Integrand size = 23, antiderivative size = 385 \[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {b^{5/3} \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \left (a^2-b^2\right )^2 d}-\frac {(a+4 b) \log (1-\sin (c+d x))}{4 (a+b)^2 d}+\frac {(a-4 b) \log (1+\sin (c+d x))}{4 (a-b)^2 d}+\frac {b^{5/3} \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \left (a^2-b^2\right )^2 d}-\frac {b^{5/3} \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \left (a^2-b^2\right )^2 d}+\frac {b \left (a^2+2 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\sin (c+d x))}-\frac {1}{4 (a-b) d (1+\sin (c+d x))} \]
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Time = 0.55 (sec) , antiderivative size = 385, 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, 2099, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {b \left (a^2+2 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 d \left (a^2-b^2\right )^2}-\frac {b^{5/3} \left (-3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} d \left (a^2-b^2\right )^2}-\frac {b^{5/3} \left (3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} d \left (a^2-b^2\right )^2}+\frac {b^{5/3} \left (3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} d \left (a^2-b^2\right )^2}+\frac {1}{4 d (a+b) (1-\sin (c+d x))}-\frac {1}{4 d (a-b) (\sin (c+d x)+1)}-\frac {(a+4 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac {(a-4 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2} \]
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Rule 31
Rule 210
Rule 266
Rule 631
Rule 642
Rule 648
Rule 1874
Rule 1885
Rule 2099
Rule 3302
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a+b x^3\right )} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{4 (a+b) (-1+x)^2}+\frac {-a-4 b}{4 (a+b)^2 (-1+x)}+\frac {1}{4 (a-b) (1+x)^2}+\frac {a-4 b}{4 (a-b)^2 (1+x)}+\frac {b^2 \left (2 a^2+b^2-3 a b x+\left (a^2+2 b^2\right ) x^2\right )}{\left (a^2-b^2\right )^2 \left (a+b x^3\right )}\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {(a+4 b) \log (1-\sin (c+d x))}{4 (a+b)^2 d}+\frac {(a-4 b) \log (1+\sin (c+d x))}{4 (a-b)^2 d}+\frac {1}{4 (a+b) d (1-\sin (c+d x))}-\frac {1}{4 (a-b) d (1+\sin (c+d x))}+\frac {b^2 \text {Subst}\left (\int \frac {2 a^2+b^2-3 a b x+\left (a^2+2 b^2\right ) x^2}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d} \\ & = -\frac {(a+4 b) \log (1-\sin (c+d x))}{4 (a+b)^2 d}+\frac {(a-4 b) \log (1+\sin (c+d x))}{4 (a-b)^2 d}+\frac {1}{4 (a+b) d (1-\sin (c+d x))}-\frac {1}{4 (a-b) d (1+\sin (c+d x))}+\frac {b^2 \text {Subst}\left (\int \frac {2 a^2+b^2-3 a b x}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d}+\frac {\left (b^2 \left (a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d} \\ & = -\frac {(a+4 b) \log (1-\sin (c+d x))}{4 (a+b)^2 d}+\frac {(a-4 b) \log (1+\sin (c+d x))}{4 (a-b)^2 d}+\frac {b \left (a^2+2 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\sin (c+d x))}-\frac {1}{4 (a-b) d (1+\sin (c+d x))}+\frac {b^{5/3} \text {Subst}\left (\int \frac {\sqrt [3]{a} \left (-3 a^{4/3} b+2 \sqrt [3]{b} \left (2 a^2+b^2\right )\right )+\sqrt [3]{b} \left (-3 a^{4/3} b-\sqrt [3]{b} \left (2 a^2+b^2\right )\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} \left (a^2-b^2\right )^2 d}+\frac {\left (b^2 \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sin (c+d x)\right )}{3 a^{2/3} \left (a^2-b^2\right )^2 d} \\ & = -\frac {(a+4 b) \log (1-\sin (c+d x))}{4 (a+b)^2 d}+\frac {(a-4 b) \log (1+\sin (c+d x))}{4 (a-b)^2 d}+\frac {b^{5/3} \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \left (a^2-b^2\right )^2 d}+\frac {b \left (a^2+2 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\sin (c+d x))}-\frac {1}{4 (a-b) d (1+\sin (c+d x))}+\frac {\left (b^2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right )\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} \left (a^2-b^2\right )^2 d}-\frac {\left (b^{5/3} \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right )\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} \left (a^2-b^2\right )^2 d} \\ & = -\frac {(a+4 b) \log (1-\sin (c+d x))}{4 (a+b)^2 d}+\frac {(a-4 b) \log (1+\sin (c+d x))}{4 (a-b)^2 d}+\frac {b^{5/3} \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \left (a^2-b^2\right )^2 d}-\frac {b^{5/3} \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\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} \left (a^2-b^2\right )^2 d}+\frac {b \left (a^2+2 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\sin (c+d x))}-\frac {1}{4 (a-b) d (1+\sin (c+d x))}+\frac {\left (b^{5/3} \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\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 )}{a^{2/3} \left (a^2-b^2\right )^2 d} \\ & = -\frac {b^{5/3} \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\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} \left (a^2-b^2\right )^2 d}-\frac {(a+4 b) \log (1-\sin (c+d x))}{4 (a+b)^2 d}+\frac {(a-4 b) \log (1+\sin (c+d x))}{4 (a-b)^2 d}+\frac {b^{5/3} \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \left (a^2-b^2\right )^2 d}-\frac {b^{5/3} \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\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} \left (a^2-b^2\right )^2 d}+\frac {b \left (a^2+2 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\sin (c+d x))}-\frac {1}{4 (a-b) d (1+\sin (c+d x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 2.26 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.94 \[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {\frac {4 \sqrt {3} b^{5/3} \left (2 a^2+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{2/3} \left (a^2-b^2\right )^2}+\frac {3 (a+4 b) \log (1-\sin (c+d x))}{(a+b)^2}-\frac {3 (a-4 b) \log (1+\sin (c+d x))}{(a-b)^2}-\frac {4 b^{5/3} \left (2 a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{a^{2/3} \left (a^2-b^2\right )^2}+\frac {2 b^{5/3} \left (2 a^2+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{a^{2/3} \left (a^2-b^2\right )^2}-\frac {4 b \left (a^2+2 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{\left (a^2-b^2\right )^2}+\frac {3}{(a+b) (-1+\sin (c+d x))}+\frac {18 b^3 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{\left (a^2-b^2\right )^2}+\frac {3}{(a-b) (1+\sin (c+d x))}}{12 d} \]
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Time = 1.89 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {-\frac {1}{\left (4 a +4 b \right ) \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-a -4 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{2}}-\frac {1}{\left (4 a -4 b \right ) \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -4 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{2}}+\frac {\left (\left (2 a^{2}+b^{2}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )-3 a b \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (a^{2}+2 b^{2}\right ) \ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{3 b}\right ) b^{2}}{\left (a -b \right )^{2} \left (a +b \right )^{2}}}{d}\) | \(372\) |
default | \(\frac {-\frac {1}{\left (4 a +4 b \right ) \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-a -4 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{2}}-\frac {1}{\left (4 a -4 b \right ) \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -4 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{2}}+\frac {\left (\left (2 a^{2}+b^{2}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )-3 a b \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (a^{2}+2 b^{2}\right ) \ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{3 b}\right ) b^{2}}{\left (a -b \right )^{2} \left (a +b \right )^{2}}}{d}\) | \(372\) |
