Integrand size = 19, antiderivative size = 118 \[ \int \csc (c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {a^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a b^3 \text {arctanh}(\sin (c+d x))}{d}+\frac {6 a^2 b^2 \sec (c+d x)}{d}-\frac {b^4 \sec (c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d}+\frac {2 a b^3 \sec (c+d x) \tan (c+d x)}{d} \]
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Time = 0.14 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3598, 3855, 2686, 8, 2691} \[ \int \csc (c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {a^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}+\frac {6 a^2 b^2 \sec (c+d x)}{d}-\frac {2 a b^3 \text {arctanh}(\sin (c+d x))}{d}+\frac {2 a b^3 \tan (c+d x) \sec (c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d}-\frac {b^4 \sec (c+d x)}{d} \]
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Rule 8
Rule 2686
Rule 2691
Rule 3598
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a^4 \csc (c+d x)+4 a^3 b \sec (c+d x)+6 a^2 b^2 \sec (c+d x) \tan (c+d x)+4 a b^3 \sec (c+d x) \tan ^2(c+d x)+b^4 \sec (c+d x) \tan ^3(c+d x)\right ) \, dx \\ & = a^4 \int \csc (c+d x) \, dx+\left (4 a^3 b\right ) \int \sec (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \sec (c+d x) \tan (c+d x) \, dx+\left (4 a b^3\right ) \int \sec (c+d x) \tan ^2(c+d x) \, dx+b^4 \int \sec (c+d x) \tan ^3(c+d x) \, dx \\ & = -\frac {a^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}+\frac {2 a b^3 \sec (c+d x) \tan (c+d x)}{d}-\left (2 a b^3\right ) \int \sec (c+d x) \, dx+\frac {\left (6 a^2 b^2\right ) \text {Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}+\frac {b^4 \text {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {a^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^3 b \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a b^3 \text {arctanh}(\sin (c+d x))}{d}+\frac {6 a^2 b^2 \sec (c+d x)}{d}-\frac {b^4 \sec (c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d}+\frac {2 a b^3 \sec (c+d x) \tan (c+d x)}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(352\) vs. \(2(118)=236\).
Time = 6.85 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.98 \[ \int \csc (c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {72 a^2 b^2-10 b^4-12 a^4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-48 a^3 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 a b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 a^4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+48 a^3 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-24 a b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {12 a b^3}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {b^4}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+2 b^2 \left (36 a^2-b^2+2 b^2 \cos (c+d x)+\left (36 a^2-5 b^2\right ) \cos (2 (c+d x))\right ) \sec ^3(c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right )-\frac {12 a b^3}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {b^4}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}}{12 d} \]
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Time = 1.37 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.44
method | result | size |
derivativedivides | \(\frac {b^{4} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+4 a \,b^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {6 a^{2} b^{2}}{\cos \left (d x +c \right )}+4 a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}\) | \(170\) |
default | \(\frac {b^{4} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+4 a \,b^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {6 a^{2} b^{2}}{\cos \left (d x +c \right )}+4 a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}\) | \(170\) |
risch | \(-\frac {2 b^{2} {\mathrm e}^{i \left (d x +c \right )} \left (-18 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+6 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}-36 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+2 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-18 a^{2}+3 b^{2}-6 i a b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {2 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {2 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}\) | \(257\) |
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Time = 0.32 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.48 \[ \int \csc (c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {3 \, a^{4} \cos \left (d x + c\right )^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, a^{4} \cos \left (d x + c\right )^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 12 \, a b^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 6 \, {\left (2 \, a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) + 6 \, {\left (2 \, a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, b^{4} - 6 \, {\left (6 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2}}{6 \, d \cos \left (d x + c\right )^{3}} \]
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\[ \int \csc (c+d x) (a+b \tan (c+d x))^4 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{4} \csc {\left (c + d x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.18 \[ \int \csc (c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {3 \, a b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 6 \, a^{3} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 3 \, a^{4} \log \left (\cot \left (d x + c\right ) + \csc \left (d x + c\right )\right ) - \frac {18 \, a^{2} b^{2}}{\cos \left (d x + c\right )} + \frac {{\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} b^{4}}{\cos \left (d x + c\right )^{3}}}{3 \, d} \]
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Time = 1.04 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.64 \[ \int \csc (c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {3 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 6 \, {\left (2 \, a^{3} b - a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 6 \, {\left (2 \, a^{3} b - a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {4 \, {\left (3 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 18 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, a^{2} b^{2} + b^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{3 \, d} \]
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Time = 6.07 (sec) , antiderivative size = 496, normalized size of antiderivative = 4.20 \[ \int \csc (c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {12\,a^2\,b^2-\frac {4\,b^4}{3}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (4\,b^4-24\,a^2\,b^2\right )+12\,a^2\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-4\,a\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {a\,b\,\mathrm {atan}\left (\frac {a\,b\,\left (2\,a^2-b^2\right )\,\left (4\,a\,b^3-8\,a^3\,b+2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-12\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2-b^2\right )\right )\,2{}\mathrm {i}+a\,b\,\left (2\,a^2-b^2\right )\,\left (4\,a\,b^3-8\,a^3\,b+2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+12\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2-b^2\right )\right )\,2{}\mathrm {i}}{2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (64\,a^6\,b^2-64\,a^4\,b^4+16\,a^2\,b^6\right )+16\,a^7\,b-8\,a^5\,b^3+2\,a\,b\,\left (2\,a^2-b^2\right )\,\left (4\,a\,b^3-8\,a^3\,b+2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-12\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2-b^2\right )\right )-2\,a\,b\,\left (2\,a^2-b^2\right )\,\left (4\,a\,b^3-8\,a^3\,b+2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+12\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2-b^2\right )\right )}\right )\,\left (2\,a^2-b^2\right )\,4{}\mathrm {i}}{d} \]
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