Optimal. Leaf size=30 \[ \frac {\tan ^6(c+d x) (a \cot (c+d x)+b)^6}{6 b d} \]
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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} \[ \frac {\tan ^6(c+d x) (a \cot (c+d x)+b)^6}{6 b d} \]
Antiderivative was successfully verified.
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Rule 37
Rule 3088
Rubi steps
\begin {align*} \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(b+a x)^5}{x^7} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {(b+a \cot (c+d x))^6 \tan ^6(c+d x)}{6 b d}\\ \end {align*}
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Mathematica [B] time = 0.49, size = 89, normalized size = 2.97 \[ \frac {\tan (c+d x) \left (6 a^5+15 a^4 b \tan (c+d x)+20 a^3 b^2 \tan ^2(c+d x)+15 a^2 b^3 \tan ^3(c+d x)+6 a b^4 \tan ^4(c+d x)+b^5 \tan ^5(c+d x)\right )}{6 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 144, normalized size = 4.80 \[ \frac {b^{5} + 3 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, a b^{4} \cos \left (d x + c\right ) + {\left (3 \, a^{5} - 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (5 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.26, size = 89, normalized size = 2.97 \[ \frac {b^{5} \tan \left (d x + c\right )^{6} + 6 \, a b^{4} \tan \left (d x + c\right )^{5} + 15 \, a^{2} b^{3} \tan \left (d x + c\right )^{4} + 20 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 15 \, a^{4} b \tan \left (d x + c\right )^{2} + 6 \, a^{5} \tan \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 120, normalized size = 4.00 \[ \frac {a^{5} \tan \left (d x +c \right )+\frac {5 a^{4} b}{2 \cos \left (d x +c \right )^{2}}+\frac {10 a^{3} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+\frac {5 a^{2} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{2 \cos \left (d x +c \right )^{4}}+\frac {a \,b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{5}}+\frac {b^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{6 \cos \left (d x +c \right )^{6}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 166, normalized size = 5.53 \[ \frac {6 \, a b^{4} \tan \left (d x + c\right )^{5} + 20 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 6 \, a^{5} \tan \left (d x + c\right ) + \frac {15 \, {\left (2 \, \sin \left (d x + c\right )^{2} - 1\right )} a^{2} b^{3}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \frac {{\left (3 \, \sin \left (d x + c\right )^{4} - 3 \, \sin \left (d x + c\right )^{2} + 1\right )} b^{5}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - \frac {15 \, a^{4} b}{\sin \left (d x + c\right )^{2} - 1}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 169, normalized size = 5.63 \[ \frac {{\cos \left (c+d\,x\right )}^4\,\left (\frac {5\,a^4\,b}{2}-5\,a^2\,b^3+\frac {b^5}{2}\right )+{\cos \left (c+d\,x\right )}^5\,\left (\sin \left (c+d\,x\right )\,a^5-\frac {10\,\sin \left (c+d\,x\right )\,a^3\,b^2}{3}+\sin \left (c+d\,x\right )\,a\,b^4\right )-{\cos \left (c+d\,x\right )}^2\,\left (\frac {b^5}{2}-\frac {5\,a^2\,b^3}{2}\right )+\frac {b^5}{6}+{\cos \left (c+d\,x\right )}^3\,\left (\frac {10\,a^3\,b^2\,\sin \left (c+d\,x\right )}{3}-2\,a\,b^4\,\sin \left (c+d\,x\right )\right )+a\,b^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^6} \]
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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