Optimal. Leaf size=177 \[ \frac {\left (a c^3+3 b c^2 d-3 a c d^2-b d^3\right ) x}{\left (c^2+d^2\right )^3}+\frac {\left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{\left (c^2+d^2\right )^3 f}+\frac {b c-a d}{2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {2 a c d-b \left (c^2-d^2\right )}{\left (c^2+d^2\right )^2 f (c+d \tan (e+f x))} \]
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Rubi [A]
time = 0.20, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3610, 3612,
3611} \begin {gather*} \frac {b c-a d}{2 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}-\frac {2 a c d-b \left (c^2-d^2\right )}{f \left (c^2+d^2\right )^2 (c+d \tan (e+f x))}+\frac {\left (a 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))}{f \left (c^2+d^2\right )^3}+\frac {x \left (a c^3-3 a c d^2+3 b c^2 d-b d^3\right )}{\left (c^2+d^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3610
Rule 3611
Rule 3612
Rubi steps
\begin {align*} \int \frac {a+b \tan (e+f x)}{(c+d \tan (e+f x))^3} \, dx &=\frac {b c-a d}{2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac {\int \frac {a c+b d+(b c-a d) \tan (e+f x)}{(c+d \tan (e+f x))^2} \, dx}{c^2+d^2}\\ &=\frac {b c-a d}{2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {2 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 {2 b c d+a \left (c^2-d^2\right )-\left (2 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+3 b c^2 d-3 a c d^2-b d^3\right ) x}{\left (c^2+d^2\right )^3}+\frac {b c-a d}{2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {2 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 (d \left (2 b c d+a \left (c^2-d^2\right )\right )-c \left (-2 a c d+b \left (c^2-d^2\right )\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+3 b c^2 d-3 a c d^2-b d^3\right ) x}{\left (c^2+d^2\right )^3}+\frac {\left (a 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 {b c-a d}{2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {2 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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 4.52, size = 244, normalized size = 1.38 \begin {gather*} \frac {-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 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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 208, normalized size = 1.18
method | result | size |
derivativedivides | \(\frac {\frac {\left (3 a \,c^{2} d -a \,d^{3}-b \,c^{3}+3 b c \,d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}-\frac {a d -b c}{2 \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {2 a c d -b \,c^{2}+b \,d^{2}}{\left (c^{2}+d^{2}\right )^{2} \left (c +d \tan \left (f x +e \right )\right )}+\frac {\frac {\left (-3 a \,c^{2} d +a \,d^{3}+b \,c^{3}-3 b c \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a \,c^{3}-3 a c \,d^{2}+3 b \,c^{2} d -b \,d^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}}{f}\) | \(208\) |
default | \(\frac {\frac {\left (3 a \,c^{2} d -a \,d^{3}-b \,c^{3}+3 b c \,d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}-\frac {a d -b c}{2 \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {2 a c d -b \,c^{2}+b \,d^{2}}{\left (c^{2}+d^{2}\right )^{2} \left (c +d \tan \left (f x +e \right )\right )}+\frac {\frac {\left (-3 a \,c^{2} d +a \,d^{3}+b \,c^{3}-3 b c \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a \,c^{3}-3 a c \,d^{2}+3 b \,c^{2} d -b \,d^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{3}}}{f}\) | \(208\) |
norman | \(\frac {\frac {\left (a \,c^{3}-3 a c \,d^{2}+3 b \,c^{2} d -b \,d^{3}\right ) c^{2} x}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (c^{2}+d^{2}\right )}+\frac {d^{2} \left (a \,c^{3}-3 a c \,d^{2}+3 b \,c^{2} d -b \,d^{3}\right ) x \left (\tan ^{2}\left (f x +e \right )\right )}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (c^{2}+d^{2}\right )}-\frac {3 a \,c^{2} d^{2}+a \,d^{4}-2 b \,c^{3} d}{2 d f \left (c^{4}+2 c^{2} d^{2}+d^{4}\right )}+\frac {d \left (2 a c \,d^{2}-b \,c^{2} d +b \,d^{3}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 f c \left (c^{4}+2 c^{2} d^{2}+d^{4}\right )}+\frac {2 d \left (a \,c^{3}-3 a c \,d^{2}+3 b \,c^{2} d -b \,d^{3}\right ) c x \tan \left (f x +e \right )}{\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (c^{2}+d^{2}\right )}}{\left (c +d \tan \left (f x +e \right )\right )^{2}}+\frac {\left (3 a \,c^{2} d -a \,d^{3}-b \,c^{3}+3 b c \,d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{f \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}-\frac {\left (3 a \,c^{2} d -a \,d^{3}-b \,c^{3}+3 b c \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}\) | \(437\) |
