Optimal. Leaf size=93 \[ \frac {a \tan ^4(c+d x)}{4 d}-\frac {a \tan ^2(c+d x)}{2 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan ^5(c+d x)}{5 d}-\frac {b \tan ^3(c+d x)}{3 d}+\frac {b \tan (c+d x)}{d}-b x \]
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Rubi [A] time = 0.10, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3528, 3525, 3475} \[ \frac {a \tan ^4(c+d x)}{4 d}-\frac {a \tan ^2(c+d x)}{2 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan ^5(c+d x)}{5 d}-\frac {b \tan ^3(c+d x)}{3 d}+\frac {b \tan (c+d x)}{d}-b x \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rule 3528
Rubi steps
\begin {align*} \int \tan ^5(c+d x) (a+b \tan (c+d x)) \, dx &=\frac {b \tan ^5(c+d x)}{5 d}+\int \tan ^4(c+d x) (-b+a \tan (c+d x)) \, dx\\ &=\frac {a \tan ^4(c+d x)}{4 d}+\frac {b \tan ^5(c+d x)}{5 d}+\int \tan ^3(c+d x) (-a-b \tan (c+d x)) \, dx\\ &=-\frac {b \tan ^3(c+d x)}{3 d}+\frac {a \tan ^4(c+d x)}{4 d}+\frac {b \tan ^5(c+d x)}{5 d}+\int \tan ^2(c+d x) (b-a \tan (c+d x)) \, dx\\ &=-\frac {a \tan ^2(c+d x)}{2 d}-\frac {b \tan ^3(c+d x)}{3 d}+\frac {a \tan ^4(c+d x)}{4 d}+\frac {b \tan ^5(c+d x)}{5 d}+\int \tan (c+d x) (a+b \tan (c+d x)) \, dx\\ &=-b x+\frac {b \tan (c+d x)}{d}-\frac {a \tan ^2(c+d x)}{2 d}-\frac {b \tan ^3(c+d x)}{3 d}+\frac {a \tan ^4(c+d x)}{4 d}+\frac {b \tan ^5(c+d x)}{5 d}+a \int \tan (c+d x) \, dx\\ &=-b x-\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan (c+d x)}{d}-\frac {a \tan ^2(c+d x)}{2 d}-\frac {b \tan ^3(c+d x)}{3 d}+\frac {a \tan ^4(c+d x)}{4 d}+\frac {b \tan ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 95, normalized size = 1.02 \[ -\frac {a \left (-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right )}{4 d}-\frac {b \tan ^{-1}(\tan (c+d x))}{d}+\frac {b \tan ^5(c+d x)}{5 d}-\frac {b \tan ^3(c+d x)}{3 d}+\frac {b \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 80, normalized size = 0.86 \[ \frac {12 \, b \tan \left (d x + c\right )^{5} + 15 \, a \tan \left (d x + c\right )^{4} - 20 \, b \tan \left (d x + c\right )^{3} - 60 \, b d x - 30 \, a \tan \left (d x + c\right )^{2} - 30 \, a \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 60 \, b \tan \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 23.20, size = 947, normalized size = 10.18 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 99, normalized size = 1.06 \[ \frac {b \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}-\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {b \tan \left (d x +c \right )}{d}+\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {b \arctan \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 81, normalized size = 0.87 \[ \frac {12 \, b \tan \left (d x + c\right )^{5} + 15 \, a \tan \left (d x + c\right )^{4} - 20 \, b \tan \left (d x + c\right )^{3} - 30 \, a \tan \left (d x + c\right )^{2} - 60 \, {\left (d x + c\right )} b + 30 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 60 \, b \tan \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.10, size = 76, normalized size = 0.82 \[ \frac {b\,\mathrm {tan}\left (c+d\,x\right )+\frac {a\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4}-\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}+\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}-b\,d\,x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 97, normalized size = 1.04 \[ \begin {cases} \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {a \tan ^{2}{\left (c + d x \right )}}{2 d} - b x + \frac {b \tan ^{5}{\left (c + d x \right )}}{5 d} - \frac {b \tan ^{3}{\left (c + d x \right )}}{3 d} + \frac {b \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\relax (c )}\right ) \tan ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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