Integrand size = 24, antiderivative size = 138 \[ \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\sqrt {a} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{5/4} d}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{5/4} d}+\frac {\cos (c+d x)}{b d} \]
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Time = 0.10 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3294, 1184, 1107, 211, 214} \[ \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\sqrt {a} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{5/4} d \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{5/4} d \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\cos (c+d x)}{b d} \]
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Rule 211
Rule 214
Rule 1107
Rule 1184
Rule 3294
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \left (-\frac {1}{b}+\frac {a}{b \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {\cos (c+d x)}{b d}-\frac {a \text {Subst}\left (\int \frac {1}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{b d} \\ & = \frac {\cos (c+d x)}{b d}+\frac {\sqrt {a} \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 \sqrt {b} d}-\frac {\sqrt {a} \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 \sqrt {b} d} \\ & = -\frac {\sqrt {a} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{5/4} d}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{5/4} d}+\frac {\cos (c+d x)}{b d} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.82 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.43 \[ \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {2 \cos (c+d x)+i a \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{-b-8 a \text {$\#$1}^2+3 b \text {$\#$1}^2-3 b \text {$\#$1}^4+b \text {$\#$1}^6}\&\right ]}{2 b d} \]
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Time = 1.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {\frac {\cos \left (d x +c \right )}{b}+a \left (-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {\arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{d}\) | \(99\) |
default | \(\frac {\frac {\cos \left (d x +c \right )}{b}+a \left (-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {\arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{d}\) | \(99\) |
risch | \(\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 b d}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 b d}-\frac {i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a \,b^{5} d^{4}-b^{6} d^{4}\right ) \textit {\_Z}^{4}-128 a \,d^{2} \textit {\_Z}^{2} b^{3}-4096 a^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {i b^{4} d^{3}}{256 a}+\frac {i b^{5} d^{3}}{256 a^{2}}\right ) \textit {\_R}^{3}+\left (\frac {i d b}{4}+\frac {i b^{2} d}{4 a}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}+1\right )\right )}{32}\) | \(154\) |
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Leaf count of result is larger than twice the leaf count of optimal. 815 vs. \(2 (98) = 196\).
Time = 0.36 (sec) , antiderivative size = 815, normalized size of antiderivative = 5.91 \[ \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {b d \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (a^{2} \cos \left (d x + c\right ) - {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a^{2} b d\right )} \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}}\right ) - b d \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (a^{2} \cos \left (d x + c\right ) - {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a^{2} b d\right )} \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}}\right ) - b d \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (-a^{2} \cos \left (d x + c\right ) - {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a^{2} b d\right )} \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}}\right ) + b d \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (-a^{2} \cos \left (d x + c\right ) - {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a^{2} b d\right )} \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}}\right ) - 4 \, \cos \left (d x + c\right )}{4 \, b d} \]
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Timed out. \[ \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\sin \left (d x + c\right )^{5}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]
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\[ \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\sin \left (d x + c\right )^{5}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]
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Time = 13.50 (sec) , antiderivative size = 1001, normalized size of antiderivative = 7.25 \[ \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\cos \left (c+d\,x\right )}{b\,d}-\frac {2\,\mathrm {atanh}\left (\frac {8\,a^2\,b^7\,\cos \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^3\,b^5}}{16\,\left (a\,b^5-b^6\right )}-\frac {a\,b^3}{16\,\left (a\,b^5-b^6\right )}}}{\frac {2\,a^3\,b^{11}}{a\,b^5-b^6}-\frac {2\,a^4\,b^{10}}{a\,b^5-b^6}+\frac {2\,a^2\,b^8\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}-\frac {2\,a^3\,b^7\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}}-\frac {8\,a^2\,b\,\cos \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^3\,b^5}}{16\,\left (a\,b^5-b^6\right )}-\frac {a\,b^3}{16\,\left (a\,b^5-b^6\right )}}}{\frac {2\,a^3\,b^5}{a\,b^5-b^6}+\frac {2\,a^2\,b^2\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}}+\frac {8\,a\,b^4\,\cos \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^3\,b^5}}{16\,\left (a\,b^5-b^6\right )}-\frac {a\,b^3}{16\,\left (a\,b^5-b^6\right )}}\,\sqrt {a^3\,b^5}}{\frac {2\,a^3\,b^{11}}{a\,b^5-b^6}-\frac {2\,a^4\,b^{10}}{a\,b^5-b^6}+\frac {2\,a^2\,b^8\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}-\frac {2\,a^3\,b^7\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}}\right )\,\sqrt {-\frac {\sqrt {a^3\,b^5}+a\,b^3}{16\,\left (a\,b^5-b^6\right )}}}{d}+\frac {2\,\mathrm {atanh}\left (\frac {8\,a^2\,b\,\cos \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^3\,b^5}}{16\,\left (a\,b^5-b^6\right )}-\frac {a\,b^3}{16\,\left (a\,b^5-b^6\right )}}}{\frac {2\,a^3\,b^5}{a\,b^5-b^6}-\frac {2\,a^2\,b^2\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}}-\frac {8\,a^2\,b^7\,\cos \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^3\,b^5}}{16\,\left (a\,b^5-b^6\right )}-\frac {a\,b^3}{16\,\left (a\,b^5-b^6\right )}}}{\frac {2\,a^3\,b^{11}}{a\,b^5-b^6}-\frac {2\,a^4\,b^{10}}{a\,b^5-b^6}-\frac {2\,a^2\,b^8\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}+\frac {2\,a^3\,b^7\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}}+\frac {8\,a\,b^4\,\cos \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^3\,b^5}}{16\,\left (a\,b^5-b^6\right )}-\frac {a\,b^3}{16\,\left (a\,b^5-b^6\right )}}\,\sqrt {a^3\,b^5}}{\frac {2\,a^3\,b^{11}}{a\,b^5-b^6}-\frac {2\,a^4\,b^{10}}{a\,b^5-b^6}-\frac {2\,a^2\,b^8\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}+\frac {2\,a^3\,b^7\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}}\right )\,\sqrt {\frac {\sqrt {a^3\,b^5}-a\,b^3}{16\,\left (a\,b^5-b^6\right )}}}{d} \]
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