Optimal. Leaf size=30 \[ \frac {\tan ^4(c+d x) (a \cot (c+d x)+b)^4}{4 b d} \]
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Rubi [A] time = 0.05, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3088, 37} \[ \frac {\tan ^4(c+d x) (a \cot (c+d x)+b)^4}{4 b d} \]
Antiderivative was successfully verified.
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Rule 37
Rule 3088
Rubi steps
\begin {align*} \int \sec ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(b+a x)^3}{x^5} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {(b+a \cot (c+d x))^4 \tan ^4(c+d x)}{4 b d}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 57, normalized size = 1.90 \[ \frac {\tan (c+d x) \left (4 a^3+6 a^2 b \tan (c+d x)+4 a b^2 \tan ^2(c+d x)+b^3 \tan ^3(c+d x)\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 78, normalized size = 2.60 \[ \frac {b^{3} + 2 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (a b^{2} \cos \left (d x + c\right ) + {\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.60, size = 57, normalized size = 1.90 \[ \frac {b^{3} \tan \left (d x + c\right )^{4} + 4 \, a b^{2} \tan \left (d x + c\right )^{3} + 6 \, a^{2} b \tan \left (d x + c\right )^{2} + 4 \, a^{3} \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 11.42, size = 72, normalized size = 2.40 \[ \frac {a^{3} \tan \left (d x +c \right )+\frac {3 a^{2} b}{2 \cos \left (d x +c \right )^{2}}+\frac {b^{2} a \left (\sin ^{3}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{3}}+\frac {b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4 \cos \left (d x +c \right )^{4}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 87, normalized size = 2.90 \[ \frac {4 \, a b^{2} \tan \left (d x + c\right )^{3} + 4 \, a^{3} \tan \left (d x + c\right ) + \frac {{\left (2 \, \sin \left (d x + c\right )^{2} - 1\right )} b^{3}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \frac {6 \, a^{2} b}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 88, normalized size = 2.93 \[ \frac {{\cos \left (c+d\,x\right )}^3\,\left (a^3\,\sin \left (c+d\,x\right )-a\,b^2\,\sin \left (c+d\,x\right )\right )+{\cos \left (c+d\,x\right )}^2\,\left (\frac {3\,a^2\,b}{2}-\frac {b^3}{2}\right )+\frac {b^3}{4}+a\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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