Integrand size = 25, antiderivative size = 140 \[ \int \frac {(c+d \tan (e+f x))^3}{a+b \tan (e+f x)} \, dx=\frac {\left (a c^3+3 b c^2 d-3 a c d^2-b d^3\right ) x}{a^2+b^2}+\frac {\left (b c^3-3 a c^2 d-3 b c d^2+a d^3\right ) \log (\cos (e+f x))}{\left (a^2+b^2\right ) f}+\frac {(b c-a d)^3 \log (a+b \tan (e+f x))}{b^2 \left (a^2+b^2\right ) f}+\frac {d^2 (c+d \tan (e+f x))}{b f} \]
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Time = 0.32 (sec) , antiderivative size = 140, 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} \[ \int \frac {(c+d \tan (e+f x))^3}{a+b \tan (e+f x)} \, dx=\frac {\left (-3 a c^2 d+a d^3+b c^3-3 b c d^2\right ) \log (\cos (e+f x))}{f \left (a^2+b^2\right )}+\frac {x \left (a c^3-3 a c d^2+3 b c^2 d-b d^3\right )}{a^2+b^2}+\frac {(b c-a d)^3 \log (a+b \tan (e+f x))}{b^2 f \left (a^2+b^2\right )}+\frac {d^2 (c+d \tan (e+f x))}{b f} \]
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Rule 31
Rule 3556
Rule 3647
Rule 3698
Rule 3707
Rubi steps \begin{align*} \text {integral}& = \frac {d^2 (c+d \tan (e+f x))}{b f}+\frac {\int \frac {b c^3-a d^3+b d \left (3 c^2-d^2\right ) \tan (e+f x)+d^2 (3 b c-a d) \tan ^2(e+f x)}{a+b \tan (e+f x)} \, dx}{b} \\ & = \frac {\left (a c^3+3 b c^2 d-3 a c d^2-b d^3\right ) x}{a^2+b^2}+\frac {d^2 (c+d \tan (e+f x))}{b f}+\frac {(b c-a d)^3 \int \frac {1+\tan ^2(e+f x)}{a+b \tan (e+f x)} \, dx}{b \left (a^2+b^2\right )}+\frac {\left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right ) \int \tan (e+f x) \, dx}{a^2+b^2} \\ & = \frac {\left (a c^3+3 b c^2 d-3 a c d^2-b d^3\right ) x}{a^2+b^2}+\frac {\left (b c^3-3 a c^2 d-3 b c d^2+a d^3\right ) \log (\cos (e+f x))}{\left (a^2+b^2\right ) f}+\frac {d^2 (c+d \tan (e+f x))}{b f}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (e+f x)\right )}{b^2 \left (a^2+b^2\right ) f} \\ & = \frac {\left (a c^3+3 b c^2 d-3 a c d^2-b d^3\right ) x}{a^2+b^2}+\frac {\left (b c^3-3 a c^2 d-3 b c d^2+a d^3\right ) \log (\cos (e+f x))}{\left (a^2+b^2\right ) f}+\frac {(b c-a d)^3 \log (a+b \tan (e+f x))}{b^2 \left (a^2+b^2\right ) f}+\frac {d^2 (c+d \tan (e+f x))}{b f} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.90 \[ \int \frac {(c+d \tan (e+f x))^3}{a+b \tan (e+f x)} \, dx=\frac {\frac {(c+i d)^3 \log (i-\tan (e+f x))}{i a-b}-\frac {(c-i d)^3 \log (i+\tan (e+f x))}{i a+b}+\frac {2 (b c-a d)^3 \log (a+b \tan (e+f x))}{b^2 \left (a^2+b^2\right )}+\frac {2 d^2 (c+d \tan (e+f x))}{b}}{2 f} \]
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Time = 0.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.17
method | result | size |
derivativedivides | \(\frac {\frac {d^{3} \tan \left (f x +e \right )}{b}+\frac {\frac {\left (3 a \,c^{2} d -a \,d^{3}-b \,c^{3}+3 b c \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a \,c^{3}-3 a c \,d^{2}+3 b \,c^{2} d -b \,d^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{a^{2}+b^{2}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{b^{2} \left (a^{2}+b^{2}\right )}}{f}\) | \(164\) |
default | \(\frac {\frac {d^{3} \tan \left (f x +e \right )}{b}+\frac {\frac {\left (3 a \,c^{2} d -a \,d^{3}-b \,c^{3}+3 b c \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a \,c^{3}-3 a c \,d^{2}+3 b \,c^{2} d -b \,d^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{a^{2}+b^{2}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{b^{2} \left (a^{2}+b^{2}\right )}}{f}\) | \(164\) |
norman | \(\frac {\left (a \,c^{3}-3 a c \,d^{2}+3 b \,c^{2} d -b \,d^{3}\right ) x}{a^{2}+b^{2}}+\frac {d^{3} \tan \left (f x +e \right )}{b f}+\frac {\left (3 a \,c^{2} d -a \,d^{3}-b \,c^{3}+3 b c \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \left (a^{2}+b^{2}\right )}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right ) b^{2} f}\) | \(171\) |
parallelrisch | \(\frac {2 a \,b^{2} c^{3} f x -6 a \,b^{2} c \,d^{2} f x +6 b^{3} c^{2} d f x -2 b^{3} d^{3} f x +3 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a \,b^{2} c^{2} d -\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a \,b^{2} d^{3}-\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) b^{3} c^{3}+3 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) b^{3} c \,d^{2}-2 \ln \left (a +b \tan \left (f x +e \right )\right ) a^{3} d^{3}+6 \ln \left (a +b \tan \left (f x +e \right )\right ) a^{2} b c \,d^{2}-6 \ln \left (a +b \tan \left (f x +e \right )\right ) a \,b^{2} c^{2} d +2 \ln \left (a +b \tan \left (f x +e \right )\right ) b^{3} c^{3}+2 a^{2} b \,d^{3} \tan \left (f x +e \right )+2 b^{3} d^{3} \tan \left (f x +e \right )}{2 \left (a^{2}+b^{2}\right ) b^{2} f}\) | \(252\) |
