Optimal. Leaf size=77 \[ \frac {a \tan ^3(c+d x)}{3 d}-\frac {a \tan (c+d x)}{d}+a x+\frac {b \tan ^4(c+d x)}{4 d}-\frac {b \tan ^2(c+d x)}{2 d}-\frac {b \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.08, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3528, 3525, 3475} \[ \frac {a \tan ^3(c+d x)}{3 d}-\frac {a \tan (c+d x)}{d}+a x+\frac {b \tan ^4(c+d x)}{4 d}-\frac {b \tan ^2(c+d x)}{2 d}-\frac {b \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rule 3528
Rubi steps
\begin {align*} \int \tan ^4(c+d x) (a+b \tan (c+d x)) \, dx &=\frac {b \tan ^4(c+d x)}{4 d}+\int \tan ^3(c+d x) (-b+a \tan (c+d x)) \, dx\\ &=\frac {a \tan ^3(c+d x)}{3 d}+\frac {b \tan ^4(c+d x)}{4 d}+\int \tan ^2(c+d x) (-a-b \tan (c+d x)) \, dx\\ &=-\frac {b \tan ^2(c+d x)}{2 d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {b \tan ^4(c+d x)}{4 d}+\int \tan (c+d x) (b-a \tan (c+d x)) \, dx\\ &=a x-\frac {a \tan (c+d x)}{d}-\frac {b \tan ^2(c+d x)}{2 d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {b \tan ^4(c+d x)}{4 d}+b \int \tan (c+d x) \, dx\\ &=a x-\frac {b \log (\cos (c+d x))}{d}-\frac {a \tan (c+d x)}{d}-\frac {b \tan ^2(c+d x)}{2 d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {b \tan ^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 79, normalized size = 1.03 \[ \frac {a \tan ^{-1}(\tan (c+d x))}{d}+\frac {a \tan ^3(c+d x)}{3 d}-\frac {a \tan (c+d x)}{d}-\frac {b \left (-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 69, normalized size = 0.90 \[ \frac {3 \, b \tan \left (d x + c\right )^{4} + 4 \, a \tan \left (d x + c\right )^{3} + 12 \, a d x - 6 \, b \tan \left (d x + c\right )^{2} - 6 \, b \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 12 \, a \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 6.46, size = 716, normalized size = 9.30 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 85, normalized size = 1.10 \[ \frac {b \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a \tan \left (d x +c \right )}{d}+\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a \arctan \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 70, normalized size = 0.91 \[ \frac {3 \, b \tan \left (d x + c\right )^{4} + 4 \, a \tan \left (d x + c\right )^{3} - 6 \, b \tan \left (d x + c\right )^{2} + 12 \, {\left (d x + c\right )} a + 6 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 12 \, a \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.01, size = 65, normalized size = 0.84 \[ \frac {\frac {b\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}-a\,\mathrm {tan}\left (c+d\,x\right )+\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}-\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4}+a\,d\,x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 83, normalized size = 1.08 \[ \begin {cases} a x + \frac {a \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {a \tan {\left (c + d x \right )}}{d} + \frac {b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {b \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {b \tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\relax (c )}\right ) \tan ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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