2 d}-\frac {a \cos (c+d x)}{\left (a^2+b^2\right ) d}+\frac {a \cos ^3(c+d x)}{3 \left (a^2+b^2\right ) d}+\frac {a^2 b \sin (c+d x)}{\left (a^2+b^2\right )^2 d}+\frac {b \sin ^3(c+d x)}{3 \left (a^2+b^2\right ) d} \\ \end{align*}
Time = 1.80 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.83 \[ \int \frac {\sin ^3(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {-24 a^3 b \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )+\sqrt {a^2+b^2} \left (\left (-9 a^3+3 a b^2\right ) \cos (c+d x)+a \left (a^2+b^2\right ) \cos (3 (c+d x))-2 b \left (-7 a^2-b^2+\left (a^2+b^2\right ) \cos (2 (c+d x))\right ) \sin (c+d x)\right )}{12 \left (a^2+b^2\right )^{5/2} d} \]
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Time = 2.10 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(\frac {-\frac {16 a^{3} b \,\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (8 a^{4}+16 a^{2} b^{2}+8 b^{4}\right ) \sqrt {a^{2}+b^{2}}}+\frac {2 a^{2} b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \,b^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\frac {10}{3} a^{2} b +\frac {4}{3} b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {4 a^{3}}{3}+\frac {2 a \,b^{2}}{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}}{d}\) | \(202\) |
default | \(\frac {-\frac {16 a^{3} b \,\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (8 a^{4}+16 a^{2} b^{2}+8 b^{4}\right ) \sqrt {a^{2}+b^{2}}}+\frac {2 a^{2} b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \,b^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\frac {10}{3} a^{2} b +\frac {4}{3} b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {4 a^{3}}{3}+\frac {2 a \,b^{2}}{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}}{d}\) | \(202\) |
risch | \(\frac {i {\mathrm e}^{i \left (d x +c \right )} b}{8 \left (-2 i a b +a^{2}-b^{2}\right ) d}-\frac {3 \,{\mathrm e}^{i \left (d x +c \right )} a}{8 \left (-2 i a b +a^{2}-b^{2}\right ) d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} b}{8 \left (i b +a \right )^{2} d}-\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )} a}{8 \left (i b +a \right )^{2} d}-\frac {i b \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right )}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )^{2} d}+\frac {i b \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right )}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )^{2} d}-\frac {a \cos \left (3 d x +3 c \right )}{12 d \left (-a^{2}-b^{2}\right )}+\frac {b \sin \left (3 d x +3 c \right )}{12 d \left (-a^{2}-b^{2}\right )}\) | \(295\) |
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Time = 0.29 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.55 \[ \int \frac {\sin ^3(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {3 \, \sqrt {a^{2} + b^{2}} a^{3} b \log \left (\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{3} - 6 \, {\left (a^{5} + a^{3} b^{2}\right )} \cos \left (d x + c\right ) + 2 \, {\left (4 \, a^{4} b + 5 \, a^{2} b^{3} + b^{5} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d} \]
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\[ \int \frac {\sin ^3(c+d x)}{a+b \tan (c+d x)} \, dx=\int \frac {\sin ^{3}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (160) = 320\).
Time = 0.30 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.17 \[ \int \frac {\sin ^3(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\frac {3 \, a^{3} b \log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (2 \, a^{3} - a b^{2} - \frac {3 \, a^{2} b \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3 \, a b^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3 \, a^{2} b \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {2 \, {\left (5 \, a^{2} b + 2 \, b^{3}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4} + \frac {3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}}{3 \, d} \]
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Time = 0.47 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.43 \[ \int \frac {\sin ^3(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\frac {3 \, a^{3} b \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (3 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 10 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a^{3} + a b^{2}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \]
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Time = 7.21 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.93 \[ \int \frac {\sin ^3(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\frac {\frac {2\,a\,b^2}{3}-\frac {4\,a^3}{3}}{a^4+2\,a^2\,b^2+b^4}-\frac {4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{a^4+2\,a^2\,b^2+b^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {20\,a^2\,b}{3}+\frac {8\,b^3}{3}\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {2\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {2\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{a^4+2\,a^2\,b^2+b^4}+\frac {2\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{a^4+2\,a^2\,b^2+b^4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {2\,a^3\,b\,\mathrm {atanh}\left (\frac {a^4\,b+b^5+2\,a^2\,b^3-a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}{{\left (a^2+b^2\right )}^{5/2}}\right )}{d\,{\left (a^2+b^2\right )}^{5/2}} \]
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