Integrand size = 23, antiderivative size = 219 \[ \int \frac {\cos ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\left (a^{4/3}-b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{5/3} d}+\frac {\left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} b^{5/3} d}-\frac {\left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} b^{5/3} d}-\frac {2 \log \left (a+b \sin ^3(c+d x)\right )}{3 b d}+\frac {\sin ^2(c+d x)}{2 b d} \]
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Time = 0.33 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3302, 1901, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {\cos ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\left (a^{4/3}-b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{5/3} d}-\frac {\left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} b^{5/3} d}+\frac {\left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} b^{5/3} d}-\frac {2 \log \left (a+b \sin ^3(c+d x)\right )}{3 b d}+\frac {\sin ^2(c+d x)}{2 b d} \]
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Rule 31
Rule 210
Rule 266
Rule 631
Rule 642
Rule 648
Rule 1874
Rule 1885
Rule 1901
Rule 3302
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {x}{b}+\frac {b-a x-2 b x^2}{b \left (a+b x^3\right )}\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\sin ^2(c+d x)}{2 b d}+\frac {\text {Subst}\left (\int \frac {b-a x-2 b x^2}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{b d} \\ & = \frac {\sin ^2(c+d x)}{2 b d}-\frac {2 \text {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{d}+\frac {\text {Subst}\left (\int \frac {b-a x}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{b d} \\ & = -\frac {2 \log \left (a+b \sin ^3(c+d x)\right )}{3 b d}+\frac {\sin ^2(c+d x)}{2 b d}+\frac {\left (\frac {1}{a^{2/3}}+\frac {a^{2/3}}{b^{4/3}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sin (c+d x)\right )}{3 d}+\frac {\text {Subst}\left (\int \frac {\sqrt [3]{a} \left (-a^{4/3}+2 b^{4/3}\right )+\sqrt [3]{b} \left (-a^{4/3}-b^{4/3}\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{3 a^{2/3} b^{4/3} d} \\ & = \frac {\left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} b^{5/3} d}-\frac {2 \log \left (a+b \sin ^3(c+d x)\right )}{3 b d}+\frac {\sin ^2(c+d x)}{2 b d}-\frac {\left (a^{4/3}-b^{4/3}\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt [3]{a} b^{4/3} d}-\frac {\left (a^{4/3}+b^{4/3}\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{6 a^{2/3} b^{5/3} d} \\ & = \frac {\left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} b^{5/3} d}-\frac {\left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} b^{5/3} d}-\frac {2 \log \left (a+b \sin ^3(c+d x)\right )}{3 b d}+\frac {\sin ^2(c+d x)}{2 b d}-\frac {\left (a^{4/3}-b^{4/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )}{a^{2/3} b^{5/3} d} \\ & = \frac {\left (a^{4/3}-b^{4/3}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3} b^{5/3} d}+\frac {\left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} b^{5/3} d}-\frac {\left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} b^{5/3} d}-\frac {2 \log \left (a+b \sin ^3(c+d x)\right )}{3 b d}+\frac {\sin ^2(c+d x)}{2 b d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.26 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {-2 \sqrt {3} b^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )+2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )-b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )-4 a^{2/3} \log \left (a+b \sin ^3(c+d x)\right )+3 a^{2/3} \sin ^2(c+d x)-3 a^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{6 a^{2/3} b d} \]
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Time = 1.07 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.21
method | result | size |
derivativedivides | \(\frac {\frac {\sin ^{2}\left (d x +c \right )}{2 b}+\frac {b \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )-a \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-\frac {2 \ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{3}}{b}}{d}\) | \(264\) |
default | \(\frac {\frac {\sin ^{2}\left (d x +c \right )}{2 b}+\frac {b \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )-a \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-\frac {2 \ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{3}}{b}}{d}\) | \(264\) |
risch | \(\frac {2 i x}{b}-\frac {{\mathrm e}^{2 i \left (d x +c \right )}}{8 b d}-\frac {{\mathrm e}^{-2 i \left (d x +c \right )}}{8 b d}+\frac {4 i c}{b d}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (27 a^{2} b^{5} d^{3} \textit {\_Z}^{3}+54 a^{2} b^{4} d^{2} \textit {\_Z}^{2}+27 a^{2} b^{3} d \textit {\_Z} -a^{4}+2 a^{2} b^{2}-b^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {18 i a^{3} b^{3} d^{2} \textit {\_R}^{2}}{a^{4}-b^{4}}+\left (\frac {24 i a^{3} b^{2} d}{a^{4}-b^{4}}-\frac {6 i a \,b^{4} d}{a^{4}-b^{4}}\right ) \textit {\_R} +\frac {4 i a^{3} b}{a^{4}-b^{4}}-\frac {4 i a \,b^{3}}{a^{4}-b^{4}}\right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {a^{4}}{a^{4}-b^{4}}+\frac {b^{4}}{a^{4}-b^{4}}\right )\right )\) | \(274\) |
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Result contains complex when optimal does not.
Time = 1.02 (sec) , antiderivative size = 3216, normalized size of antiderivative = 14.68 \[ \int \frac {\cos ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cos ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\frac {9 \, \sin \left (d x + c\right )^{2}}{b} - \frac {2 \, \sqrt {3} {\left (a {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} - 4\right )} - b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} - \frac {4 \, a}{b}\right )}\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b} - \frac {3 \, {\left (b {\left (4 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 1\right )} + a \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \log \left (\sin \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {6 \, {\left (b {\left (2 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} - 1\right )} - a \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{18 \, d} \]
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Time = 0.39 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.01 \[ \int \frac {\cos ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\frac {3 \, \sin \left (d x + c\right )^{2}}{b} - \frac {4 \, \log \left ({\left | b \sin \left (d x + c\right )^{3} + a \right |}\right )}{b} + \frac {2 \, \sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} + \left (-a b^{2}\right )^{\frac {2}{3}} a\right )} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b^{3}} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} - \left (-a b^{2}\right )^{\frac {2}{3}} a\right )} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a b^{3}} + \frac {2 \, {\left (a b^{4} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - b^{5}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{a b^{5}}}{6 \, d} \]
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Time = 14.54 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.05 \[ \int \frac {\cos ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\left (\sum _{k=1}^3\ln \left (3\,a+\mathrm {root}\left (27\,a^2\,b^5\,d^3+54\,a^2\,b^4\,d^2+27\,a^2\,b^3\,d+2\,a^2\,b^2-b^4-a^4,d,k\right )\,\left (12\,a\,b+3\,b^2\,\sin \left (c+d\,x\right )+\mathrm {root}\left (27\,a^2\,b^5\,d^3+54\,a^2\,b^4\,d^2+27\,a^2\,b^3\,d+2\,a^2\,b^2-b^4-a^4,d,k\right )\,a\,b^2\,9\right )+\frac {\sin \left (c+d\,x\right )\,\left (a^2+2\,b^2\right )}{b}\right )\,\mathrm {root}\left (27\,a^2\,b^5\,d^3+54\,a^2\,b^4\,d^2+27\,a^2\,b^3\,d+2\,a^2\,b^2-b^4-a^4,d,k\right )\right )+\frac {{\sin \left (c+d\,x\right )}^2}{2\,b}}{d} \]
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