risch | \(-\frac {6 i a^{2} c \,d^{2} e}{\left (a^{2}+b^{2}\right ) b f}+\frac {6 i a \,c^{2} d x}{a^{2}+b^{2}}-\frac {x \,c^{3}}{i b -a}+\frac {3 x c \,d^{2}}{i b -a}-\frac {2 i b \,c^{3} x}{a^{2}+b^{2}}+\frac {3 i x \,c^{2} d}{i b -a}-\frac {6 i a^{2} c \,d^{2} x}{\left (a^{2}+b^{2}\right ) b}-\frac {2 i b \,c^{3} e}{\left (a^{2}+b^{2}\right ) f}+\frac {6 i d^{2} c e}{b f}+\frac {6 i a \,c^{2} d e}{\left (a^{2}+b^{2}\right ) f}-\frac {2 i d^{3} a e}{b^{2} f}+\frac {2 i a^{3} d^{3} e}{\left (a^{2}+b^{2}\right ) b^{2} f}+\frac {6 i d^{2} c x}{b}+\frac {2 i a^{3} d^{3} x}{\left (a^{2}+b^{2}\right ) b^{2}}-\frac {i x \,d^{3}}{i b -a}-\frac {2 i d^{3} a x}{b^{2}}+\frac {2 i d^{3}}{f b \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}+\frac {d^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a}{b^{2} f}-\frac {3 d^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) c}{b f}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i b +a}{i b -a}\right ) a^{3} d^{3}}{\left (a^{2}+b^{2}\right ) b^{2} f}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i b +a}{i b -a}\right ) a^{2} c \,d^{2}}{\left (a^{2}+b^{2}\right ) b f}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i b +a}{i b -a}\right ) a \,c^{2} d}{\left (a^{2}+b^{2}\right ) f}+\frac {b \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i b +a}{i b -a}\right ) c^{3}}{\left (a^{2}+b^{2}\right ) f}\) | \(563\) |
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Time = 0.28 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.44 \[ \int \frac {(c+d \tan (e+f x))^3}{a+b \tan (e+f x)} \, dx=\frac {2 \, {\left (a^{2} b + b^{3}\right )} d^{3} \tan \left (f x + e\right ) + 2 \, {\left (a b^{2} c^{3} + 3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} - b^{3} d^{3}\right )} f x + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\frac {b^{2} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) - {\left (3 \, {\left (a^{2} b + b^{3}\right )} c d^{2} - {\left (a^{3} + a b^{2}\right )} d^{3}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right )}{2 \, {\left (a^{2} b^{2} + b^{4}\right )} f} \]
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Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 1712, normalized size of antiderivative = 12.23 \[ \int \frac {(c+d \tan (e+f x))^3}{a+b \tan (e+f x)} \, dx=\text {Too large to display} \]
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Time = 0.38 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.22 \[ \int \frac {(c+d \tan (e+f x))^3}{a+b \tan (e+f x)} \, dx=\frac {\frac {2 \, d^{3} \tan \left (f x + e\right )}{b} + \frac {2 \, {\left (a c^{3} + 3 \, b c^{2} d - 3 \, a c d^{2} - b d^{3}\right )} {\left (f x + e\right )}}{a^{2} + b^{2}} + \frac {2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{2} b^{2} + b^{4}} - \frac {{\left (b c^{3} - 3 \, a c^{2} d - 3 \, b c d^{2} + a d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{2} + b^{2}}}{2 \, f} \]
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Time = 0.64 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.23 \[ \int \frac {(c+d \tan (e+f x))^3}{a+b \tan (e+f x)} \, dx=\frac {\frac {2 \, d^{3} \tan \left (f x + e\right )}{b} + \frac {2 \, {\left (a c^{3} + 3 \, b c^{2} d - 3 \, a c d^{2} - b d^{3}\right )} {\left (f x + e\right )}}{a^{2} + b^{2}} - \frac {{\left (b c^{3} - 3 \, a c^{2} d - 3 \, b c d^{2} + a d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{2} b^{2} + b^{4}}}{2 \, f} \]
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Time = 6.64 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.27 \[ \int \frac {(c+d \tan (e+f x))^3}{a+b \tan (e+f x)} \, dx=\frac {d^3\,\mathrm {tan}\left (e+f\,x\right )}{b\,f}-\frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{f\,\left (a^2\,b^2+b^4\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (-c^3\,1{}\mathrm {i}+3\,c^2\,d+c\,d^2\,3{}\mathrm {i}-d^3\right )}{2\,f\,\left (a+b\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (-c^3+c^2\,d\,3{}\mathrm {i}+3\,c\,d^2-d^3\,1{}\mathrm {i}\right )}{2\,f\,\left (b+a\,1{}\mathrm {i}\right )} \]
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