Integrand size = 21, antiderivative size = 95 \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {a^2 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {b^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {2 a b \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a b \csc (c+d x)}{d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {b^2 \sec (c+d x)}{d} \]
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Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3598, 3853, 3855, 2701, 327, 213, 2702} \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {a^2 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {2 a b \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a b \csc (c+d x)}{d}-\frac {b^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {b^2 \sec (c+d x)}{d} \]
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Rule 213
Rule 327
Rule 2701
Rule 2702
Rule 3598
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 \csc ^3(c+d x)+2 a b \csc ^2(c+d x) \sec (c+d x)+b^2 \csc (c+d x) \sec ^2(c+d x)\right ) \, dx \\ & = a^2 \int \csc ^3(c+d x) \, dx+(2 a b) \int \csc ^2(c+d x) \sec (c+d x) \, dx+b^2 \int \csc (c+d x) \sec ^2(c+d x) \, dx \\ & = -\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {1}{2} a^2 \int \csc (c+d x) \, dx-\frac {(2 a b) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {b^2 \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {a^2 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {2 a b \csc (c+d x)}{d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {b^2 \sec (c+d x)}{d}-\frac {(2 a b) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {b^2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {a^2 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {b^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {2 a b \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a b \csc (c+d x)}{d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {b^2 \sec (c+d x)}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(250\) vs. \(2(95)=190\).
Time = 2.78 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.63 \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {8 b^2-8 a b \cot \left (\frac {1}{2} (c+d x)\right )-a^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )-4 \left (a^2+2 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-16 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 \left (a^2+2 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+16 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )+\frac {8 b^2 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {8 b^2 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}-8 a b \tan \left (\frac {1}{2} (c+d x)\right )}{8 d} \]
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Time = 1.81 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {b^{2} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+2 a b \left (-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(101\) |
default | \(\frac {b^{2} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+2 a b \left (-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(101\) |
risch | \(\frac {{\mathrm e}^{i \left (d x +c \right )} \left (a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+2 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-4 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}+2 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-4 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+a^{2}+2 b^{2}+4 i a b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{d}+\frac {2 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {2 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{d}\) | \(257\) |
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Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (91) = 182\).
Time = 0.31 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.42 \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {8 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, {\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 4 \, b^{2} - {\left ({\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left ({\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 4 \, {\left (a b \cos \left (d x + c\right )^{3} - a b \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 4 \, {\left (a b \cos \left (d x + c\right )^{3} - a b \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{4 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )\right )}} \]
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\[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^2 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{2} \csc ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.28 \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {a^{2} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2 \, b^{2} {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 4 \, a b {\left (\frac {2}{\sin \left (d x + c\right )} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{4 \, d} \]
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Time = 0.51 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.81 \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 16 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 16 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, {\left (a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {16 \, b^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \frac {6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
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Time = 4.16 (sec) , antiderivative size = 292, normalized size of antiderivative = 3.07 \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {a^2}{2}+b^2\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{2}+8\,b^2\right )-\frac {a^2}{2}+4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {4\,a\,b\,\mathrm {atanh}\left (\frac {8\,a\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,a^3\,b-16\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2+8\,a\,b^3}-\frac {16\,a^2\,b^2}{4\,a^3\,b-16\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2+8\,a\,b^3}+\frac {4\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,a^3\,b-16\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2+8\,a\,b^3}\right )}{d}-\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d} \]
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