Integrand size = 21, antiderivative size = 290 \[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {\sqrt [3]{b} \left (a^{4/3}-b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \left (a^2-b^2\right ) d}-\frac {\log (1-\sin (c+d x))}{2 (a+b) d}+\frac {\log (1+\sin (c+d x))}{2 (a-b) d}-\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \left (a^2-b^2\right ) d}+\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \left (a^2-b^2\right ) d}-\frac {b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right ) d} \]
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Time = 0.36 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3302, 2099, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {b \log \left (a+b \sin ^3(c+d x)\right )}{3 d \left (a^2-b^2\right )}-\frac {\sqrt [3]{b} \left (a^{4/3}-b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} d \left (a^2-b^2\right )}+\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} d \left (a^2-b^2\right )}-\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} d \left (a^2-b^2\right )}-\frac {\log (1-\sin (c+d x))}{2 d (a+b)}+\frac {\log (\sin (c+d x)+1)}{2 d (a-b)} \]
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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 ) \left (a+b x^3\right )} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{2 (a+b) (-1+x)}+\frac {1}{2 (a-b) (1+x)}+\frac {b \left (b-a x+b x^2\right )}{\left (-a^2+b^2\right ) \left (a+b x^3\right )}\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {\log (1-\sin (c+d x))}{2 (a+b) d}+\frac {\log (1+\sin (c+d x))}{2 (a-b) d}-\frac {b \text {Subst}\left (\int \frac {b-a x+b x^2}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right ) d} \\ & = -\frac {\log (1-\sin (c+d x))}{2 (a+b) d}+\frac {\log (1+\sin (c+d x))}{2 (a-b) d}-\frac {b \text {Subst}\left (\int \frac {b-a x}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right ) d}-\frac {b^2 \text {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right ) d} \\ & = -\frac {\log (1-\sin (c+d x))}{2 (a+b) d}+\frac {\log (1+\sin (c+d x))}{2 (a-b) d}-\frac {b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right ) d}-\frac {b^{2/3} \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} \left (a^2-b^2\right ) d}-\frac {\left (b^{2/3} \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 )}{3 a^{2/3} \left (a^2-b^2\right ) d} \\ & = -\frac {\log (1-\sin (c+d x))}{2 (a+b) d}+\frac {\log (1+\sin (c+d x))}{2 (a-b) d}-\frac {\sqrt [3]{b} \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} \left (a^2-b^2\right ) d}-\frac {b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right ) d}+\frac {\left (b^{2/3} \left (a^{4/3}-b^{4/3}\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 ) d}+\frac {\left (\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\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 ) d} \\ & = -\frac {\log (1-\sin (c+d x))}{2 (a+b) d}+\frac {\log (1+\sin (c+d x))}{2 (a-b) d}-\frac {\sqrt [3]{b} \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} \left (a^2-b^2\right ) d}+\frac {\sqrt [3]{b} \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} \left (a^2-b^2\right ) d}-\frac {b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right ) d}+\frac {\left (\sqrt [3]{b} \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 )}{a^{2/3} \left (a^2-b^2\right ) d} \\ & = -\frac {\sqrt [3]{b} \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} \left (a^2-b^2\right ) d}-\frac {\log (1-\sin (c+d x))}{2 (a+b) d}+\frac {\log (1+\sin (c+d x))}{2 (a-b) d}-\frac {\sqrt [3]{b} \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} \left (a^2-b^2\right ) d}+\frac {\sqrt [3]{b} \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} \left (a^2-b^2\right ) d}-\frac {b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right ) d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.23 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.92 \[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {2 \sqrt {3} b^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )-3 a^{5/3} \log (1-\sin (c+d x))+3 a^{2/3} b \log (1-\sin (c+d x))+3 a^{5/3} \log (1+\sin (c+d x))+3 a^{2/3} b \log (1+\sin (c+d x))-2 b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )+b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )-2 a^{2/3} b \log \left (a+b \sin ^3(c+d x)\right )+3 a^{2/3} b \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{6 a^{2/3} (a-b) (a+b) d} \]
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Time = 0.90 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(\frac {-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 a +2 b}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 a -2 b}+\frac {\left (-b \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+a \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-\frac {\ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{3}\right ) b}{\left (a +b \right ) \left (a -b \right )}}{d}\) | \(300\) |
