Integrand size = 33, antiderivative size = 209 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^3} \, dx=-\frac {\left (c^3 C-3 B c^2 d-3 c C d^2+B d^3-A \left (c^3-3 c d^2\right )\right ) x}{\left (c^2+d^2\right )^3}+\frac {\left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{\left (c^2+d^2\right )^3 f}-\frac {c^2 C-B c d+A d^2}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {2 c (A-C) d-B \left (c^2-d^2\right )}{\left (c^2+d^2\right )^2 f (c+d \tan (e+f x))} \]
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Time = 0.45 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3709, 3610, 3612, 3611} \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^3} \, dx=-\frac {A d^2-B c d+c^2 C}{2 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}-\frac {2 c d (A-C)-B \left (c^2-d^2\right )}{f \left (c^2+d^2\right )^2 (c+d \tan (e+f x))}+\frac {\left (d (A-C) \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^3}-\frac {x \left (-A \left (c^3-3 c d^2\right )-3 B c^2 d+B d^3+c^3 C-3 c C d^2\right )}{\left (c^2+d^2\right )^3} \]
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Rule 3610
Rule 3611
Rule 3612
Rule 3709
Rubi steps \begin{align*} \text {integral}& = -\frac {c^2 C-B c d+A d^2}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac {\int \frac {A c-c C+B d+(B c-(A-C) d) \tan (e+f x)}{(c+d \tan (e+f x))^2} \, dx}{c^2+d^2} \\ & = -\frac {c^2 C-B c d+A d^2}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {2 c (A-C) d-B \left (c^2-d^2\right )}{\left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac {\int \frac {-c^2 C+2 B c d+C d^2+A \left (c^2-d^2\right )-\left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{\left (c^2+d^2\right )^2} \\ & = \frac {\left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right ) x}{\left (c^2+d^2\right )^3}-\frac {c^2 C-B c d+A d^2}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {2 c (A-C) d-B \left (c^2-d^2\right )}{\left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac {\left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{\left (c^2+d^2\right )^3} \\ & = \frac {\left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right ) x}{\left (c^2+d^2\right )^3}+\frac {\left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{\left (c^2+d^2\right )^3 f}-\frac {c^2 C-B c d+A d^2}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {2 c (A-C) d-B \left (c^2-d^2\right )}{\left (c^2+d^2\right )^2 f (c+d \tan (e+f x))} \\ \end{align*}
Result contains complex when optimal does not.
Time = 5.52 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.25 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^3} \, dx=-\frac {\frac {C}{(c+d \tan (e+f x))^2}+B \left (\frac {i \log (i-\tan (e+f x))}{(c+i d)^2}-\frac {i \log (i+\tan (e+f x))}{(c-i d)^2}+\frac {2 d \left (-2 c \log (c+d \tan (e+f x))+\frac {c^2+d^2}{c+d \tan (e+f x)}\right )}{\left (c^2+d^2\right )^2}\right )-(B c+(-A+C) d) \left (\frac {i \log (i-\tan (e+f x))}{(c+i d)^3}-\frac {\log (i+\tan (e+f x))}{(i c+d)^3}+\frac {d \left (\left (-6 c^2+2 d^2\right ) \log (c+d \tan (e+f x))+\frac {\left (c^2+d^2\right ) \left (5 c^2+d^2+4 c d \tan (e+f x)\right )}{(c+d \tan (e+f x))^2}\right )}{\left (c^2+d^2\right )^3}\right )}{2 d f} \]
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Time = 0.14 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.25
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (-3 A \,c^{2} d +A \,d^{3}+B \,c^{3}-3 B c \,d^{2}+3 C \,c^{2} d -C \,d^{3}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (A \,c^{3}-3 A c \,d^{2}+3 B \,c^{2} d -B \,d^{3}-c^{3} C +3 C c \,d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}-\frac {A \,d^{2}-B c d +c^{2} C}{2 \left (c^{2}+d^{2}\right ) d \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {2 A d c -B \,c^{2}+d^{2} B -2 C c d}{\left (c^{2}+d^{2}\right )^{2} \left (c +d \tan \left (f x +e \right )\right )}+\frac {\left (3 A \,c^{2} d -A \,d^{3}-B \,c^{3}+3 B c \,d^{2}-3 C \,c^{2} d +C \,d^{3}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}}{f}\) | \(262\) |
default | \(\frac {\frac {\frac {\left (-3 A \,c^{2} d +A \,d^{3}+B \,c^{3}-3 B c \,d^{2}+3 C \,c^{2} d -C \,d^{3}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (A \,c^{3}-3 A c \,d^{2}+3 B \,c^{2} d -B \,d^{3}-c^{3} C +3 C c \,d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}-\frac {A \,d^{2}-B c d +c^{2} C}{2 \left (c^{2}+d^{2}\right ) d \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {2 A d c -B \,c^{2}+d^{2} B -2 C c d}{\left (c^{2}+d^{2}\right )^{2} \left (c +d \tan \left (f x +e \right )\right )}+\frac {\left (3 A \,c^{2} d -A \,d^{3}-B \,c^{3}+3 B c \,d^{2}-3 C \,c^{2} d +C \,d^{3}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}}{f}\) | \(262\) |
