Integrand size = 24, antiderivative size = 95 \[ \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\left (\sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4} d}-\frac {\left (\sqrt {a}-\sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4} d} \]
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Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3302, 1181, 211, 214} \[ \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\left (\sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4} d}-\frac {\left (\sqrt {a}-\sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4} d} \]
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Rule 211
Rule 214
Rule 1181
Rule 3302
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1-x^2}{a-b x^4} \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx,x,\sin (c+d x)\right )}{2 d}-\frac {\left (1+\frac {\sqrt {b}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx,x,\sin (c+d x)\right )}{2 d} \\ & = \frac {\left (\sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4} d}-\frac {\left (\sqrt {a}-\sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4} d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.68 \[ \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\left (\sqrt {a}-\sqrt {b}\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} \sin (c+d x)\right )+i \left (\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt [4]{a}-i \sqrt [4]{b} \sin (c+d x)\right )-i \left (\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt [4]{a}+i \sqrt [4]{b} \sin (c+d x)\right )-\left (\sqrt {a}-\sqrt {b}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} \sin (c+d x)\right )}{4 a^{3/4} b^{3/4} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(135\) vs. \(2(67)=134\).
Time = 0.72 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.43
method | result | size |
derivativedivides | \(\frac {\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{d}\) | \(136\) |
default | \(\frac {\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{d}\) | \(136\) |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (256 a^{3} b^{3} d^{4} \textit {\_Z}^{4}+64 a^{2} b^{2} d^{2} \textit {\_Z}^{2}-a^{2}+2 a b -b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (-\frac {128 i a^{3} b^{2} d^{3} \textit {\_R}^{3}}{a^{2}-b^{2}}+\left (-\frac {24 i d b \,a^{2}}{a^{2}-b^{2}}-\frac {8 i a \,b^{2} d}{a^{2}-b^{2}}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {a^{2}}{a^{2}-b^{2}}+\frac {b^{2}}{a^{2}-b^{2}}\right )\) | \(170\) |
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Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (67) = 134\).
Time = 0.37 (sec) , antiderivative size = 631, normalized size of antiderivative = 6.64 \[ \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {1}{4} \, \sqrt {-\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + 2}{a b d^{2}}} \log \left (\frac {1}{2} \, {\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right ) + \frac {1}{2} \, {\left (a^{3} b^{2} d^{3} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - {\left (a^{2} b + a b^{2}\right )} d\right )} \sqrt {-\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + 2}{a b d^{2}}}\right ) - \frac {1}{4} \, \sqrt {\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - 2}{a b d^{2}}} \log \left (\frac {1}{2} \, {\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right ) + \frac {1}{2} \, {\left (a^{3} b^{2} d^{3} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + {\left (a^{2} b + a b^{2}\right )} d\right )} \sqrt {\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - 2}{a b d^{2}}}\right ) - \frac {1}{4} \, \sqrt {-\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + 2}{a b d^{2}}} \log \left (-\frac {1}{2} \, {\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right ) + \frac {1}{2} \, {\left (a^{3} b^{2} d^{3} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - {\left (a^{2} b + a b^{2}\right )} d\right )} \sqrt {-\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + 2}{a b d^{2}}}\right ) + \frac {1}{4} \, \sqrt {\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - 2}{a b d^{2}}} \log \left (-\frac {1}{2} \, {\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right ) + \frac {1}{2} \, {\left (a^{3} b^{2} d^{3} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + {\left (a^{2} b + a b^{2}\right )} d\right )} \sqrt {\frac {a b d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - 2}{a b d^{2}}}\right ) \]
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Timed out. \[ \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Timed out} \]
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none
Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.27 \[ \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {2 \, {\left (\sqrt {a} + \sqrt {b}\right )} \arctan \left (\frac {\sqrt {b} \sin \left (d x + c\right )}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (\sqrt {a} - \sqrt {b}\right )} \log \left (\frac {\sqrt {b} \sin \left (d x + c\right ) - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} \sin \left (d x + c\right ) + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (67) = 134\).
Time = 0.86 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.95 \[ \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {2 \, \sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} - \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a b^{3}} + \frac {2 \, \sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} - \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a b^{3}} + \frac {\sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} + \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \log \left (\sin \left (d x + c\right )^{2} + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{a b^{3}} - \frac {\sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} + \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \log \left (\sin \left (d x + c\right )^{2} - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{a b^{3}}}{8 \, d} \]
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Time = 14.79 (sec) , antiderivative size = 489, normalized size of antiderivative = 5.15 \[ \int \frac {\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {2\,\mathrm {atanh}\left (\frac {8\,b^3\,\sin \left (c+d\,x\right )\,\sqrt {-\frac {1}{8\,a\,b}-\frac {\sqrt {a^3\,b^3}}{16\,a^2\,b^3}-\frac {\sqrt {a^3\,b^3}}{16\,a^3\,b^2}}}{2\,a\,b+\frac {2\,\sqrt {a^3\,b^3}}{a}+2\,b^2+\frac {2\,b\,\sqrt {a^3\,b^3}}{a^2}}+\frac {8\,a\,b^2\,\sin \left (c+d\,x\right )\,\sqrt {-\frac {1}{8\,a\,b}-\frac {\sqrt {a^3\,b^3}}{16\,a^2\,b^3}-\frac {\sqrt {a^3\,b^3}}{16\,a^3\,b^2}}}{2\,a\,b+\frac {2\,\sqrt {a^3\,b^3}}{a}+2\,b^2+\frac {2\,b\,\sqrt {a^3\,b^3}}{a^2}}\right )\,\sqrt {-\frac {a\,\sqrt {a^3\,b^3}+b\,\sqrt {a^3\,b^3}+2\,a^2\,b^2}{16\,a^3\,b^3}}}{d}-\frac {2\,\mathrm {atanh}\left (\frac {8\,b^3\,\sin \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^3\,b^3}}{16\,a^2\,b^3}-\frac {1}{8\,a\,b}+\frac {\sqrt {a^3\,b^3}}{16\,a^3\,b^2}}}{2\,a\,b-\frac {2\,\sqrt {a^3\,b^3}}{a}+2\,b^2-\frac {2\,b\,\sqrt {a^3\,b^3}}{a^2}}+\frac {8\,a\,b^2\,\sin \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^3\,b^3}}{16\,a^2\,b^3}-\frac {1}{8\,a\,b}+\frac {\sqrt {a^3\,b^3}}{16\,a^3\,b^2}}}{2\,a\,b-\frac {2\,\sqrt {a^3\,b^3}}{a}+2\,b^2-\frac {2\,b\,\sqrt {a^3\,b^3}}{a^2}}\right )\,\sqrt {\frac {a\,\sqrt {a^3\,b^3}+b\,\sqrt {a^3\,b^3}-2\,a^2\,b^2}{16\,a^3\,b^3}}}{d} \]
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