Integrand size = 24, antiderivative size = 27 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {i (a+i a \tan (c+d x))^4}{4 a d} \]
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Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 32} \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {i (a+i a \tan (c+d x))^4}{4 a d} \]
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Rule 32
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int (a+x)^3 \, dx,x,i a \tan (c+d x)\right )}{a d} \\ & = -\frac {i (a+i a \tan (c+d x))^4}{4 a d} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {a^3 \tan (c+d x) \left (4+6 i \tan (c+d x)-4 \tan ^2(c+d x)-i \tan ^3(c+d x)\right )}{4 d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (23 ) = 46\).
Time = 4.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.15
method | result | size |
risch | \(\frac {4 i a^{3} \left (4 \,{\mathrm e}^{6 i \left (d x +c \right )}+6 \,{\mathrm e}^{4 i \left (d x +c \right )}+4 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}\) | \(58\) |
derivativedivides | \(\frac {-\frac {i a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4 \cos \left (d x +c \right )^{4}}-\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{3}}+\frac {3 i a^{3}}{2 \cos \left (d x +c \right )^{2}}+a^{3} \tan \left (d x +c \right )}{d}\) | \(73\) |
default | \(\frac {-\frac {i a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4 \cos \left (d x +c \right )^{4}}-\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{3}}+\frac {3 i a^{3}}{2 \cos \left (d x +c \right )^{2}}+a^{3} \tan \left (d x +c \right )}{d}\) | \(73\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (21) = 42\).
Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 3.74 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {4 \, {\left (-4 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 6 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 4 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3}\right )}}{d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]
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\[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx=- i a^{3} \left (\int i \sec ^{2}{\left (c + d x \right )}\, dx + \int \left (- 3 \tan {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int \tan ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \left (- 3 i \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}{4 \, a d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (21) = 42\).
Time = 0.49 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.07 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {i \, a^{3} \tan \left (d x + c\right )^{4} + 4 \, a^{3} \tan \left (d x + c\right )^{3} - 6 i \, a^{3} \tan \left (d x + c\right )^{2} - 4 \, a^{3} \tan \left (d x + c\right )}{4 \, d} \]
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Time = 3.85 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.07 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {-\frac {a^3\,{\mathrm {tan}\left (c+d\,x\right )}^4\,1{}\mathrm {i}}{4}-a^3\,{\mathrm {tan}\left (c+d\,x\right )}^3+\frac {a^3\,{\mathrm {tan}\left (c+d\,x\right )}^2\,3{}\mathrm {i}}{2}+a^3\,\mathrm {tan}\left (c+d\,x\right )}{d} \]
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