Integrand size = 25, antiderivative size = 144 \[ \int \frac {(a+b \tan (e+f x))^3}{c+d \tan (e+f x)} \, dx=\frac {\left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right ) x}{c^2+d^2}-\frac {\left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) \log (\cos (e+f x))}{\left (c^2+d^2\right ) f}-\frac {(b c-a d)^3 \log (c+d \tan (e+f x))}{d^2 \left (c^2+d^2\right ) f}+\frac {b^2 (a+b \tan (e+f x))}{d f} \]
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Time = 0.30 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3647, 3707, 3698, 31, 3556} \[ \int \frac {(a+b \tan (e+f x))^3}{c+d \tan (e+f x)} \, dx=-\frac {\left (a^3 (-d)+3 a^2 b c+3 a b^2 d-b^3 c\right ) \log (\cos (e+f x))}{f \left (c^2+d^2\right )}+\frac {x \left (a^3 c+3 a^2 b d-3 a b^2 c-b^3 d\right )}{c^2+d^2}+\frac {b^2 (a+b \tan (e+f x))}{d f}-\frac {(b c-a d)^3 \log (c+d \tan (e+f x))}{d^2 f \left (c^2+d^2\right )} \]
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Rule 31
Rule 3556
Rule 3647
Rule 3698
Rule 3707
Rubi steps \begin{align*} \text {integral}& = \frac {b^2 (a+b \tan (e+f x))}{d f}+\frac {\int \frac {-b^3 c+a^3 d+b \left (3 a^2-b^2\right ) d \tan (e+f x)-b^2 (b c-3 a d) \tan ^2(e+f x)}{c+d \tan (e+f x)} \, dx}{d} \\ & = \frac {\left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right ) x}{c^2+d^2}+\frac {b^2 (a+b \tan (e+f x))}{d f}-\frac {(b c-a d)^3 \int \frac {1+\tan ^2(e+f x)}{c+d \tan (e+f x)} \, dx}{d \left (c^2+d^2\right )}+\frac {\left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) \int \tan (e+f x) \, dx}{c^2+d^2} \\ & = \frac {\left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right ) x}{c^2+d^2}-\frac {\left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) \log (\cos (e+f x))}{\left (c^2+d^2\right ) f}+\frac {b^2 (a+b \tan (e+f x))}{d f}-\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{c+x} \, dx,x,d \tan (e+f x)\right )}{d^2 \left (c^2+d^2\right ) f} \\ & = \frac {\left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right ) x}{c^2+d^2}-\frac {\left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) \log (\cos (e+f x))}{\left (c^2+d^2\right ) f}-\frac {(b c-a d)^3 \log (c+d \tan (e+f x))}{d^2 \left (c^2+d^2\right ) f}+\frac {b^2 (a+b \tan (e+f x))}{d f} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.80 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b \tan (e+f x))^3}{c+d \tan (e+f x)} \, dx=\frac {\frac {(a+i b)^3 \log (i-\tan (e+f x))}{i c-d}-\frac {(i a+b)^3 \log (i+\tan (e+f x))}{c-i d}+\frac {2 (-b c+a d)^3 \log (c+d \tan (e+f x))}{d^2 \left (c^2+d^2\right )}+\frac {2 b^2 (a+b \tan (e+f x))}{d}}{2 f} \]
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Time = 0.32 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(\frac {\frac {b^{3} \tan \left (f x +e \right )}{d}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{d^{2} \left (c^{2}+d^{2}\right )}+\frac {\frac {\left (-a^{3} d +3 a^{2} b c +3 a \,b^{2} d -b^{3} c \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a^{3} c +3 a^{2} b d -3 a \,b^{2} c -b^{3} d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{c^{2}+d^{2}}}{f}\) | \(164\) |
default | \(\frac {\frac {b^{3} \tan \left (f x +e \right )}{d}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{d^{2} \left (c^{2}+d^{2}\right )}+\frac {\frac {\left (-a^{3} d +3 a^{2} b c +3 a \,b^{2} d -b^{3} c \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a^{3} c +3 a^{2} b d -3 a \,b^{2} c -b^{3} d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{c^{2}+d^{2}}}{f}\) | \(164\) |
norman | \(\frac {\left (a^{3} c +3 a^{2} b d -3 a \,b^{2} c -b^{3} d \right ) x}{c^{2}+d^{2}}+\frac {b^{3} \tan \left (f x +e \right )}{d f}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right ) d^{2} f}-\frac {\left (a^{3} d -3 a^{2} b c -3 a \,b^{2} d +b^{3} c \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \left (c^{2}+d^{2}\right )}\) | \(168\) |
parallelrisch | \(-\frac {-2 a^{3} c \,d^{2} f x -6 a^{2} b \,d^{3} f x +6 a \,b^{2} c \,d^{2} f x +2 b^{3} d^{3} f x +\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a^{3} d^{3}-3 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a^{2} b c \,d^{2}-3 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a \,b^{2} d^{3}+\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) b^{3} c \,d^{2}-2 \ln \left (c +d \tan \left (f x +e \right )\right ) a^{3} d^{3}+6 \ln \left (c +d \tan \left (f x +e \right )\right ) a^{2} b c \,d^{2}-6 \ln \left (c +d \tan \left (f x +e \right )\right ) a \,b^{2} c^{2} d +2 \ln \left (c +d \tan \left (f x +e \right )\right ) b^{3} c^{3}-2 b^{3} c^{2} d \tan \left (f x +e \right )-2 b^{3} d^{3} \tan \left (f x +e \right )}{2 \left (c^{2}+d^{2}\right ) d^{2} f}\) | \(250\) |
