Optimal. Leaf size=120 \[ \frac {a^3 \tan (c+d x)}{d}+\frac {b \left (3 a^2+b^2\right ) \tan ^4(c+d x)}{4 d}+\frac {a \left (a^2+3 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {3 a^2 b \tan ^2(c+d x)}{2 d}+\frac {3 a b^2 \tan ^5(c+d x)}{5 d}+\frac {b^3 \tan ^6(c+d x)}{6 d} \]
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Rubi [A] time = 0.10, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3088, 894} \[ \frac {b \left (3 a^2+b^2\right ) \tan ^4(c+d x)}{4 d}+\frac {a \left (a^2+3 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {3 a^2 b \tan ^2(c+d x)}{2 d}+\frac {a^3 \tan (c+d x)}{d}+\frac {3 a b^2 \tan ^5(c+d x)}{5 d}+\frac {b^3 \tan ^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 894
Rule 3088
Rubi steps
\begin {align*} \int \sec ^7(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 \left (1+x^2\right )}{x^7} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {b^3}{x^7}+\frac {3 a b^2}{x^6}+\frac {3 a^2 b+b^3}{x^5}+\frac {a^3+3 a b^2}{x^4}+\frac {3 a^2 b}{x^3}+\frac {a^3}{x^2}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {a^3 \tan (c+d x)}{d}+\frac {3 a^2 b \tan ^2(c+d x)}{2 d}+\frac {a \left (a^2+3 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {b \left (3 a^2+b^2\right ) \tan ^4(c+d x)}{4 d}+\frac {3 a b^2 \tan ^5(c+d x)}{5 d}+\frac {b^3 \tan ^6(c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 54, normalized size = 0.45 \[ \frac {(a+b \tan (c+d x))^4 \left (a^2-4 a b \tan (c+d x)+10 b^2 \tan ^2(c+d x)+15 b^2\right )}{60 b^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 105, normalized size = 0.88 \[ \frac {10 \, b^{3} + 15 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, {\left (5 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 9 \, a b^{2} \cos \left (d x + c\right ) + {\left (5 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{60 \, d \cos \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 112, normalized size = 0.93 \[ \frac {10 \, b^{3} \tan \left (d x + c\right )^{6} + 36 \, a b^{2} \tan \left (d x + c\right )^{5} + 45 \, a^{2} b \tan \left (d x + c\right )^{4} + 15 \, b^{3} \tan \left (d x + c\right )^{4} + 20 \, a^{3} \tan \left (d x + c\right )^{3} + 60 \, a b^{2} \tan \left (d x + c\right )^{3} + 90 \, a^{2} b \tan \left (d x + c\right )^{2} + 60 \, a^{3} \tan \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 20.80, size = 127, normalized size = 1.06 \[ \frac {-a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+\frac {3 a^{2} b}{4 \cos \left (d x +c \right )^{4}}+3 b^{2} a \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 122, normalized size = 1.02 \[ \frac {20 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} + 12 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 5 \, \tan \left (d x + c\right )^{3}\right )} a b^{2} - \frac {5 \, {\left (3 \, \sin \left (d x + c\right )^{2} - 1\right )} b^{3}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} + \frac {45 \, a^{2} b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.84, size = 123, normalized size = 1.02 \[ \frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {a^3\,\sin \left (c+d\,x\right )}{3}-\frac {a\,b^2\,\sin \left (c+d\,x\right )}{5}\right )+{\cos \left (c+d\,x\right )}^5\,\left (\frac {2\,a^3\,\sin \left (c+d\,x\right )}{3}-\frac {2\,a\,b^2\,\sin \left (c+d\,x\right )}{5}\right )+{\cos \left (c+d\,x\right )}^2\,\left (\frac {3\,a^2\,b}{4}-\frac {b^3}{4}\right )+\frac {b^3}{6}+\frac {3\,a\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{5}}{d\,{\cos \left (c+d\,x\right )}^6} \]
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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