Optimal. Leaf size=183 \[ \frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a^2 \left (a^4+3 a^2 b^2+6 b^4\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^3 d}-\frac {a^2 \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a^3 \left (a^2+3 b^2\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
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Rubi [A]
time = 0.22, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3646, 3716,
3707, 3698, 31, 3556} \begin {gather*} -\frac {a^2 \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {a x \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3}+\frac {a^2 \left (a^4+3 a^2 b^2+6 b^4\right ) \log (a+b \tan (c+d x))}{b^3 d \left (a^2+b^2\right )^3}+\frac {a^3 \left (a^2+3 b^2\right )}{b^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 3556
Rule 3646
Rule 3698
Rule 3707
Rule 3716
Rubi steps
\begin {align*} \int \frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=-\frac {a^2 \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {\tan (c+d x) \left (2 a^2-2 a b \tan (c+d x)+2 \left (a^2+b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=-\frac {a^2 \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a^3 \left (a^2+3 b^2\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {2 a^2 \left (a^2+3 b^2\right )-4 a b^3 \tan (c+d x)+2 \left (a^2+b^2\right )^2 \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {a^2 \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a^3 \left (a^2+3 b^2\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left (b \left (3 a^2-b^2\right )\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^3}+\frac {\left (a^2 \left (a^4+3 a^2 b^2+6 b^4\right )\right ) \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^2 \left (a^2+b^2\right )^3}\\ &=\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac {a^2 \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a^3 \left (a^2+3 b^2\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left (a^2 \left (a^4+3 a^2 b^2+6 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^3 \left (a^2+b^2\right )^3 d}\\ &=\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a^2 \left (a^4+3 a^2 b^2+6 b^4\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^3 d}-\frac {a^2 \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a^3 \left (a^2+3 b^2\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.20, size = 351, normalized size = 1.92 \begin {gather*} \frac {\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (-a^4 b^2 \left (a^2+b^2\right )-2 a^2 b \left (a^2+b^2\right ) \left (a^2+4 b^2\right ) \sin (c+d x) (a \cos (c+d x)+b \sin (c+d x))+2 a b^3 \left (a^2-3 b^2\right ) (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2+2 i a^2 \left (a^4+3 a^2 b^2+6 b^4\right ) (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2-2 i a^2 \left (a^4+3 a^2 b^2+6 b^4\right ) \text {ArcTan}(\tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2-2 \left (a^2+b^2\right )^3 \log (\cos (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2+a^2 \left (a^4+3 a^2 b^2+6 b^4\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2\right )}{2 b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 169, normalized size = 0.92
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (-3 a^{2} b +b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (a^{3}-3 b^{2} a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {a^{4}}{2 b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{2} \left (a^{4}+3 a^{2} b^{2}+6 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} b^{3}}+\frac {2 a^{3} \left (a^{2}+2 b^{2}\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(169\) |
default | \(\frac {\frac {\frac {\left (-3 a^{2} b +b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (a^{3}-3 b^{2} a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {a^{4}}{2 b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{2} \left (a^{4}+3 a^{2} b^{2}+6 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} b^{3}}+\frac {2 a^{3} \left (a^{2}+2 b^{2}\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(169\) |
norman | \(\frac {\frac {\left (a^{2}-3 b^{2}\right ) a^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {b^{2} \left (a^{2}-3 b^{2}\right ) a x \left (\tan ^{2}\left (d x +c \right )\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {a^{2} \left (3 a^{4}+7 a^{2} b^{2}\right )}{2 d \,b^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 a \left (a^{4}+2 a^{2} b^{2}\right ) \tan \left (d x +c \right )}{d \,b^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b \left (a^{2}-3 b^{2}\right ) a^{2} x \tan \left (d x +c \right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{2} \left (a^{4}+3 a^{2} b^{2}+6 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d \,b^{3}}-\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(352\) |
risch | \(-\frac {x}{3 i b \,a^{2}-i b^{3}-a^{3}+3 b^{2} a}+\frac {2 i x}{b^{3}}+\frac {2 i c}{d \,b^{3}}-\frac {2 i a^{6} x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b^{3}}-\frac {2 i a^{6} c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b^{3} d}-\frac {6 i a^{4} x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b}-\frac {6 i a^{4} c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b d}-\frac {12 i a^{2} b x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {12 i a^{2} b c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {2 i a^{3} \left (i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+2 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+4 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+i a^{3}+4 i a \,b^{2}-a^{2} b -4 b^{3}\right )}{\left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{2} \left (-i a +b \right )^{2} b^{2} d \left (i a +b \right )^{3}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{3} d}+\frac {a^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b^{3} d}+\frac {3 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b d}+\frac {6 a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}\) | \(623\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 279, normalized size = 1.52 \begin {gather*} \frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 6 \, a^{2} b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {3 \, a^{6} + 7 \, a^{4} b^{2} + 4 \, {\left (a^{5} b + 2 \, a^{3} b^{3}\right )} \tan \left (d x + c\right )}{a^{6} b^{3} + 2 \, a^{4} b^{5} + a^{2} b^{7} + {\left (a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{4} + 2 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )}}{2 \, d} \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. 481 vs.
\(2 (181) = 362\).
time = 1.35, size = 481, normalized size = 2.63 \begin {gather*} \frac {a^{6} b^{2} + 7 \, a^{4} b^{4} + 2 \, {\left (a^{5} b^{3} - 3 \, a^{3} b^{5}\right )} d x - {\left (3 \, a^{6} b^{2} + 9 \, a^{4} b^{4} - 2 \, {\left (a^{3} b^{5} - 3 \, a b^{7}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (a^{8} + 3 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + {\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 6 \, a^{2} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 6 \, a^{3} b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6} + {\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} - 4 \, a^{3} b^{5} - 2 \, {\left (a^{4} b^{4} - 3 \, a^{2} b^{6}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{4} + 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right ) + {\left (a^{8} b^{3} + 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} + a^{2} b^{9}\right )} d\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 [A]
time = 1.43, size = 304, normalized size = 1.66 \begin {gather*} \frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 6 \, a^{2} b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}} - \frac {3 \, a^{6} b \tan \left (d x + c\right )^{2} + 9 \, a^{4} b^{3} \tan \left (d x + c\right )^{2} + 18 \, a^{2} b^{5} \tan \left (d x + c\right )^{2} + 2 \, a^{7} \tan \left (d x + c\right ) + 6 \, a^{5} b^{2} \tan \left (d x + c\right ) + 28 \, a^{3} b^{4} \tan \left (d x + c\right ) - a^{6} b + 11 \, a^{4} b^{3}}{{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.21, size = 240, normalized size = 1.31 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}+\frac {\frac {3\,a^6+7\,a^4\,b^2}{2\,b^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {2\,a^3\,\mathrm {tan}\left (c+d\,x\right )\,\left (a^2+2\,b^2\right )}{b^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {a^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^4+3\,a^2\,b^2+6\,b^4\right )}{b^3\,d\,{\left (a^2+b^2\right )}^3}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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