default | \(\frac {-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 a +2 b}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 a -2 b}+\frac {\left (-b \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+a \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-\frac {\ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{3}\right ) b}{\left (a +b \right ) \left (a -b \right )}}{d}\) | \(300\) |
risch | \(-\frac {i x}{a -b}-\frac {i c}{d \left (a -b \right )}+\frac {i x}{a +b}+\frac {i c}{d \left (a +b \right )}+\frac {2 i a^{2} b \,d^{3} x}{a^{4} d^{3}-a^{2} b^{2} d^{3}}+\frac {2 i a^{2} b \,d^{2} c}{a^{4} d^{3}-a^{2} b^{2} d^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \left (a -b \right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d \left (a +b \right )}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (216 a^{4} d^{3}-216 a^{2} b^{2} d^{3}\right ) \textit {\_Z}^{3}+108 a^{2} b \,d^{2} \textit {\_Z}^{2}+b \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (\frac {72 i a^{5} d^{2}}{a^{2} b +b^{3}}-\frac {72 i a^{3} b^{2} d^{2}}{a^{2} b +b^{3}}\right ) \textit {\_R}^{2}+\left (\frac {24 i a^{3} b d}{a^{2} b +b^{3}}+\frac {12 i a \,b^{3} d}{a^{2} b +b^{3}}\right ) \textit {\_R} -\frac {2 i a \,b^{2}}{a^{2} b +b^{3}}\right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {a^{2} b}{a^{2} b +b^{3}}-\frac {b^{3}}{a^{2} b +b^{3}}\right )\right )\) | \(367\) |
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Result contains complex when optimal does not.
Time = 1.22 (sec) , antiderivative size = 4396, normalized size of antiderivative = 15.16 \[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int \frac {\sec {\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d x \right )}}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.99 \[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\frac {2 \, \sqrt {3} {\left (a {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2\right )} - b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {2 \, a}{b}\right )}\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{{\left (a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {3 \, {\left (b {\left (2 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} - 1\right )} - a \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \log \left (\sin \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {6 \, {\left (b {\left (\left (\frac {a}{b}\right )^{\frac {2}{3}} + 1\right )} + a \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {9 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a - b} - \frac {9 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a + b}}{18 \, d} \]
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Time = 0.39 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.07 \[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {\frac {2 \, {\left (a^{3} b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{4} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{3} + b^{5}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{a^{5} b - 2 \, a^{3} b^{3} + a b^{5}} + \frac {6 \, {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} + \left (-a b^{2}\right )^{\frac {2}{3}} a\right )} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a^{3} b - \sqrt {3} a b^{3}} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} - \left (-a b^{2}\right )^{\frac {2}{3}} a\right )} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{3} b - a b^{3}} + \frac {2 \, b \log \left ({\left | b \sin \left (d x + c\right )^{3} + a \right |}\right )}{a^{2} - b^{2}} - \frac {3 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a - b} + \frac {3 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a + b}}{6 \, d} \]
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Time = 0.26 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.07 \[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\left (\sum _{k=1}^3\ln \left (-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^2\,a\,b^4\,13-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^3\,a\,b^5\,36-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^4\,a\,b^6\,36-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^2\,b^5\,\sin \left (c+d\,x\right )\,16-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^3\,b^6\,\sin \left (c+d\,x\right )\,12-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^3\,a^3\,b^3\,27-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^4\,a^3\,b^4\,180-\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )\,b^4\,\sin \left (c+d\,x\right )\,5-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^3\,a^2\,b^4\,\sin \left (c+d\,x\right )\,69-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^4\,a^2\,b^5\,\sin \left (c+d\,x\right )\,162-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^4\,a^4\,b^3\,\sin \left (c+d\,x\right )\,54\right )\,\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )\right )-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )}{2\,a+2\,b}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{2\,a-2\,b}}{d} \]
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