Optimal. Leaf size=239 \[ -\frac {a^2 \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {a \left (a^2-3 b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {b x \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3}+\frac {\left (3 a^4+6 a^2 b^2+b^4\right ) \tan (c+d x)}{b^3 d \left (a^2+b^2\right )^2}-\frac {a^3 \left (3 a^4+9 a^2 b^2+10 b^4\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^3} \]
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Rubi [A] time = 0.56, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3565, 3645, 3647, 3626, 3617, 31, 3475} \[ -\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {a^2 \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\left (6 a^2 b^2+3 a^4+b^4\right ) \tan (c+d x)}{b^3 d \left (a^2+b^2\right )^2}-\frac {a^3 \left (9 a^2 b^2+3 a^4+10 b^4\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^3}-\frac {a \left (a^2-3 b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {b x \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3475
Rule 3565
Rule 3617
Rule 3626
Rule 3645
Rule 3647
Rubi steps
\begin {align*} \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=-\frac {a^2 \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {\tan ^2(c+d x) \left (3 a^2-2 a b \tan (c+d x)+\left (3 a^2+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 ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\tan (c+d x) \left (2 a^2 \left (3 a^2+7 b^2\right )-4 a b^3 \tan (c+d x)+2 \left (3 a^4+6 a^2 b^2+b^4\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=\frac {\left (3 a^4+6 a^2 b^2+b^4\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {-2 a \left (3 a^4+6 a^2 b^2+b^4\right )+2 b^3 \left (a^2-b^2\right ) \tan (c+d x)-6 a \left (a^2+b^2\right )^2 \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{2 b^3 \left (a^2+b^2\right )^2}\\ &=\frac {b \left (3 a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (3 a^4+6 a^2 b^2+b^4\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left (a \left (a^2-3 b^2\right )\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^3}-\frac {\left (a^3 \left (3 a^4+9 a^2 b^2+10 b^4\right )\right ) \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^3 \left (a^2+b^2\right )^3}\\ &=\frac {b \left (3 a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {a \left (a^2-3 b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {\left (3 a^4+6 a^2 b^2+b^4\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left (a^3 \left (3 a^4+9 a^2 b^2+10 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^4 \left (a^2+b^2\right )^3 d}\\ &=\frac {b \left (3 a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {a \left (a^2-3 b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac {a^3 \left (3 a^4+9 a^2 b^2+10 b^4\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3 d}+\frac {\left (3 a^4+6 a^2 b^2+b^4\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+7 b^2\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] time = 6.53, size = 694, normalized size = 2.90 \[ \frac {a^5 \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))}{2 b^2 d (a-i b)^2 (a+i b)^2 (a+b \tan (c+d x))^3}+\frac {b \left (3 a^2-b^2\right ) (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3}{d (a-i b)^3 (a+i b)^3 (a+b \tan (c+d x))^3}+\frac {\sec ^3(c+d x) \left (2 a^5 \sin (c+d x)+5 a^3 b^2 \sin (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{b^3 d (a-i b)^2 (a+i b)^2 (a+b \tan (c+d x))^3}-\frac {i \left (-3 a^7-9 a^5 b^2-10 a^3 b^4\right ) \tan ^{-1}(\tan (c+d x)) \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3}{b^4 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^3}+\frac {\left (-3 a^7-9 a^5 b^2-10 a^3 b^4\right ) \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )}{2 b^4 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^3}+\frac {3 a \sec ^3(c+d x) \log (\cos (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{b^4 d (a+b \tan (c+d x))^3}+\frac {\tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3}{b^3 d (a+b \tan (c+d x))^3}-\frac {i \left (3 a^{12} b^3-3 i a^{11} b^4+15 a^{10} b^5-15 i a^9 b^6+31 a^8 b^7-31 i a^7 b^8+29 a^6 b^9-29 i a^5 b^{10}+10 a^4 b^{11}-10 i a^3 b^{12}\right ) (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3}{b^7 d (a-i b)^6 (a+i b)^5 (a+b \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 549, normalized size = 2.30 \[ -\frac {3 \, a^{7} b^{2} + 