Optimal. Leaf size=103 \[ \frac {a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {3 a^2 b \sec (c+d x)}{d}-\frac {3 a b^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {3 a b^2 \tan (c+d x) \sec (c+d x)}{2 d}+\frac {b^3 \sec ^3(c+d x)}{3 d}-\frac {b^3 \sec (c+d x)}{d} \]
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Rubi [A] time = 0.12, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3090, 3770, 2606, 8, 2611} \[ \frac {3 a^2 b \sec (c+d x)}{d}+\frac {a^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {3 a b^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {3 a b^2 \tan (c+d x) \sec (c+d x)}{2 d}+\frac {b^3 \sec ^3(c+d x)}{3 d}-\frac {b^3 \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2606
Rule 2611
Rule 3090
Rule 3770
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx &=\int \left (a^3 \sec (c+d x)+3 a^2 b \sec (c+d x) \tan (c+d x)+3 a b^2 \sec (c+d x) \tan ^2(c+d x)+b^3 \sec (c+d x) \tan ^3(c+d x)\right ) \, dx\\ &=a^3 \int \sec (c+d x) \, dx+\left (3 a^2 b\right ) \int \sec (c+d x) \tan (c+d x) \, dx+\left (3 a b^2\right ) \int \sec (c+d x) \tan ^2(c+d x) \, dx+b^3 \int \sec (c+d x) \tan ^3(c+d x) \, dx\\ &=\frac {a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{2} \left (3 a b^2\right ) \int \sec (c+d x) \, dx+\frac {\left (3 a^2 b\right ) \operatorname {Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}+\frac {b^3 \operatorname {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {a^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {3 a b^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {3 a^2 b \sec (c+d x)}{d}-\frac {b^3 \sec (c+d x)}{d}+\frac {b^3 \sec ^3(c+d x)}{3 d}+\frac {3 a b^2 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [B] time = 1.63, size = 293, normalized size = 2.84 \[ \frac {12 a^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )-6 a \left (2 a^2-3 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 b \sin ^2\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (\left (18 a^2-5 b^2\right ) \cos (2 (c+d x))+18 a^2+2 b^2 \cos (c+d x)-b^2\right )+36 a^2 b+\frac {9 a b^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {9 a b^2}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}-18 a b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {b^3}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {b^3}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}-10 b^3}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 123, normalized size = 1.19 \[ \frac {3 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 18 \, a b^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 4 \, b^{3} + 12 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2}}{12 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 171, normalized size = 1.66 \[ \frac {3 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (9 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 18 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 18 \, a^{2} b + 4 \, b^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 10.25, size = 187, normalized size = 1.82 \[ \frac {a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 a^{2} b}{d \cos \left (d x +c \right )}+\frac {3 b^{2} a \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {3 a \,b^{2} \sin \left (d x +c \right )}{2 d}-\frac {3 b^{2} a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )^{3}}-\frac {b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )}-\frac {b^{3} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{3 d}-\frac {2 b^{3} \cos \left (d x +c \right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 118, normalized size = 1.15 \[ -\frac {9 \, a b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 6 \, a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {36 \, a^{2} b}{\cos \left (d x + c\right )} + \frac {4 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} b^{3}}{\cos \left (d x + c\right )^{3}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.38, size = 160, normalized size = 1.55 \[ -\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (3\,a\,b^2-2\,a^3\right )}{d}-\frac {6\,a^2\,b-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (12\,a^2\,b-4\,b^3\right )-\frac {4\,b^3}{3}+3\,a\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-3\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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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