norman | \(\frac {\frac {\left (A \,c^{3}-3 A c \,d^{2}+3 B \,c^{2} d -B \,d^{3}-c^{3} C +3 C c \,d^{2}\right ) c^{2} x}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (c^{2}+d^{2}\right )}+\frac {d^{2} \left (A \,c^{3}-3 A c \,d^{2}+3 B \,c^{2} d -B \,d^{3}-c^{3} C +3 C c \,d^{2}\right ) x \tan \left (f x +e \right )^{2}}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (c^{2}+d^{2}\right )}-\frac {5 A \,c^{2} d^{3}+A \,d^{5}-3 B \,c^{3} d^{2}+B c \,d^{4}+C \,c^{4} d -3 C \,c^{2} d^{3}}{2 f \,d^{2} \left (c^{4}+2 c^{2} d^{2}+d^{4}\right )}-\frac {\left (2 A c \,d^{3}-B \,c^{2} d^{2}+d^{4} B -2 c C \,d^{3}\right ) \tan \left (f x +e \right )}{f d \left (c^{4}+2 c^{2} d^{2}+d^{4}\right )}+\frac {2 d \left (A \,c^{3}-3 A c \,d^{2}+3 B \,c^{2} d -B \,d^{3}-c^{3} C +3 C c \,d^{2}\right ) c x \tan \left (f x +e \right )}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (c^{2}+d^{2}\right )}}{\left (c +d \tan \left (f x +e \right )\right )^{2}}+\frac {\left (3 A \,c^{2} d -A \,d^{3}-B \,c^{3}+3 B c \,d^{2}-3 C \,c^{2} d +C \,d^{3}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{f \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}-\frac {\left (3 A \,c^{2} d -A \,d^{3}-B \,c^{3}+3 B c \,d^{2}-3 C \,c^{2} d +C \,d^{3}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}\) | \(529\) |
risch | \(\text {Expression too large to display}\) | \(1176\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1348\) |
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Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (207) = 414\).
Time = 0.30 (sec) , antiderivative size = 566, normalized size of antiderivative = 2.71 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^3} \, dx=-\frac {3 \, C c^{4} d - 5 \, B c^{3} d^{2} + {\left (7 \, A - 3 \, C\right )} c^{2} d^{3} + B c d^{4} + A d^{5} - 2 \, {\left ({\left (A - C\right )} c^{5} + 3 \, B c^{4} d - 3 \, {\left (A - C\right )} c^{3} d^{2} - B c^{2} d^{3}\right )} f x - {\left (C c^{4} d - 3 \, B c^{3} d^{2} + 5 \, {\left (A - C\right )} c^{2} d^{3} + 3 \, B c d^{4} - A d^{5} + 2 \, {\left ({\left (A - C\right )} c^{3} d^{2} + 3 \, B c^{2} d^{3} - 3 \, {\left (A - C\right )} c d^{4} - B d^{5}\right )} f x\right )} \tan \left (f x + e\right )^{2} + {\left (B c^{5} - 3 \, {\left (A - C\right )} c^{4} d - 3 \, B c^{3} d^{2} + {\left (A - C\right )} c^{2} d^{3} + {\left (B c^{3} d^{2} - 3 \, {\left (A - C\right )} c^{2} d^{3} - 3 \, B c d^{4} + {\left (A - C\right )} d^{5}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (B c^{4} d - 3 \, {\left (A - C\right )} c^{3} d^{2} - 3 \, B c^{2} d^{3} + {\left (A - C\right )} c d^{4}\right )} \tan \left (f x + e\right )\right )} \log \left (\frac {d^{2} \tan \left (f x + e\right )^{2} + 2 \, c d \tan \left (f x + e\right ) + c^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 2 \, {\left (C c^{5} - 2 \, B c^{4} d + 3 \, {\left (A - C\right )} c^{3} d^{2} + 3 \, B c^{2} d^{3} - {\left (3 \, A - 2 \, C\right )} c d^{4} - B d^{5} + 2 \, {\left ({\left (A - C\right )} c^{4} d + 3 \, B c^{3} d^{2} - 3 \, {\left (A - C\right )} c^{2} d^{3} - B c d^{4}\right )} f x\right )} \tan \left (f x + e\right )}{2 \, {\left ({\left (c^{6} d^{2} + 3 \, c^{4} d^{4} + 3 \, c^{2} d^{6} + d^{8}\right )} f \tan \left (f x + e\right )^{2} + 2 \, {\left (c^{7} d + 3 \, c^{5} d^{3} + 3 \, c^{3} d^{5} + c d^{7}\right )} f \tan \left (f x + e\right ) + {\left (c^{8} + 3 \, c^{6} d^{2} + 3 \, c^{4} d^{4} + c^{2} d^{6}\right )} f\right )}} \]
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Exception generated. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^3} \, dx=\text {Exception raised: AttributeError} \]
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none
Time = 0.33 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.76 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^3} \, dx=\frac {\frac {2 \, {\left ({\left (A - C\right )} c^{3} + 3 \, B c^{2} d - 3 \, {\left (A - C\right )} c d^{2} - B d^{3}\right )} {\left (f x + e\right )}}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {2 \, {\left (B c^{3} - 3 \, {\left (A - C\right )} c^{2} d - 3 \, B c d^{2} + {\left (A - C\right )} d^{3}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} + \frac {{\left (B c^{3} - 3 \, {\left (A - C\right )} c^{2} d - 3 \, B c d^{2} + {\left (A - C\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {C c^{4} - 3 \, B c^{3} d + {\left (5 \, A - 3 \, C\right )} c^{2} d^{2} + B c d^{3} + A d^{4} - 2 \, {\left (B c^{2} d^{2} - 2 \, {\left (A - C\right )} c d^{3} - B d^{4}\right )} \tan \left (f x + e\right )}{c^{6} d + 2 \, c^{4} d^{3} + c^{2} d^{5} + {\left (c^{4} d^{3} + 2 \, c^{2} d^{5} + d^{7}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (c^{5} d^{2} + 2 \, c^{3} d^{4} + c d^{6}\right )} \tan \left (f x + e\right )}}{2 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (207) = 414\).