risch | \(\frac {6 i b^{2} a x}{d}+\frac {6 i a^{2} b c e}{\left (c^{2}+d^{2}\right ) f}-\frac {a^{3} x}{i d -c}+\frac {3 x a \,b^{2}}{i d -c}-\frac {2 i d \,a^{3} e}{\left (c^{2}+d^{2}\right ) f}-\frac {6 i a \,b^{2} c^{2} x}{\left (c^{2}+d^{2}\right ) d}+\frac {6 i a^{2} b c x}{c^{2}+d^{2}}-\frac {2 i b^{3} c x}{d^{2}}-\frac {2 i b^{3} c e}{d^{2} f}-\frac {2 i d \,a^{3} x}{c^{2}+d^{2}}+\frac {3 i x \,a^{2} b}{i d -c}+\frac {2 i b^{3} c^{3} e}{\left (c^{2}+d^{2}\right ) d^{2} f}+\frac {6 i b^{2} a e}{d f}-\frac {i x \,b^{3}}{i d -c}+\frac {2 i b^{3} c^{3} x}{\left (c^{2}+d^{2}\right ) d^{2}}+\frac {2 i b^{3}}{f d \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}-\frac {6 i a \,b^{2} c^{2} e}{\left (c^{2}+d^{2}\right ) d f}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a}{d f}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) c}{d^{2} f}+\frac {d \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right ) a^{3}}{\left (c^{2}+d^{2}\right ) f}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right ) a^{2} b c}{\left (c^{2}+d^{2}\right ) f}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right ) a \,b^{2} c^{2}}{\left (c^{2}+d^{2}\right ) d f}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right ) b^{3} c^{3}}{\left (c^{2}+d^{2}\right ) d^{2} f}\) | \(563\) |
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Time = 0.30 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b \tan (e+f x))^3}{c+d \tan (e+f x)} \, dx=\frac {2 \, {\left ({\left (a^{3} - 3 \, a b^{2}\right )} c d^{2} + {\left (3 \, a^{2} b - b^{3}\right )} d^{3}\right )} f x - {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\frac {d^{2} \tan \left (f x + e\right )^{2} + 2 \, c d \tan \left (f x + e\right ) + c^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \, {\left (b^{3} c^{2} d + b^{3} d^{3}\right )} \tan \left (f x + e\right )}{2 \, {\left (c^{2} d^{2} + d^{4}\right )} f} \]
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Result contains complex when optimal does not.
Time = 0.78 (sec) , antiderivative size = 1712, normalized size of antiderivative = 11.89 \[ \int \frac {(a+b \tan (e+f x))^3}{c+d \tan (e+f x)} \, dx=\text {Too large to display} \]
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Time = 0.37 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b \tan (e+f x))^3}{c+d \tan (e+f x)} \, dx=\frac {\frac {2 \, b^{3} \tan \left (f x + e\right )}{d} + \frac {2 \, {\left ({\left (a^{3} - 3 \, a b^{2}\right )} c + {\left (3 \, a^{2} b - b^{3}\right )} d\right )} {\left (f x + e\right )}}{c^{2} + d^{2}} - \frac {2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{2} d^{2} + d^{4}} + \frac {{\left ({\left (3 \, a^{2} b - b^{3}\right )} c - {\left (a^{3} - 3 \, a b^{2}\right )} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{2} + d^{2}}}{2 \, f} \]
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Time = 0.64 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b \tan (e+f x))^3}{c+d \tan (e+f x)} \, dx=\frac {\frac {2 \, b^{3} \tan \left (f x + e\right )}{d} + \frac {2 \, {\left (a^{3} c - 3 \, a b^{2} c + 3 \, a^{2} b d - b^{3} d\right )} {\left (f x + e\right )}}{c^{2} + d^{2}} + \frac {{\left (3 \, a^{2} b c - b^{3} c - a^{3} d + 3 \, a b^{2} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{2} + d^{2}} - \frac {2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{c^{2} d^{2} + d^{4}}}{2 \, f} \]
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Time = 6.53 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.23 \[ \int \frac {(a+b \tan (e+f x))^3}{c+d \tan (e+f x)} \, dx=\frac {b^3\,\mathrm {tan}\left (e+f\,x\right )}{d\,f}+\frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{f\,\left (c^2\,d^2+d^4\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )}{2\,f\,\left (c+d\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}{2\,f\,\left (d+c\,1{}\mathrm {i}\right )} \]
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