Integrand size = 23, antiderivative size = 238 \[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {2 \left (a^{4/3}+b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{5/3} d}-\frac {2 \left (a^{4/3}-b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{5/3} d}+\frac {\left (a^{4/3}-b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{5/3} d}+\frac {\sin (c+d x) \left (b-a \sin (c+d x)-2 b \sin ^2(c+d x)\right )}{3 a b d \left (a+b \sin ^3(c+d x)\right )} \]
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Time = 0.26 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3302, 1872, 1874, 31, 648, 631, 210, 642} \[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {2 \left (a^{4/3}+b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{5/3} d}+\frac {\left (a^{4/3}-b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{5/3} d}-\frac {2 \left (a^{4/3}-b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{5/3} d}+\frac {\sin (c+d x) \left (-a \sin (c+d x)-2 b \sin ^2(c+d x)+b\right )}{3 a b d \left (a+b \sin ^3(c+d x)\right )} \]
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Rule 31
Rule 210
Rule 631
Rule 642
Rule 648
Rule 1872
Rule 1874
Rule 3302
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a+b x^3\right )^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\sin (c+d x) \left (b-a \sin (c+d x)-2 b \sin ^2(c+d x)\right )}{3 a b d \left (a+b \sin ^3(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-2 b^2-2 a b x}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{3 a b^2 d} \\ & = \frac {\sin (c+d x) \left (b-a \sin (c+d x)-2 b \sin ^2(c+d x)\right )}{3 a b d \left (a+b \sin ^3(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {\sqrt [3]{a} \left (-2 a^{4/3} b-4 b^{7/3}\right )+\sqrt [3]{b} \left (-2 a^{4/3} b+2 b^{7/3}\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\left (2 \left (a^{4/3}-b^{4/3}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} b^{4/3} d} \\ & = -\frac {2 \left (a^{4/3}-b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{5/3} d}+\frac {\sin (c+d x) \left (b-a \sin (c+d x)-2 b \sin ^2(c+d x)\right )}{3 a b d \left (a+b \sin ^3(c+d x)\right )}+\frac {\left (\frac {1}{a^{4/3}}+\frac {1}{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 )}{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 )}{9 a^{5/3} b^{5/3} d} \\ & = -\frac {2 \left (a^{4/3}-b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} 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 )}{9 a^{5/3} b^{5/3} d}+\frac {\sin (c+d x) \left (b-a \sin (c+d x)-2 b \sin ^2(c+d x)\right )}{3 a b d \left (a+b \sin ^3(c+d x)\right )}+\frac {\left (2 \left (a^{4/3}+b^{4/3}\right )\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 )}{3 a^{5/3} b^{5/3} d} \\ & = -\frac {2 \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 )}{3 \sqrt {3} a^{5/3} b^{5/3} d}-\frac {2 \left (a^{4/3}-b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} 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 )}{9 a^{5/3} b^{5/3} d}+\frac {\sin (c+d x) \left (b-a \sin (c+d x)-2 b \sin ^2(c+d x)\right )}{3 a b d \left (a+b \sin ^3(c+d x)\right )} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.96 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.08 \[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {-\frac {4 \sqrt {3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3} \sqrt [3]{b}}+\frac {4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{a^{5/3} \sqrt [3]{b}}-\frac {2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{a^{5/3} \sqrt [3]{b}}+\frac {9 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{a b}-\frac {9 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},2,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{a b}+\frac {12}{b \left (a+b \sin ^3(c+d x)\right )}+\frac {6 \sin (c+d x)}{a \left (a+b \sin ^3(c+d x)\right )}}{18 d} \]
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Time = 1.06 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\frac {\frac {-\frac {\sin ^{2}\left (d x +c \right )}{3 b}+\frac {\sin \left (d x +c \right )}{3 a}+\frac {2}{3 b}}{a +b \left (\sin ^{3}\left (d x +c \right )\right )}+\frac {\frac {2 b \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 a \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}}{a b}}{d}\) | \(284\) |
default | \(\frac {\frac {-\frac {\sin ^{2}\left (d x +c \right )}{3 b}+\frac {\sin \left (d x +c \right )}{3 a}+\frac {2}{3 b}}{a +b \left (\sin ^{3}\left (d x +c \right )\right )}+\frac {\frac {2 b \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 a \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}}{a b}}{d}\) | \(284\) |
risch | \(-\frac {2 i \left (a \,{\mathrm e}^{5 i \left (d x +c \right )}+6 a \,{\mathrm e}^{3 i \left (d x +c \right )}-2 i b \,{\mathrm e}^{4 i \left (d x +c \right )}+a \,{\mathrm e}^{i \left (d x +c \right )}+2 i b \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 a b d \left (b \,{\mathrm e}^{6 i \left (d x +c \right )}-3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i a \,{\mathrm e}^{3 i \left (d x +c \right )}-b \right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (729 a^{5} b^{5} d^{3} \textit {\_Z}^{3}+108 a^{3} b^{3} d \textit {\_Z} +8 a^{4}-8 b^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {81 i a^{5} b^{3} d^{2} \textit {\_R}^{2}}{2 a^{4}+2 b^{4}}+\frac {18 i a^{2} b^{4} d \textit {\_R}}{2 a^{4}+2 b^{4}}+\frac {8 i a^{3} b}{2 a^{4}+2 b^{4}}\right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {2 a^{4}}{2 a^{4}+2 b^{4}}-\frac {2 b^{4}}{2 a^{4}+2 b^{4}}\right )\right )\) | \(300\) |
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Result contains complex when optimal does not.
Time = 1.21 (sec) , antiderivative size = 2255, normalized size of antiderivative = 9.47 \[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.89 \[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {\frac {3 \, {\left (a \sin \left (d x + c\right )^{2} - b \sin \left (d x + c\right ) - 2 \, a\right )}}{a b^{2} \sin \left (d x + c\right )^{3} + a^{2} b} - \frac {2 \, \sqrt {3} {\left (a \left (\frac {a}{b}\right )^{\frac {1}{3}} + b\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (a \left (\frac {a}{b}\right )^{\frac {1}{3}} - b\right )} \log \left (\sin \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, {\left (a \left (\frac {a}{b}\right )^{\frac {1}{3}} - b\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{9 \, d} \]
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\[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int { \frac {\cos \left (d x + c\right )^{5}}{{\left (b \sin \left (d x + c\right )^{3} + a\right )}^{2}} \,d x } \]
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Time = 13.70 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^5(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\sum _{k=1}^3\ln \left (\frac {4\,b+4\,a\,\sin \left (c+d\,x\right )+{\mathrm {root}\left (729\,a^5\,b^5\,d^3+108\,a^3\,b^3\,d-8\,b^4+8\,a^4,d,k\right )}^2\,a^2\,b^3\,81+\mathrm {root}\left (729\,a^5\,b^5\,d^3+108\,a^3\,b^3\,d-8\,b^4+8\,a^4,d,k\right )\,b^3\,\sin \left (c+d\,x\right )\,18}{a\,b\,9}\right )\,\mathrm {root}\left (729\,a^5\,b^5\,d^3+108\,a^3\,b^3\,d-8\,b^4+8\,a^4,d,k\right )}{d}+\frac {\frac {\sin \left (c+d\,x\right )}{3\,a}+\frac {2}{3\,b}-\frac {{\sin \left (c+d\,x\right )}^2}{3\,b}}{d\,\left (b\,{\sin \left (c+d\,x\right )}^3+a\right )} \]
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