Integrand size = 23, antiderivative size = 335 \[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {3 x}{8 b}+\frac {2 (-1)^{2/3} a^{5/3} \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} b^{7/3} d}-\frac {2 a^{5/3} \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-b^{2/3}} b^{7/3} d}+\frac {2 \sqrt [3]{-1} a^{5/3} \arctan \left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} b^{7/3} d}+\frac {a \cos (c+d x)}{b^2 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 b d} \]
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Time = 0.51 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3299, 2718, 2715, 8, 2739, 632, 210} \[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {2 (-1)^{2/3} a^{5/3} \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 b^{7/3} d \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac {2 a^{5/3} \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 b^{7/3} d \sqrt {a^{2/3}-b^{2/3}}}+\frac {2 \sqrt [3]{-1} a^{5/3} \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+(-1)^{2/3} \sqrt [3]{b}}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 b^{7/3} d \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}+\frac {a \cos (c+d x)}{b^2 d}-\frac {\sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \sin (c+d x) \cos (c+d x)}{8 b d}+\frac {3 x}{8 b} \]
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Rule 8
Rule 210
Rule 632
Rule 2715
Rule 2718
Rule 2739
Rule 3299
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a \sin (c+d x)}{b^2}+\frac {\sin ^4(c+d x)}{b}+\frac {a^2 \sin (c+d x)}{b^2 \left (a+b \sin ^3(c+d x)\right )}\right ) \, dx \\ & = -\frac {a \int \sin (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {\sin (c+d x)}{a+b \sin ^3(c+d x)} \, dx}{b^2}+\frac {\int \sin ^4(c+d x) \, dx}{b} \\ & = \frac {a \cos (c+d x)}{b^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 b d}+\frac {a^2 \int \left (-\frac {1}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}-\frac {(-1)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)\right )}+\frac {\sqrt [3]{-1}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)\right )}\right ) \, dx}{b^2}+\frac {3 \int \sin ^2(c+d x) \, dx}{4 b} \\ & = \frac {a \cos (c+d x)}{b^2 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 b d}-\frac {a^{5/3} \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^{7/3}}+\frac {\left (\sqrt [3]{-1} a^{5/3}\right ) \int \frac {1}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^{7/3}}-\frac {\left ((-1)^{2/3} a^{5/3}\right ) \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^{7/3}}+\frac {3 \int 1 \, dx}{8 b} \\ & = \frac {3 x}{8 b}+\frac {a \cos (c+d x)}{b^2 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 b d}-\frac {\left (2 a^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+2 \sqrt [3]{b} x+\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{7/3} d}+\frac {\left (2 \sqrt [3]{-1} a^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+2 (-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{7/3} d}-\frac {\left (2 (-1)^{2/3} a^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-2 \sqrt [3]{-1} \sqrt [3]{b} x+\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{7/3} d} \\ & = \frac {3 x}{8 b}+\frac {a \cos (c+d x)}{b^2 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 b d}+\frac {\left (4 a^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{b}+2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{7/3} d}-\frac {\left (4 \sqrt [3]{-1} a^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}+\sqrt [3]{-1} b^{2/3}\right )-x^2} \, dx,x,2 (-1)^{2/3} \sqrt [3]{b}+2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{7/3} d}+\frac {\left (4 (-1)^{2/3} a^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}-(-1)^{2/3} b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{-1} \sqrt [3]{b}+2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{7/3} d} \\ & = \frac {3 x}{8 b}+\frac {2 (-1)^{2/3} a^{5/3} \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} b^{7/3} d}-\frac {2 a^{5/3} \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-b^{2/3}} b^{7/3} d}+\frac {2 \sqrt [3]{-1} a^{5/3} \arctan \left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} b^{7/3} d}+\frac {a \cos (c+d x)}{b^2 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 b d} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.56 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.65 \[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {96 a \cos (c+d x)-32 a^2 \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {-2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2}{b-4 i a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]+3 b (12 (c+d x)-8 \sin (2 (c+d x))+\sin (4 (c+d x)))}{96 b^2 d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.92 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {-\frac {4 \left (\frac {-\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{16}-\frac {a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {11 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{16}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{16}-\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}+\frac {3 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16}-\frac {a}{2}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {3 b \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}\right )}{b^{2}}+\frac {2 a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R} \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 b^{2}}}{d}\) | \(221\) |
default | \(\frac {-\frac {4 \left (\frac {-\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{16}-\frac {a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {11 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{16}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{16}-\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}+\frac {3 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16}-\frac {a}{2}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {3 b \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}\right )}{b^{2}}+\frac {2 a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R} \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 b^{2}}}{d}\) | \(221\) |
risch | \(\frac {3 x}{8 b}+\frac {a \,{\mathrm e}^{i \left (d x +c \right )}}{2 d \,b^{2}}+\frac {a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{2} d}+\frac {i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (729 a^{2} b^{14} d^{6}-729 b^{16} d^{6}\right ) \textit {\_Z}^{6}-3981312 a^{4} b^{10} d^{4} \textit {\_Z}^{4}-4398046511104 a^{10}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\left (\frac {243 i d^{5} b^{12} a^{2}}{34359738368 a^{9}+34359738368 a^{7} b^{2}}-\frac {243 i d^{5} b^{14}}{34359738368 a^{9}+34359738368 a^{7} b^{2}}\right ) \textit {\_R}^{5}+\left (\frac {10368 i d^{4} b^{9} a^{4}}{34359738368 a^{9}+34359738368 a^{7} b^{2}}-\frac {10368 i d^{4} b^{11} a^{2}}{34359738368 a^{9}+34359738368 a^{7} b^{2}}\right ) \textit {\_R}^{4}+\left (-\frac {884736 i d^{3} b^{8} a^{4}}{34359738368 a^{9}+34359738368 a^{7} b^{2}}-\frac {442368 i d^{3} b^{10} a^{2}}{34359738368 a^{9}+34359738368 a^{7} b^{2}}\right ) \textit {\_R}^{3}+\left (-\frac {37748736 i d^{2} b^{5} a^{6}}{34359738368 a^{9}+34359738368 a^{7} b^{2}}-\frac {18874368 i d^{2} b^{7} a^{4}}{34359738368 a^{9}+34359738368 a^{7} b^{2}}\right ) \textit {\_R}^{2}+\left (-\frac {805306368 i d \,b^{2} a^{8}}{34359738368 a^{9}+34359738368 a^{7} b^{2}}-\frac {1610612736 i d \,b^{4} a^{6}}{34359738368 a^{9}+34359738368 a^{7} b^{2}}\right ) \textit {\_R} -\frac {34359738368 i b \,a^{8}}{34359738368 a^{9}+34359738368 a^{7} b^{2}}\right )\right )}{128}+\frac {\sin \left (4 d x +4 c \right )}{32 d b}-\frac {\sin \left (2 d x +2 c \right )}{4 d b}\) | \(461\) |
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Result contains complex when optimal does not.
Time = 1.43 (sec) , antiderivative size = 21338, normalized size of antiderivative = 63.70 \[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int { \frac {\sin \left (d x + c\right )^{7}}{b \sin \left (d x + c\right )^{3} + a} \,d x } \]
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\[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int { \frac {\sin \left (d x + c\right )^{7}}{b \sin \left (d x + c\right )^{3} + a} \,d x } \]
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Time = 15.45 (sec) , antiderivative size = 1978, normalized size of antiderivative = 5.90 \[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]
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