risch | \(\frac {2 i b x}{a^{2}-2 a b +b^{2}}-\frac {i a c}{2 d \left (a^{2}-2 a b +b^{2}\right )}+\frac {2 i b c}{d \left (a^{2}-2 a b +b^{2}\right )}-\frac {i a x}{2 \left (a^{2}-2 a b +b^{2}\right )}-\frac {4 i d^{2} b^{3} a^{2} c}{a^{6} d^{3}-2 a^{4} b^{2} d^{3}+d^{3} b^{4} a^{2}}+\frac {i a x}{2 a^{2}+4 a b +2 b^{2}}+\frac {2 i b x}{a^{2}+2 a b +b^{2}}-\frac {2 i a^{4} b \,d^{2} c}{a^{6} d^{3}-2 a^{4} b^{2} d^{3}+d^{3} b^{4} a^{2}}-\frac {2 i a^{4} b \,d^{3} x}{a^{6} d^{3}-2 a^{4} b^{2} d^{3}+d^{3} b^{4} a^{2}}+\frac {i a c}{2 \left (a^{2}+2 a b +b^{2}\right ) d}-\frac {4 i d^{3} b^{3} a^{2} x}{a^{6} d^{3}-2 a^{4} b^{2} d^{3}+d^{3} b^{4} a^{2}}+\frac {2 i b c}{d \left (a^{2}+2 a b +b^{2}\right )}+\frac {i a \,{\mathrm e}^{3 i \left (d x +c \right )}-i a \,{\mathrm e}^{i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{\left (-a^{2}+b^{2}\right ) d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a}{2 \left (a^{2}-2 a b +b^{2}\right ) d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b}{\left (a^{2}-2 a b +b^{2}\right ) d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a}{2 \left (a^{2}+2 a b +b^{2}\right ) d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b}{\left (a^{2}+2 a b +b^{2}\right ) d}+8 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (13824 a^{6} d^{3}-27648 a^{4} b^{2} d^{3}+13824 d^{3} b^{4} a^{2}\right ) \textit {\_Z}^{3}+\left (-1728 a^{4} b \,d^{2}-3456 d^{2} b^{3} a^{2}\right ) \textit {\_Z}^{2}+72 a^{2} b^{2} d \textit {\_Z} -b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {3456 i a^{7} d^{2}}{8 a^{2} b^{3}+b^{5}}+\frac {6912 i a^{5} b^{2} d^{2}}{8 a^{2} b^{3}+b^{5}}-\frac {3456 i a^{3} b^{4} d^{2}}{8 a^{2} b^{3}+b^{5}}\right ) \textit {\_R}^{2}+\left (\frac {480 i a^{5} b d}{8 a^{2} b^{3}+b^{5}}+\frac {768 i a^{3} b^{3} d}{8 a^{2} b^{3}+b^{5}}+\frac {48 i a \,b^{5} d}{8 a^{2} b^{3}+b^{5}}\right ) \textit {\_R} -\frac {14 i a^{3} b^{2}}{8 a^{2} b^{3}+b^{5}}-\frac {4 i a \,b^{4}}{8 a^{2} b^{3}+b^{5}}\right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {8 a^{2} b^{3}}{8 a^{2} b^{3}+b^{5}}-\frac {b^{5}}{8 a^{2} b^{3}+b^{5}}\right )\right )\) | \(867\) |
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Result contains complex when optimal does not.
Time = 2.13 (sec) , antiderivative size = 10135, normalized size of antiderivative = 26.32 \[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int \frac {\sec ^{3}{\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d x \right )}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.22 \[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {\frac {4 \, \sqrt {3} {\left (a b^{2} {\left (9 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 4\right )} - b^{3} {\left (3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {4 \, a}{b}\right )} - 2 \, a^{2} b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {a}{b}\right )} + 2 \, a^{3}\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 )}{{\left (a^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {6 \, {\left (b^{3} {\left (4 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} - 1\right )} + 2 \, a^{2} b {\left (\left (\frac {a}{b}\right )^{\frac {2}{3}} - 1\right )} - 3 \, a b^{2} \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 )}{a^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {12 \, {\left (b^{3} {\left (2 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 1\right )} + a^{2} b {\left (\left (\frac {a}{b}\right )^{\frac {2}{3}} + 2\right )} + 3 \, a b^{2} \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 )}{a^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {9 \, {\left (a - 4 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {9 \, {\left (a + 4 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}} + \frac {18 \, {\left (a \sin \left (d x + c\right ) - b\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2} - a^{2} + b^{2}}}{36 \, d} \]
