Integrand size = 21, antiderivative size = 122 \[ \int \frac {\csc ^3(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {\text {arctanh}(\cos (c+d x))}{2 a d}-\frac {b^2 \text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \csc (c+d x)}{a^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d} \]
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Time = 0.37 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3599, 3189, 3853, 3855, 2701, 327, 213, 2702, 3183, 3153, 212} \[ \int \frac {\csc ^3(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {b^2 \text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {b \csc (c+d x)}{a^2 d}+\frac {b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d} \]
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Rule 212
Rule 213
Rule 327
Rule 2701
Rule 2702
Rule 3153
Rule 3183
Rule 3189
Rule 3599
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot (c+d x) \csc ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx \\ & = \int \left (\frac {\csc ^3(c+d x)}{a}-\frac {b \csc ^2(c+d x) \sec (c+d x)}{a^2}+\frac {b^2 \csc (c+d x) \sec ^2(c+d x)}{a^3}-\frac {b^3 \sec ^2(c+d x)}{a^3 (a \cos (c+d x)+b \sin (c+d x))}\right ) \, dx \\ & = \frac {\int \csc ^3(c+d x) \, dx}{a}-\frac {b \int \csc ^2(c+d x) \sec (c+d x) \, dx}{a^2}+\frac {b^2 \int \csc (c+d x) \sec ^2(c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {\sec ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{a^3} \\ & = -\frac {\cot (c+d x) \csc (c+d x)}{2 a d}-\frac {b^2 \sec (c+d x)}{a^3 d}+\frac {\int \csc (c+d x) \, dx}{2 a}+\frac {b \int \sec (c+d x) \, dx}{a^2}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{a^3}+\frac {b \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{a^2 d}+\frac {b^2 \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^3 d} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{2 a d}+\frac {b \text {arctanh}(\sin (c+d x))}{a^2 d}+\frac {b \csc (c+d x)}{a^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}+\frac {b \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{a^2 d}+\frac {b^2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^3 d}+\frac {\left (b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{a^3 d} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{2 a d}-\frac {b^2 \text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \csc (c+d x)}{a^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d} \\ \end{align*}
Time = 1.31 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.47 \[ \int \frac {\csc ^3(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {-16 b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )+4 a b \cot \left (\frac {1}{2} (c+d x)\right )-a^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )-4 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-8 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+4 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )+4 a b \tan \left (\frac {1}{2} (c+d x)\right )}{8 a^3 d} \]
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Time = 1.51 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.15
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {2 b \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{3}}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (2 a^{2}+4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(140\) |
default | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {2 b \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{3}}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (2 a^{2}+4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(140\) |
risch | \(\frac {i {\mathrm e}^{i \left (d x +c \right )} \left (-i a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}-i a -2 b \right )}{d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 a d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{a^{3} d}-\frac {i \sqrt {-a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {\sqrt {-a^{2}-b^{2}}\, \left (i b +a \right )}{a^{2}+b^{2}}\right )}{d \,a^{3}}+\frac {i \sqrt {-a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {\sqrt {-a^{2}-b^{2}}\, \left (i b +a \right )}{a^{2}+b^{2}}\right )}{d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 a d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{a^{3} d}\) | \(284\) |
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Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (114) = 228\).
Time = 0.31 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.21 \[ \int \frac {\csc ^3(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {2 \, a^{2} \cos \left (d x + c\right ) - 4 \, a b \sin \left (d x + c\right ) + 2 \, {\left (b \cos \left (d x + c\right )^{2} - b\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) - {\left ({\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} - 2 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left ({\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} - 2 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )}} \]
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\[ \int \frac {\csc ^3(c+d x)}{a+b \tan (c+d x)} \, dx=\int \frac {\csc ^{3}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.76 \[ \int \frac {\csc ^3(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\frac {\frac {4 \, b \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a^{2}} + \frac {4 \, {\left (a^{2} + 2 \, b^{2}\right )} \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {{\left (a - \frac {4 \, b \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a^{2} \sin \left (d x + c\right )^{2}} + \frac {8 \, {\left (a^{2} b + b^{3}\right )} \log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{3}}}{8 \, d} \]
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Time = 0.50 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.71 \[ \int \frac {\csc ^3(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{2}} + \frac {4 \, {\left (a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {8 \, {\left (a^{2} b + b^{3}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{3}} - \frac {6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
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Time = 5.55 (sec) , antiderivative size = 764, normalized size of antiderivative = 6.26 \[ \int \frac {\csc ^3(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {b^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2\,\left (\frac {a^3\,d}{2}-\frac {a^3\,d\,\cos \left (2\,c+2\,d\,x\right )}{2}\right )}-\frac {a^2\,\left (\frac {\cos \left (c+d\,x\right )}{2}-\frac {\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4}+\frac {\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (2\,c+2\,d\,x\right )}{4}\right )}{\frac {a^3\,d}{2}-\frac {a^3\,d\,\cos \left (2\,c+2\,d\,x\right )}{2}}+\frac {a\,b\,\sin \left (c+d\,x\right )}{\frac {a^3\,d}{2}-\frac {a^3\,d\,\cos \left (2\,c+2\,d\,x\right )}{2}}-\frac {b^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (2\,c+2\,d\,x\right )}{2\,\left (\frac {a^3\,d}{2}-\frac {a^3\,d\,\cos \left (2\,c+2\,d\,x\right )}{2}\right )}+\frac {b\,\mathrm {atan}\left (\frac {a^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2+b^2}\,1{}\mathrm {i}+b^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2+b^2}\,8{}\mathrm {i}+a\,b^3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2+b^2}\,4{}\mathrm {i}+a^3\,b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2+b^2}\,3{}\mathrm {i}+a^2\,b^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2+b^2}\,8{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^5+4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^4\,b+5\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3\,b^2+12\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^3+4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^4+8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^5}\right )\,\sqrt {a^2+b^2}\,1{}\mathrm {i}}{\frac {a^3\,d}{2}-\frac {a^3\,d\,\cos \left (2\,c+2\,d\,x\right )}{2}}-\frac {b\,\cos \left (2\,c+2\,d\,x\right )\,\mathrm {atan}\left (\frac {a^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2+b^2}\,1{}\mathrm {i}+b^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2+b^2}\,8{}\mathrm {i}+a\,b^3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2+b^2}\,4{}\mathrm {i}+a^3\,b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2+b^2}\,3{}\mathrm {i}+a^2\,b^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2+b^2}\,8{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^5+4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^4\,b+5\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3\,b^2+12\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^3+4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^4+8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^5}\right )\,\sqrt {a^2+b^2}\,1{}\mathrm {i}}{\frac {a^3\,d}{2}-\frac {a^3\,d\,\cos \left (2\,c+2\,d\,x\right )}{2}} \]
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