Integrand size = 14, antiderivative size = 89 \[ \int \left (a+b \tan ^3(c+d x)\right )^2 \, dx=\left (a^2-b^2\right ) x+\frac {2 a b \log (\cos (c+d x))}{d}+\frac {b^2 \tan (c+d x)}{d}+\frac {a b \tan ^2(c+d x)}{d}-\frac {b^2 \tan ^3(c+d x)}{3 d}+\frac {b^2 \tan ^5(c+d x)}{5 d} \]
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Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3742, 1824, 649, 209, 266} \[ \int \left (a+b \tan ^3(c+d x)\right )^2 \, dx=x \left (a^2-b^2\right )+\frac {a b \tan ^2(c+d x)}{d}+\frac {2 a b \log (\cos (c+d x))}{d}+\frac {b^2 \tan ^5(c+d x)}{5 d}-\frac {b^2 \tan ^3(c+d x)}{3 d}+\frac {b^2 \tan (c+d x)}{d} \]
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Rule 209
Rule 266
Rule 649
Rule 1824
Rule 3742
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x^3\right )^2}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (b^2+2 a b x-b^2 x^2+b^2 x^4+\frac {a^2-b^2-2 a b x}{1+x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {b^2 \tan (c+d x)}{d}+\frac {a b \tan ^2(c+d x)}{d}-\frac {b^2 \tan ^3(c+d x)}{3 d}+\frac {b^2 \tan ^5(c+d x)}{5 d}+\frac {\text {Subst}\left (\int \frac {a^2-b^2-2 a b x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {b^2 \tan (c+d x)}{d}+\frac {a b \tan ^2(c+d x)}{d}-\frac {b^2 \tan ^3(c+d x)}{3 d}+\frac {b^2 \tan ^5(c+d x)}{5 d}-\frac {(2 a b) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (a^2-b^2\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \left (a^2-b^2\right ) x+\frac {2 a b \log (\cos (c+d x))}{d}+\frac {b^2 \tan (c+d x)}{d}+\frac {a b \tan ^2(c+d x)}{d}-\frac {b^2 \tan ^3(c+d x)}{3 d}+\frac {b^2 \tan ^5(c+d x)}{5 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.20 \[ \int \left (a+b \tan ^3(c+d x)\right )^2 \, dx=\frac {-15 i \left ((a-i b)^2 \log (i-\tan (c+d x))-(a+i b)^2 \log (i+\tan (c+d x))\right )+30 b^2 \tan (c+d x)+30 a b \tan ^2(c+d x)-10 b^2 \tan ^3(c+d x)+6 b^2 \tan ^5(c+d x)}{30 d} \]
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Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.92
method | result | size |
parts | \(x \,a^{2}+\frac {b^{2} \left (\frac {\tan \left (d x +c \right )^{5}}{5}-\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {a b \tan \left (d x +c \right )^{2}}{d}-\frac {a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d}\) | \(82\) |
derivativedivides | \(\frac {\frac {b^{2} \tan \left (d x +c \right )^{5}}{5}-\frac {b^{2} \tan \left (d x +c \right )^{3}}{3}+a b \tan \left (d x +c \right )^{2}+b^{2} \tan \left (d x +c \right )-a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(85\) |
default | \(\frac {\frac {b^{2} \tan \left (d x +c \right )^{5}}{5}-\frac {b^{2} \tan \left (d x +c \right )^{3}}{3}+a b \tan \left (d x +c \right )^{2}+b^{2} \tan \left (d x +c \right )-a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(85\) |
parallelrisch | \(-\frac {-3 b^{2} \tan \left (d x +c \right )^{5}+5 b^{2} \tan \left (d x +c \right )^{3}-15 a^{2} d x +15 b^{2} d x -15 a b \tan \left (d x +c \right )^{2}+15 a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )-15 b^{2} \tan \left (d x +c \right )}{15 d}\) | \(85\) |
norman | \(\left (a^{2}-b^{2}\right ) x +\frac {b^{2} \tan \left (d x +c \right )}{d}+\frac {a b \tan \left (d x +c \right )^{2}}{d}-\frac {b^{2} \tan \left (d x +c \right )^{3}}{3 d}+\frac {b^{2} \tan \left (d x +c \right )^{5}}{5 d}-\frac {a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d}\) | \(90\) |
risch | \(-2 i a b x +x \,a^{2}-x \,b^{2}-\frac {4 i a b c}{d}+\frac {2 b \left (45 i b \,{\mathrm e}^{8 i \left (d x +c \right )}+30 a \,{\mathrm e}^{8 i \left (d x +c \right )}+90 i b \,{\mathrm e}^{6 i \left (d x +c \right )}+90 a \,{\mathrm e}^{6 i \left (d x +c \right )}+140 i b \,{\mathrm e}^{4 i \left (d x +c \right )}+90 a \,{\mathrm e}^{4 i \left (d x +c \right )}+70 i b \,{\mathrm e}^{2 i \left (d x +c \right )}+30 a \,{\mathrm e}^{2 i \left (d x +c \right )}+23 i b \right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {2 a b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(171\) |
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Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96 \[ \int \left (a+b \tan ^3(c+d x)\right )^2 \, dx=\frac {3 \, b^{2} \tan \left (d x + c\right )^{5} - 5 \, b^{2} \tan \left (d x + c\right )^{3} + 15 \, a b \tan \left (d x + c\right )^{2} + 15 \, {\left (a^{2} - b^{2}\right )} d x + 15 \, a b \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 15 \, b^{2} \tan \left (d x + c\right )}{15 \, d} \]
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Time = 0.15 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.06 \[ \int \left (a+b \tan ^3(c+d x)\right )^2 \, dx=\begin {cases} a^{2} x - \frac {a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {a b \tan ^{2}{\left (c + d x \right )}}{d} - b^{2} x + \frac {b^{2} \tan ^{5}{\left (c + d x \right )}}{5 d} - \frac {b^{2} \tan ^{3}{\left (c + d x \right )}}{3 d} + \frac {b^{2} \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan ^{3}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.93 \[ \int \left (a+b \tan ^3(c+d x)\right )^2 \, dx=a^{2} x + \frac {{\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} b^{2}}{15 \, d} - \frac {a b {\left (\frac {1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1005 vs. \(2 (85) = 170\).
Time = 1.46 (sec) , antiderivative size = 1005, normalized size of antiderivative = 11.29 \[ \int \left (a+b \tan ^3(c+d x)\right )^2 \, dx=\text {Too large to display} \]
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Time = 11.83 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.31 \[ \int \left (a+b \tan ^3(c+d x)\right )^2 \, dx=\frac {b^2\,\mathrm {tan}\left (c+d\,x\right )}{d}-\frac {b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,d}+\frac {b^2\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5\,d}+\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (a+b\right )\,\left (a-b\right )}{a^2-b^2}\right )\,\left (a+b\right )\,\left (a-b\right )}{d}-\frac {a\,b\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{d}+\frac {a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{d} \]
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