Integrand size = 25, antiderivative size = 233 \[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx=-\frac {(A-C) \arctan \left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d}+\frac {(A-C) \arctan \left (1+\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d}-\frac {(A-C) \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d}+\frac {(A-C) \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d}-\frac {2 b C}{3 d (b \tan (c+d x))^{3/2}} \]
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Time = 0.37 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3754, 3710, 12, 16, 3557, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx=-\frac {(A-C) \arctan \left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d}+\frac {(A-C) \arctan \left (\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}+1\right )}{\sqrt {2} \sqrt {b} d}-\frac {(A-C) \log \left (\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}+\sqrt {b}\right )}{2 \sqrt {2} \sqrt {b} d}+\frac {(A-C) \log \left (\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}+\sqrt {b}\right )}{2 \sqrt {2} \sqrt {b} d}-\frac {2 b C}{3 d (b \tan (c+d x))^{3/2}} \]
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Rule 12
Rule 16
Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3557
Rule 3710
Rule 3754
Rubi steps \begin{align*} \text {integral}& = b^2 \int \frac {C+A \tan ^2(c+d x)}{(b \tan (c+d x))^{5/2}} \, dx \\ & = -\frac {2 b C}{3 d (b \tan (c+d x))^{3/2}}+\int \frac {b (A-C) \tan (c+d x)}{(b \tan (c+d x))^{3/2}} \, dx \\ & = -\frac {2 b C}{3 d (b \tan (c+d x))^{3/2}}+(b (A-C)) \int \frac {\tan (c+d x)}{(b \tan (c+d x))^{3/2}} \, dx \\ & = -\frac {2 b C}{3 d (b \tan (c+d x))^{3/2}}+(A-C) \int \frac {1}{\sqrt {b \tan (c+d x)}} \, dx \\ & = -\frac {2 b C}{3 d (b \tan (c+d x))^{3/2}}+\frac {(b (A-C)) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = -\frac {2 b C}{3 d (b \tan (c+d x))^{3/2}}+\frac {(2 b (A-C)) \text {Subst}\left (\int \frac {1}{b^2+x^4} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{d} \\ & = -\frac {2 b C}{3 d (b \tan (c+d x))^{3/2}}+\frac {(A-C) \text {Subst}\left (\int \frac {b-x^2}{b^2+x^4} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{d}+\frac {(A-C) \text {Subst}\left (\int \frac {b+x^2}{b^2+x^4} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{d} \\ & = -\frac {2 b C}{3 d (b \tan (c+d x))^{3/2}}+\frac {(A-C) \text {Subst}\left (\int \frac {1}{b-\sqrt {2} \sqrt {b} x+x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 d}+\frac {(A-C) \text {Subst}\left (\int \frac {1}{b+\sqrt {2} \sqrt {b} x+x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 d}-\frac {(A-C) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {b}+2 x}{-b-\sqrt {2} \sqrt {b} x-x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d}-\frac {(A-C) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {b}-2 x}{-b+\sqrt {2} \sqrt {b} x-x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d} \\ & = -\frac {(A-C) \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d}+\frac {(A-C) \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d}-\frac {2 b C}{3 d (b \tan (c+d x))^{3/2}}+\frac {(A-C) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d}-\frac {(A-C) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d} \\ & = -\frac {(A-C) \arctan \left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d}+\frac {(A-C) \arctan \left (1+\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d}-\frac {(A-C) \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d}+\frac {(A-C) \log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d}-\frac {2 b C}{3 d (b \tan (c+d x))^{3/2}} \\ \end{align*}
Time = 0.91 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.64 \[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx=\frac {-8 C \cot (c+d x)-3 \sqrt {2} (A-C) \left (2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right ) \sqrt {\tan (c+d x)}}{12 d \sqrt {b \tan (c+d x)}} \]
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Time = 2.08 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(\frac {2 b \left (-\frac {C}{3 \left (b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {\left (A -C \right ) \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 b^{2}}\right )}{d}\) | \(160\) |
default | \(\frac {2 b \left (-\frac {C}{3 \left (b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {\left (A -C \right ) \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 b^{2}}\right )}{d}\) | \(160\) |
parts | \(\frac {A \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d b}+\frac {C \sqrt {2}\, \left (2 \left (1-\cos \left (d x +c \right )\right )^{2} \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \csc \left (d x +c \right )-3 \ln \left (-\frac {-\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+2 \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sin \left (d x +c \right )-2+2 \cos \left (d x +c \right )+\sin \left (d x +c \right )}{1-\cos \left (d x +c \right )}\right ) \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )-6 \arctan \left (\frac {\sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sin \left (d x +c \right )-1+\cos \left (d x +c \right )}{1-\cos \left (d x +c \right )}\right ) \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+3 \ln \left (\frac {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+2 \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sin \left (d x +c \right )+2-2 \cos \left (d x +c \right )-\sin \left (d x +c \right )}{1-\cos \left (d x +c \right )}\right ) \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )-6 \arctan \left (\frac {\sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sin \left (d x +c \right )+1-\cos \left (d x +c \right )}{1-\cos \left (d x +c \right )}\right ) \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )-2 \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \sin \left (d x +c \right )\right )}{12 d \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right ) \csc \left (d x +c \right )}\, \left (1-\cos \left (d x +c \right )\right ) \sqrt {-\frac {b \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}}\) | \(763\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.00 \[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx=-\frac {3 \, b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} \log \left (b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} - \sqrt {b \tan \left (d x + c\right )} {\left (A - C\right )}\right ) \tan \left (d x + c\right )^{2} + 3 i \, b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} \log \left (i \, b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} - \sqrt {b \tan \left (d x + c\right )} {\left (A - C\right )}\right ) \tan \left (d x + c\right )^{2} - 3 i \, b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} \log \left (-i \, b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} - \sqrt {b \tan \left (d x + c\right )} {\left (A - C\right )}\right ) \tan \left (d x + c\right )^{2} - 3 \, b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} \log \left (-b d \left (-\frac {A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right )^{\frac {1}{4}} - \sqrt {b \tan \left (d x + c\right )} {\left (A - C\right )}\right ) \tan \left (d x + c\right )^{2} + 4 \, \sqrt {b \tan \left (d x + c\right )} C}{6 \, b d \tan \left (d x + c\right )^{2}} \]
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\[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx=\int \frac {A + C \cot ^{2}{\left (c + d x \right )}}{\sqrt {b \tan {\left (c + d x \right )}}}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.77 \[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx=\frac {3 \, {\left (2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {b} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {b}}\right ) + 2 \, \sqrt {2} \sqrt {b} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {b} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {b}}\right ) + \sqrt {2} \sqrt {b} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {b} + b\right ) - \sqrt {2} \sqrt {b} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {b} + b\right )\right )} {\left (A - C\right )} - \frac {8 \, C b^{2}}{\left (b \tan \left (d x + c\right )\right )^{\frac {3}{2}}}}{12 \, b d} \]
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Time = 0.47 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.06 \[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx=\frac {\sqrt {2} {\left (A \sqrt {{\left | b \right |}} - C \sqrt {{\left | b \right |}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{2 \, b d} + \frac {\sqrt {2} {\left (A \sqrt {{\left | b \right |}} - C \sqrt {{\left | b \right |}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{2 \, b d} + \frac {\sqrt {2} {\left (A \sqrt {{\left | b \right |}} - C \sqrt {{\left | b \right |}}\right )} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{4 \, b d} - \frac {\sqrt {2} {\left (A \sqrt {{\left | b \right |}} - C \sqrt {{\left | b \right |}}\right )} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{4 \, b d} - \frac {2 \, C}{3 \, \sqrt {b \tan \left (d x + c\right )} d \tan \left (d x + c\right )} \]
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Time = 13.65 (sec) , antiderivative size = 828, normalized size of antiderivative = 3.55 \[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx=-\frac {2\,C\,b}{3\,d\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )-\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )}{2\,\sqrt {b}\,d}\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}+\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )+\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )}{2\,\sqrt {b}\,d}\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}}{\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )-\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )}{2\,\sqrt {b}\,d}\right )}{2\,\sqrt {b}\,d}-\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )+\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )}{2\,\sqrt {b}\,d}\right )}{2\,\sqrt {b}\,d}}\right )\,\left (A-C\right )\,1{}\mathrm {i}}{\sqrt {b}\,d}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )-\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}\right )}{2\,\sqrt {b}\,d}+\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )+\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}\right )}{2\,\sqrt {b}\,d}}{\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )-\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}-\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (16\,A^2\,b^2\,d^3-32\,A\,C\,b^2\,d^3+16\,C^2\,b^2\,d^3\right )+\frac {{\left (-1\right )}^{1/4}\,\left (A-C\right )\,\left (32\,A\,b^3\,d^4-32\,C\,b^3\,d^4\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}\right )\,1{}\mathrm {i}}{2\,\sqrt {b}\,d}}\right )\,\left (A-C\right )}{\sqrt {b}\,d} \]
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