Integrand size = 19, antiderivative size = 77 \[ \int \tan ^4(c+d x) (a+b \tan (c+d x)) \, dx=a x-\frac {b \log (\cos (c+d x))}{d}-\frac {a \tan (c+d x)}{d}-\frac {b \tan ^2(c+d x)}{2 d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {b \tan ^4(c+d x)}{4 d} \]
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Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3609, 3606, 3556} \[ \int \tan ^4(c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \tan ^3(c+d x)}{3 d}-\frac {a \tan (c+d x)}{d}+a x+\frac {b \tan ^4(c+d x)}{4 d}-\frac {b \tan ^2(c+d x)}{2 d}-\frac {b \log (\cos (c+d x))}{d} \]
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Rule 3556
Rule 3606
Rule 3609
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03 \[ \int \tan ^4(c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \arctan (\tan (c+d x))}{d}-\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d}-\frac {b \left (4 \log (\cos (c+d x))+2 \tan ^2(c+d x)-\tan ^4(c+d x)\right )}{4 d} \]
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Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88
method | result | size |
parallelrisch | \(\frac {3 \left (\tan ^{4}\left (d x +c \right )\right ) b +4 a \left (\tan ^{3}\left (d x +c \right )\right )+12 a d x -6 b \left (\tan ^{2}\left (d x +c \right )\right )+6 b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )-12 a \tan \left (d x +c \right )}{12 d}\) | \(68\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{4}\left (d x +c \right )\right ) b}{4}+\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}-a \tan \left (d x +c \right )+\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+a \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(71\) |
default | \(\frac {\frac {\left (\tan ^{4}\left (d x +c \right )\right ) b}{4}+\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}-a \tan \left (d x +c \right )+\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+a \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(71\) |
parts | \(\frac {a \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {b \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\) | \(72\) |
norman | \(a x -\frac {a \tan \left (d x +c \right )}{d}+\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {b \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(76\) |
risch | \(i b x +a x +\frac {2 i b c}{d}-\frac {4 \left (3 i a \,{\mathrm e}^{6 i \left (d x +c \right )}+3 b \,{\mathrm e}^{6 i \left (d x +c \right )}+6 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+5 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+3 b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 i a \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(134\) |
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Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.90 \[ \int \tan ^4(c+d x) (a+b \tan (c+d x)) \, dx=\frac {3 \, b \tan \left (d x + c\right )^{4} + 4 \, a \tan \left (d x + c\right )^{3} + 12 \, a d x - 6 \, b \tan \left (d x + c\right )^{2} - 6 \, b \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 12 \, a \tan \left (d x + c\right )}{12 \, d} \]
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Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.08 \[ \int \tan ^4(c+d x) (a+b \tan (c+d x)) \, dx=\begin {cases} a x + \frac {a \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {a \tan {\left (c + d x \right )}}{d} + \frac {b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {b \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {b \tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right ) \tan ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.42 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.91 \[ \int \tan ^4(c+d x) (a+b \tan (c+d x)) \, dx=\frac {3 \, b \tan \left (d x + c\right )^{4} + 4 \, a \tan \left (d x + c\right )^{3} - 6 \, b \tan \left (d x + c\right )^{2} + 12 \, {\left (d x + c\right )} a + 6 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 12 \, a \tan \left (d x + c\right )}{12 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (71) = 142\).
Time = 1.34 (sec) , antiderivative size = 666, normalized size of antiderivative = 8.65 \[ \int \tan ^4(c+d x) (a+b \tan (c+d x)) \, dx=\text {Too large to display} \]
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Time = 4.91 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84 \[ \int \tan ^4(c+d x) (a+b \tan (c+d x)) \, dx=\frac {\frac {b\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}-a\,\mathrm {tan}\left (c+d\,x\right )+\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}-\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4}+a\,d\,x}{d} \]
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