Optimal. Leaf size=125 \[ -\frac {a \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {b x}{a^2+b^2}+\frac {\left (a^2-b^2\right ) \tan (c+d x)}{b^3 d}-\frac {a^5 \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )}-\frac {a \tan ^2(c+d x)}{2 b^2 d}+\frac {\tan ^3(c+d x)}{3 b d} \]
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Rubi [A] time = 0.38, antiderivative size = 125, 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 = {3566, 3647, 3648, 3626, 3617, 31, 3475} \[ \frac {\left (a^2-b^2\right ) \tan (c+d x)}{b^3 d}-\frac {a^5 \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )}-\frac {a \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {b x}{a^2+b^2}-\frac {a \tan ^2(c+d x)}{2 b^2 d}+\frac {\tan ^3(c+d x)}{3 b d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3475
Rule 3566
Rule 3617
Rule 3626
Rule 3647
Rule 3648
Rubi steps
\begin {align*} \int \frac {\tan ^5(c+d x)}{a+b \tan (c+d x)} \, dx &=\frac {\tan ^3(c+d x)}{3 b d}+\frac {\int \frac {\tan ^2(c+d x) \left (-3 a-3 b \tan (c+d x)-3 a \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{3 b}\\ &=-\frac {a \tan ^2(c+d x)}{2 b^2 d}+\frac {\tan ^3(c+d x)}{3 b d}+\frac {\int \frac {\tan (c+d x) \left (6 a^2+6 \left (a^2-b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 b^2}\\ &=\frac {\left (a^2-b^2\right ) \tan (c+d x)}{b^3 d}-\frac {a \tan ^2(c+d x)}{2 b^2 d}+\frac {\tan ^3(c+d x)}{3 b d}+\frac {\int \frac {-6 a \left (a^2-b^2\right )+6 b^3 \tan (c+d x)-6 a \left (a^2-b^2\right ) \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{6 b^3}\\ &=\frac {b x}{a^2+b^2}+\frac {\left (a^2-b^2\right ) \tan (c+d x)}{b^3 d}-\frac {a \tan ^2(c+d x)}{2 b^2 d}+\frac {\tan ^3(c+d x)}{3 b d}+\frac {a \int \tan (c+d x) \, dx}{a^2+b^2}-\frac {a^5 \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^3 \left (a^2+b^2\right )}\\ &=\frac {b x}{a^2+b^2}-\frac {a \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}+\frac {\left (a^2-b^2\right ) \tan (c+d x)}{b^3 d}-\frac {a \tan ^2(c+d x)}{2 b^2 d}+\frac {\tan ^3(c+d x)}{3 b d}-\frac {a^5 \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^4 \left (a^2+b^2\right ) d}\\ &=\frac {b x}{a^2+b^2}-\frac {a \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}-\frac {a^5 \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right ) d}+\frac {\left (a^2-b^2\right ) \tan (c+d x)}{b^3 d}-\frac {a \tan ^2(c+d x)}{2 b^2 d}+\frac {\tan ^3(c+d x)}{3 b d}\\ \end {align*}
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Mathematica [C] time = 0.61, size = 155, normalized size = 1.24 \[ \frac {a^2 \tan (c+d x)}{b^3 d}-\frac {a^5 \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )}-\frac {a \tan ^2(c+d x)}{2 b^2 d}+\frac {\log (-\tan (c+d x)+i)}{2 d (a+i b)}+\frac {\log (\tan (c+d x)+i)}{2 d (a-i b)}+\frac {\tan ^3(c+d x)}{3 b d}-\frac {\tan (c+d x)}{b d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 159, normalized size = 1.27 \[ \frac {6 \, b^{5} d x - 3 \, a^{5} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (a^{2} b^{3} + b^{5}\right )} \tan \left (d x + c\right )^{3} - 3 \, {\left (a^{3} b^{2} + a b^{4}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{5} - a b^{4}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 6 \, {\left (a^{4} b - b^{5}\right )} \tan \left (d x + c\right )}{6 \, {\left (a^{2} b^{4} + b^{6}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 7.41, size = 129, normalized size = 1.03 \[ -\frac {\frac {6 \, a^{5} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{2} b^{4} + b^{6}} - \frac {6 \, {\left (d x + c\right )} b}{a^{2} + b^{2}} - \frac {3 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, b^{2} \tan \left (d x + c\right )^{3} - 3 \, a b \tan \left (d x + c\right )^{2} + 6 \, a^{2} \tan \left (d x + c\right ) - 6 \, b^{2} \tan \left (d x + c\right )}{b^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 143, normalized size = 1.14 \[ \frac {\tan ^{3}\left (d x +c \right )}{3 b d}-\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2 b^{2} d}+\frac {a^{2} \tan \left (d x +c \right )}{d \,b^{3}}-\frac {\tan \left (d x +c \right )}{b d}-\frac {a^{5} \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{4} \left (a^{2}+b^{2}\right ) d}+\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{2}+b^{2}\right )}+\frac {b \arctan \left (\tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 123, normalized size = 0.98 \[ -\frac {\frac {6 \, a^{5} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{2} b^{4} + b^{6}} - \frac {6 \, {\left (d x + c\right )} b}{a^{2} + b^{2}} - \frac {3 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, b^{2} \tan \left (d x + c\right )^{3} - 3 \, a b \tan \left (d x + c\right )^{2} + 6 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )}{b^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.08, size = 138, normalized size = 1.10 \[ \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (a-b\,1{}\mathrm {i}\right )}-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {1}{b}-\frac {a^2}{b^3}\right )}{d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,b\,d}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,b^2\,d}-\frac {a^5\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d\,\left (a^2\,b^4+b^6\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-b+a\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.34, size = 835, normalized size = 6.68 \[ \begin {cases} \tilde {\infty } x \tan ^{4}{\relax (c )} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {\frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {\tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {\tan ^{2}{\left (c + d x \right )}}{2 d}}{a} & \text {for}\: b = 0 \\- \frac {15 d x \tan {\left (c + d x \right )}}{- 6 b d \tan {\left (c + d x \right )} + 6 i b d} + \frac {15 i d x}{- 6 b d \tan {\left (c + d x \right )} + 6 i b d} + \frac {6 i \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{- 6 b d \tan {\left (c + d x \right )} + 6 i b d} + \frac {6 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{- 6 b d \tan {\left (c + d x \right )} + 6 i b d} - \frac {2 \tan ^{4}{\left (c + d x \right )}}{- 6 b d \tan {\left (c + d x \right )} + 6 i b d} - \frac {i \tan ^{3}{\left (c + d x \right )}}{- 6 b d \tan {\left (c + d x \right )} + 6 i b d} + \frac {9 \tan ^{2}{\left (c + d x \right )}}{- 6 b d \tan {\left (c + d x \right )} + 6 i b d} + \frac {15}{- 6 b d \tan {\left (c + d x \right )} + 6 i b d} & \text {for}\: a = - i b \\- \frac {15 d x \tan {\left (c + d x \right )}}{- 6 b d \tan {\left (c + d x \right )} - 6 i b d} - \frac {15 i d x}{- 6 b d \tan {\left (c + d x \right )} - 6 i b d} - \frac {6 i \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{- 6 b d \tan {\left (c + d x \right )} - 6 i b d} + \frac {6 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{- 6 b d \tan {\left (c + d x \right )} - 6 i b d} - \frac {2 \tan ^{4}{\left (c + d x \right )}}{- 6 b d \tan {\left (c + d x \right )} - 6 i b d} + \frac {i \tan ^{3}{\left (c + d x \right )}}{- 6 b d \tan {\left (c + d x \right )} - 6 i b d} + \frac {9 \tan ^{2}{\left (c + d x \right )}}{- 6 b d \tan {\left (c + d x \right )} - 6 i b d} + \frac {15}{- 6 b d \tan {\left (c + d x \right )} - 6 i b d} & \text {for}\: a = i b \\\frac {x \tan ^{5}{\relax (c )}}{a + b \tan {\relax (c )}} & \text {for}\: d = 0 \\- \frac {6 a^{5} \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{6 a^{2} b^{4} d + 6 b^{6} d} + \frac {6 a^{4} b \tan {\left (c + d x \right )}}{6 a^{2} b^{4} d + 6 b^{6} d} - \frac {3 a^{3} b^{2} \tan ^{2}{\left (c + d x \right )}}{6 a^{2} b^{4} d + 6 b^{6} d} + \frac {2 a^{2} b^{3} \tan ^{3}{\left (c + d x \right )}}{6 a^{2} b^{4} d + 6 b^{6} d} + \frac {3 a b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{6 a^{2} b^{4} d + 6 b^{6} d} - \frac {3 a b^{4} \tan ^{2}{\left (c + d x \right )}}{6 a^{2} b^{4} d + 6 b^{6} d} + \frac {6 b^{5} d x}{6 a^{2} b^{4} d + 6 b^{6} d} + \frac {2 b^{5} \tan ^{3}{\left (c + d x \right )}}{6 a^{2} b^{4} d + 6 b^{6} d} - \frac {6 b^{5} \tan {\left (c + d x \right )}}{6 a^{2} b^{4} d + 6 b^{6} d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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