Optimal. Leaf size=30 \[ \frac {\tan ^5(c+d x) (a \cot (c+d x)+b)^5}{5 b d} \]
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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} \[ \frac {\tan ^5(c+d x) (a \cot (c+d x)+b)^5}{5 b d} \]
Antiderivative was successfully verified.
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Rule 37
Rule 3088
Rubi steps
\begin {align*} \int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(b+a x)^4}{x^6} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {(b+a \cot (c+d x))^5 \tan ^5(c+d x)}{5 b d}\\ \end {align*}
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Mathematica [B] time = 0.32, size = 73, normalized size = 2.43 \[ \frac {\tan (c+d x) \left (5 a^4+10 a^3 b \tan (c+d x)+10 a^2 b^2 \tan ^2(c+d x)+5 a b^3 \tan ^3(c+d x)+b^4 \tan ^4(c+d x)\right )}{5 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 109, normalized size = 3.63 \[ \frac {5 \, a b^{3} \cos \left (d x + c\right ) + 10 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left ({\left (5 \, a^{4} - 10 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + b^{4} + 2 \, {\left (5 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{5 \, d \cos \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 73, normalized size = 2.43 \[ \frac {b^{4} \tan \left (d x + c\right )^{5} + 5 \, a b^{3} \tan \left (d x + c\right )^{4} + 10 \, a^{2} b^{2} \tan \left (d x + c\right )^{3} + 10 \, a^{3} b \tan \left (d x + c\right )^{2} + 5 \, a^{4} \tan \left (d x + c\right )}{5 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 27.47, size = 96, normalized size = 3.20 \[ \frac {a^{4} \tan \left (d x +c \right )+\frac {2 a^{3} b}{\cos \left (d x +c \right )^{2}}+\frac {2 a^{2} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{3}}+\frac {a \,b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{4}}+\frac {b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{5 \cos \left (d x +c \right )^{5}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 103, normalized size = 3.43 \[ \frac {b^{4} \tan \left (d x + c\right )^{5} + 10 \, a^{2} b^{2} \tan \left (d x + c\right )^{3} + 5 \, a^{4} \tan \left (d x + c\right ) + \frac {5 \, {\left (2 \, \sin \left (d x + c\right )^{2} - 1\right )} a b^{3}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \frac {10 \, a^{3} b}{\sin \left (d x + c\right )^{2} - 1}}{5 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 139, normalized size = 4.63 \[ \frac {\frac {b^4\,\sin \left (c+d\,x\right )}{5}-{\cos \left (c+d\,x\right )}^3\,\left (2\,a\,b^3-2\,a^3\,b\right )-{\cos \left (c+d\,x\right )}^2\,\left (\frac {2\,b^4\,\sin \left (c+d\,x\right )}{5}-2\,a^2\,b^2\,\sin \left (c+d\,x\right )\right )+{\cos \left (c+d\,x\right )}^4\,\left (\sin \left (c+d\,x\right )\,a^4-2\,\sin \left (c+d\,x\right )\,a^2\,b^2+\frac {\sin \left (c+d\,x\right )\,b^4}{5}\right )+a\,b^3\,\cos \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^5} \]
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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