9 \, a^{5} b^{4} - 2 \, {\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{3} - 2 \, {\left (3 \, a^{4} b^{5} - a^{2} b^{7}\right )} d x - {\left (9 \, a^{7} b^{2} + 23 \, a^{5} b^{4} + 12 \, a^{3} b^{6} + 4 \, a b^{8} + 2 \, {\left (3 \, a^{2} b^{7} - b^{9}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (3 \, a^{9} + 9 \, a^{7} b^{2} + 10 \, a^{5} b^{4} + {\left (3 \, a^{7} b^{2} + 9 \, a^{5} b^{4} + 10 \, a^{3} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{8} b + 9 \, a^{6} b^{3} + 10 \, a^{4} b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6} + {\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{8} b + 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} + a^{2} b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (3 \, a^{8} b + 6 \, a^{6} b^{3} - 2 \, a^{4} b^{5} + a^{2} b^{7} + 2 \, {\left (3 \, a^{3} b^{6} - a b^{8}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{6} + 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} + b^{12}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{5} + 3 \, a^{5} b^{7} + 3 \, a^{3} b^{9} + a b^{11}\right )} d \tan \left (d x + c\right ) + {\left (a^{8} b^{4} + 3 \, a^{6} b^{6} + 3 \, a^{4} b^{8} + a^{2} b^{10}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 7.94, size = 325, normalized size = 1.36 \[ \frac {\frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (3 \, a^{7} + 9 \, a^{5} b^{2} + 10 \, a^{3} b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac {9 \, a^{7} b^{2} \tan \left (d x + c\right )^{2} + 27 \, a^{5} b^{4} \tan \left (d x + c\right )^{2} + 30 \, a^{3} b^{6} \tan \left (d x + c\right )^{2} + 12 \, a^{8} b \tan \left (d x + c\right ) + 38 \, a^{6} b^{3} \tan \left (d x + c\right ) + 50 \, a^{4} b^{5} \tan \left (d x + c\right ) + 4 \, a^{9} + 13 \, a^{7} b^{2} + 21 \, a^{5} b^{4}}{{\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}} + \frac {2 \, \tan \left (d x + c\right )}{b^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 307, normalized size = 1.28 \[ \frac {\tan \left (d x +c \right )}{d \,b^{3}}-\frac {3 a^{6}}{d \,b^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {5 a^{4}}{d \,b^{2} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {a^{5}}{2 d \,b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {3 a^{7} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,b^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {9 a^{5} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,b^{2} \left (a^{2}+b^{2}\right )^{3}}-\frac {10 a^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3}}{2 d \left (a^{2}+b^{2}\right )^{3}}-\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{2} a}{2 d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 \arctan \left (\tan \left (d x +c \right )\right ) a^{2} b}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {\arctan \left (\tan \left (d x +c \right )\right ) b^{3}}{d \left (a^{2}+b^{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 293, normalized size = 1.23 \[ \frac {\frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (3 \, a^{7} + 9 \, a^{5} b^{2} + 10 \, a^{3} b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {5 \, a^{7} + 9 \, a^{5} b^{2} + 2 \, {\left (3 \, a^{6} b + 5 \, a^{4} b^{3}\right )} \tan \left (d x + c\right )}{a^{6} b^{4} + 2 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{4} b^{6} + 2 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{5} + 2 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )} + \frac {2 \, \tan \left (d x + c\right )}{b^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.25, size = 263, normalized size = 1.10 \[ \frac {\mathrm {tan}\left (c+d\,x\right )}{b^3\,d}-\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (3\,a^6+5\,a^4\,b^2\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {5\,a^7+9\,a^5\,b^2}{2\,b\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,b^3+2\,a\,b^4\,\mathrm {tan}\left (c+d\,x\right )+b^5\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}-\frac {a^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,a^4+9\,a^2\,b^2+10\,b^4\right )}{b^4\,d\,{\left (a^2+b^2\right )}^3}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )} \]
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 \[ \text {Exception raised: AttributeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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