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Time = 0.37 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.32 \[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\frac {4 \, {\left (3 \, a^{5} b^{4} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 6 \, a^{3} b^{6} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 3 \, a b^{8} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - b^{9}\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^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}} + \frac {12 \, {\left (3 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b + {\left (2 \, a^{2} b + b^{3}\right )} \left (-a b^{2}\right )^{\frac {1}{3}}\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 )}{\sqrt {3} a^{5} - 2 \, \sqrt {3} a^{3} b^{2} + \sqrt {3} a b^{4}} - \frac {2 \, {\left (3 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b - {\left (2 \, a^{2} b + b^{3}\right )} \left (-a b^{2}\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 )}{a^{5} - 2 \, a^{3} b^{2} + a b^{4}} + \frac {4 \, {\left (a^{2} b + 2 \, b^{3}\right )} \log \left ({\left | b \sin \left (d x + c\right )^{3} + a \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac {3 \, {\left (a - 4 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {3 \, {\left (a + 4 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2} + 2 \, a b + b^{2}} + \frac {6 \, {\left (a^{2} b \sin \left (d x + c\right )^{2} + 2 \, b^{3} \sin \left (d x + c\right )^{2} - a^{3} \sin \left (d x + c\right ) + a b^{2} \sin \left (d x + c\right ) - 3 \, b^{3}\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}}}{12 \, d} \]
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Time = 14.96 (sec) , antiderivative size = 898, normalized size of antiderivative = 2.33 \[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\left (\sum _{k=1}^3\ln \left (-\mathrm {root}\left (54\,a^4\,b^2\,z^3-27\,a^2\,b^4\,z^3-27\,a^6\,z^3+54\,a^2\,b^3\,z^2+27\,a^4\,b\,z^2-9\,a^2\,b^2\,z+b^3,z,k\right )\,\left (-\frac {28\,a\,b^7-6\,a^3\,b^5}{a^4-2\,a^2\,b^2+b^4}+\mathrm {root}\left (54\,a^4\,b^2\,z^3-27\,a^2\,b^4\,z^3-27\,a^6\,z^3+54\,a^2\,b^3\,z^2+27\,a^4\,b\,z^2-9\,a^2\,b^2\,z+b^3,z,k\right )\,\left (-\frac {18\,a^5\,b^4-\frac {219\,a^3\,b^6}{4}+12\,a\,b^8}{a^4-2\,a^2\,b^2+b^4}+\mathrm {root}\left (54\,a^4\,b^2\,z^3-27\,a^2\,b^4\,z^3-27\,a^6\,z^3+54\,a^2\,b^3\,z^2+27\,a^4\,b\,z^2-9\,a^2\,b^2\,z+b^3,z,k\right )\,\left (\frac {\frac {27\,a^7\,b^3}{2}-87\,a^5\,b^5+\frac {51\,a^3\,b^7}{2}+48\,a\,b^9}{a^4-2\,a^2\,b^2+b^4}+\mathrm {root}\left (54\,a^4\,b^2\,z^3-27\,a^2\,b^4\,z^3-27\,a^6\,z^3+54\,a^2\,b^3\,z^2+27\,a^4\,b\,z^2-9\,a^2\,b^2\,z+b^3,z,k\right )\,\left (\frac {180\,a^7\,b^4-324\,a^5\,b^6+108\,a^3\,b^8+36\,a\,b^{10}}{a^4-2\,a^2\,b^2+b^4}+\frac {\sin \left (c+d\,x\right )\,\left (216\,a^8\,b^3+216\,a^6\,b^5-1080\,a^4\,b^7+648\,a^2\,b^9\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}\right )+\frac {\sin \left (c+d\,x\right )\,\left (-600\,a^4\,b^6+552\,a^2\,b^8+48\,b^{10}\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}\right )+\frac {\sin \left (c+d\,x\right )\,\left (-171\,a^4\,b^5+120\,a^2\,b^7+96\,b^9\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}\right )+\frac {\sin \left (c+d\,x\right )\,\left (61\,a^2\,b^6+40\,b^8\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}\right )+\frac {a\,b^6}{2\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {2\,b^7\,\sin \left (c+d\,x\right )}{a^4-2\,a^2\,b^2+b^4}\right )\,\mathrm {root}\left (54\,a^4\,b^2\,z^3-27\,a^2\,b^4\,z^3-27\,a^6\,z^3+54\,a^2\,b^3\,z^2+27\,a^4\,b\,z^2-9\,a^2\,b^2\,z+b^3,z,k\right )\right )+\frac {\frac {b}{2\,\left (a^2-b^2\right )}-\frac {a\,\sin \left (c+d\,x\right )}{2\,\left (a^2-b^2\right )}}{{\sin \left (c+d\,x\right )}^2-1}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (a+4\,b\right )}{4\,a^2+8\,a\,b+4\,b^2}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (a-4\,b\right )}{4\,a^2-8\,a\,b+4\,b^2}}{d} \]
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