Time = 0.69 (sec) , antiderivative size = 531, normalized size of antiderivative = 2.54 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^3} \, dx=\frac {\frac {2 \, {\left (A c^{3} - C c^{3} + 3 \, B c^{2} d - 3 \, A c d^{2} + 3 \, C c d^{2} - B d^{3}\right )} {\left (f x + e\right )}}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} + \frac {{\left (B c^{3} - 3 \, A c^{2} d + 3 \, C c^{2} d - 3 \, B c d^{2} + A d^{3} - C d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {2 \, {\left (B c^{3} d - 3 \, A c^{2} d^{2} + 3 \, C c^{2} d^{2} - 3 \, B c d^{3} + A d^{4} - C d^{4}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{c^{6} d + 3 \, c^{4} d^{3} + 3 \, c^{2} d^{5} + d^{7}} + \frac {3 \, B c^{3} d^{3} \tan \left (f x + e\right )^{2} - 9 \, A c^{2} d^{4} \tan \left (f x + e\right )^{2} + 9 \, C c^{2} d^{4} \tan \left (f x + e\right )^{2} - 9 \, B c d^{5} \tan \left (f x + e\right )^{2} + 3 \, A d^{6} \tan \left (f x + e\right )^{2} - 3 \, C d^{6} \tan \left (f x + e\right )^{2} + 8 \, B c^{4} d^{2} \tan \left (f x + e\right ) - 22 \, A c^{3} d^{3} \tan \left (f x + e\right ) + 22 \, C c^{3} d^{3} \tan \left (f x + e\right ) - 18 \, B c^{2} d^{4} \tan \left (f x + e\right ) + 2 \, A c d^{5} \tan \left (f x + e\right ) - 2 \, C c d^{5} \tan \left (f x + e\right ) - 2 \, B d^{6} \tan \left (f x + e\right ) - C c^{6} + 6 \, B c^{5} d - 14 \, A c^{4} d^{2} + 11 \, C c^{4} d^{2} - 7 \, B c^{3} d^{3} - 3 \, A c^{2} d^{4} - B c d^{5} - A d^{6}}{{\left (c^{6} d + 3 \, c^{4} d^{3} + 3 \, c^{2} d^{5} + d^{7}\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{2}}}{2 \, f} \]
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Time = 10.86 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.56 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^3} \, dx=-\frac {\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (B\,d^3+2\,A\,c\,d^2-B\,c^2\,d-2\,C\,c\,d^2\right )}{c^4+2\,c^2\,d^2+d^4}+\frac {A\,d^4+C\,c^4+5\,A\,c^2\,d^2-3\,C\,c^2\,d^2+B\,c\,d^3-3\,B\,c^3\,d}{2\,d\,\left (c^4+2\,c^2\,d^2+d^4\right )}}{f\,\left (c^2+2\,c\,d\,\mathrm {tan}\left (e+f\,x\right )+d^2\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (B-A\,1{}\mathrm {i}+C\,1{}\mathrm {i}\right )}{2\,f\,\left (-c^3-c^2\,d\,3{}\mathrm {i}+3\,c\,d^2+d^3\,1{}\mathrm {i}\right )}-\frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (B\,c^3+\left (3\,C-3\,A\right )\,c^2\,d-3\,B\,c\,d^2+\left (A-C\right )\,d^3\right )}{f\,\left (c^6+3\,c^4\,d^2+3\,c^2\,d^4+d^6\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (C-A+B\,1{}\mathrm {i}\right )}{2\,f\,\left (-c^3\,1{}\mathrm {i}-3\,c^2\,d+c\,d^2\,3{}\mathrm {i}+d^3\right )} \]
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