Optimal. Leaf size=22 \[ \frac {(a+b \tan (c+d x))^4}{4 b d} \]
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Rubi [A] time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3506, 32} \[ \frac {(a+b \tan (c+d x))^4}{4 b d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3506
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+b \tan (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^3 \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {(a+b \tan (c+d x))^4}{4 b d}\\ \end {align*}
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Mathematica [B] time = 0.16, size = 57, normalized size = 2.59 \[ \frac {\tan (c+d x) \left (4 a^3+6 a^2 b \tan (c+d x)+4 a b^2 \tan ^2(c+d x)+b^3 \tan ^3(c+d x)\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 78, normalized size = 3.55 \[ \frac {b^{3} + 2 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (a b^{2} \cos \left (d x + c\right ) + {\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.96, size = 57, normalized size = 2.59 \[ \frac {b^{3} \tan \left (d x + c\right )^{4} + 4 \, a b^{2} \tan \left (d x + c\right )^{3} + 6 \, a^{2} b \tan \left (d x + c\right )^{2} + 4 \, a^{3} \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.43, size = 72, normalized size = 3.27 \[ \frac {a^{3} \tan \left (d x +c \right )+\frac {3 a^{2} b}{2 \cos \left (d x +c \right )^{2}}+\frac {b^{2} a \left (\sin ^{3}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{3}}+\frac {b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4 \cos \left (d x +c \right )^{4}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 20, normalized size = 0.91 \[ \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{4}}{4 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.57, size = 55, normalized size = 2.50 \[ \frac {a^3\,\mathrm {tan}\left (c+d\,x\right )+\frac {3\,a^2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3+\frac {b^3\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{3} \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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