Integrand size = 25, antiderivative size = 133 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 b \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} d}-\frac {a \sec (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d} \]
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Time = 0.14 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2943, 2945, 12, 2739, 632, 210} \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 b \left (2 a^2+b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{5/2}}+\frac {\sec (c+d x) \left (2 a^2-3 a b \sin (c+d x)+b^2\right )}{d \left (a^2-b^2\right )^2}-\frac {a \sec (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))} \]
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Rule 12
Rule 210
Rule 632
Rule 2739
Rule 2943
Rule 2945
Rubi steps \begin{align*} \text {integral}& = -\frac {a \sec (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {\sec ^2(c+d x) (b-2 a \sin (c+d x))}{a+b \sin (c+d x)} \, dx}{-a^2+b^2} \\ & = -\frac {a \sec (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d}+\frac {\int \frac {2 a^2 b+b^3}{a+b \sin (c+d x)} \, dx}{\left (a^2-b^2\right )^2} \\ & = -\frac {a \sec (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d}+\frac {\left (b \left (2 a^2+b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{\left (a^2-b^2\right )^2} \\ & = -\frac {a \sec (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d}+\frac {\left (2 b \left (2 a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2-b^2\right )^2 d} \\ & = -\frac {a \sec (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d}-\frac {\left (4 b \left (2 a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2-b^2\right )^2 d} \\ & = \frac {2 b \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} d}-\frac {a \sec (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d} \\ \end{align*}
Time = 0.78 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.27 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {2 b \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\sin \left (\frac {1}{2} (c+d x)\right ) \left (\frac {1}{(a+b)^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {1}{(a-b)^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right )+\frac {a b^2 \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}}{d} \]
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Time = 0.83 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.23
method | result | size |
derivativedivides | \(\frac {\frac {4 b \left (\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}+\frac {a b}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (2 a^{2}+b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}+\frac {1}{\left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{\left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(164\) |
default | \(\frac {\frac {4 b \left (\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}+\frac {a b}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (2 a^{2}+b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}+\frac {1}{\left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{\left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(164\) |
risch | \(\frac {4 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-4 i a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-2 i b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-8 i a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}+2 i b^{3} {\mathrm e}^{i \left (d x +c \right )}+6 a \,b^{2}+2 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right ) \left (a^{2}-b^{2}\right )^{2} d}-\frac {2 i b \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {i b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {2 i b \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {i b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}\) | \(490\) |
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Time = 0.29 (sec) , antiderivative size = 557, normalized size of antiderivative = 4.19 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\left [\frac {2 \, a^{5} - 4 \, a^{3} b^{2} + 2 \, a b^{4} + 6 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d \cos \left (d x + c\right )\right )}}, \frac {a^{5} - 2 \, a^{3} b^{2} + a b^{4} + 3 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )}{{\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d \cos \left (d x + c\right )}\right ] \]
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\[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\int \frac {\sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.34 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.83 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 \, {\left (\frac {{\left (2 \, a^{2} b + b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3} - 2 \, a b^{2}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )} {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}}\right )}}{d} \]
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Time = 16.17 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.33 \[ \int \frac {\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {6\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{{\left (a^2-b^2\right )}^2}+\frac {2\,a\,\left (a^2+2\,b^2\right )}{{\left (a^2-b^2\right )}^2}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (4\,a\,b^2-a^3\right )}{a^4-2\,a^2\,b^2+b^4}-\frac {2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,a^2+b^2\right )}{{\left (a^2-b^2\right )}^2}}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}+\frac {2\,b\,\mathrm {atan}\left (\frac {\frac {b\,\left (2\,a^2+b^2\right )\,\left (2\,a^4\,b-4\,a^2\,b^3+2\,b^5\right )}{{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}+\frac {2\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2+b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}{{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}}{4\,a^2\,b+2\,b^3}\right )\,\left (2\,a^2+b^2\right )}{d\,{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}} \]
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