risch | \(\frac {i x b}{3 i c^{2} d -i d^{3}-c^{3}+3 c \,d^{2}}-\frac {a x}{3 i c^{2} d -i d^{3}-c^{3}+3 c \,d^{2}}-\frac {6 i a \,c^{2} d x}{c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}}+\frac {2 i a \,d^{3} x}{c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}}+\frac {2 i b \,c^{3} x}{c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}}-\frac {6 i b c \,d^{2} x}{c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}}-\frac {6 i a \,c^{2} d e}{f \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}+\frac {2 i a \,d^{3} e}{f \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}+\frac {2 i b \,c^{3} e}{f \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}-\frac {6 i b c \,d^{2} e}{f \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}-\frac {2 i \left (2 i a c \,d^{3} {\mathrm e}^{2 i \left (f x +e \right )}-i b \,c^{2} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-a \,d^{4} {\mathrm e}^{2 i \left (f x +e \right )}-3 i a c \,d^{3}+2 i b \,c^{2} d^{2}-b c \,d^{3}+i b \,d^{4} {\mathrm e}^{2 i \left (f x +e \right )}-3 a \,c^{2} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+2 b \,c^{3} d \,{\mathrm e}^{2 i \left (f x +e \right )}-i b \,d^{4}-3 a \,c^{2} d^{2}+2 b \,c^{3} d \right )}{\left (i d +c \right )^{2} \left (-i {\mathrm e}^{2 i \left (f x +e \right )} d +i d +{\mathrm e}^{2 i \left (f x +e \right )} c +c \right )^{2} f \left (-i d +c \right )^{3}}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right ) a \,c^{2} d}{f \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right ) a \,d^{3}}{f \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right ) b \,c^{3}}{f \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right ) b c \,d^{2}}{f \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}\) | \(803\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.67, size = 327, normalized size = 1.85 \begin {gather*} \frac {\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 )}}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {2 \, {\left (b c^{3} - 3 \, a c^{2} d - 3 \, b c d^{2} + a 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 \, a c^{2} d - 3 \, b c d^{2} + a 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 {3 \, b c^{3} - 5 \, a c^{2} d - b c d^{2} - a d^{3} + 2 \, {\left (b c^{2} d - 2 \, a c d^{2} - b d^{3}\right )} \tan \left (f x + e\right )}{c^{6} + 2 \, c^{4} d^{2} + c^{2} d^{4} + {\left (c^{4} d^{2} + 2 \, c^{2} d^{4} + d^{6}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (c^{5} d + 2 \, c^{3} d^{3} + c d^{5}\right )} \tan \left (f x + e\right )}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 491 vs.
\(2 (179) = 358\).
time = 1.06, size = 491, normalized size = 2.77 \begin {gather*} \frac {5 \, b c^{3} d^{2} - 7 \, a c^{2} d^{3} - b c d^{4} - a d^{5} + 2 \, {\left (a c^{5} + 3 \, b c^{4} d - 3 \, a c^{3} d^{2} - b c^{2} d^{3}\right )} f x - {\left (3 \, b c^{3} d^{2} - 5 \, a c^{2} d^{3} - 3 \, b c d^{4} + a d^{5} - 2 \, {\left (a c^{3} d^{2} + 3 \, b c^{2} d^{3} - 3 \, a c d^{4} - b d^{5}\right )} f x\right )} \tan \left (f x + e\right )^{2} - {\left (b c^{5} - 3 \, a c^{4} d - 3 \, b c^{3} d^{2} + a c^{2} d^{3} + {\left (b c^{3} d^{2} - 3 \, a c^{2} d^{3} - 3 \, b c d^{4} + a d^{5}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (b c^{4} d - 3 \, a c^{3} d^{2} - 3 \, b c^{2} d^{3} + a 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 (2 \, b c^{4} d - 3 \, a c^{3} d^{2} - 3 \, b c^{2} d^{3} + 3 \, a c d^{4} + b d^{5} - 2 \, {\left (a c^{4} d + 3 \, b c^{3} d^{2} - 3 \, a 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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 422 vs.
\(2 (179) = 358\).
time = 0.65, size = 422, normalized size = 2.38 \begin {gather*} \frac {\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 )}}{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 \, b c d^{2} + a 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 \, b c d^{3} + a 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^{2} \tan \left (f x + e\right )^{2} - 9 \, a c^{2} d^{3} \tan \left (f x + e\right )^{2} - 9 \, b c d^{4} \tan \left (f x + e\right )^{2} + 3 \, a d^{5} \tan \left (f x + e\right )^{2} + 8 \, b c^{4} d \tan \left (f x + e\right ) - 22 \, a c^{3} d^{2} \tan \left (f x + e\right ) - 18 \, b c^{2} d^{3} \tan \left (f x + e\right ) + 2 \, a c d^{4} \tan \left (f x + e\right ) - 2 \, b d^{5} \tan \left (f x + e\right ) + 6 \, b c^{5} - 14 \, a c^{4} d - 7 \, b c^{3} d^{2} - 3 \, a c^{2} d^{3} - b c d^{4} - a d^{5}}{{\left (c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{2}}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.66, size = 279, normalized size = 1.58 \begin {gather*} \frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (\frac {3\,a\,d-b\,c}{{\left (c^2+d^2\right )}^2}-\frac {4\,d^2\,\left (a\,d-b\,c\right )}{{\left (c^2+d^2\right )}^3}\right )}{f}-\frac {\frac {-3\,b\,c^3+5\,a\,c^2\,d+b\,c\,d^2+a\,d^3}{2\,\left (c^4+2\,c^2\,d^2+d^4\right )}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (-b\,c^2\,d+2\,a\,c\,d^2+b\,d^3\right )}{c^4+2\,c^2\,d^2+d^4}}{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}\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 (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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