Optimal. Leaf size=151 \[ \frac {a^4 \sin (c+d x)}{d}-\frac {4 a^3 b \cos (c+d x)}{d}-\frac {6 a^2 b^2 \sin (c+d x)}{d}+\frac {6 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {4 a b^3 \cos (c+d x)}{d}+\frac {4 a b^3 \sec (c+d x)}{d}+\frac {3 b^4 \sin (c+d x)}{2 d}+\frac {b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac {3 b^4 \tanh ^{-1}(\sin (c+d x))}{2 d} \]
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Rubi [A] time = 0.16, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {3090, 2637, 2638, 2592, 321, 206, 2590, 14, 288} \[ -\frac {6 a^2 b^2 \sin (c+d x)}{d}+\frac {6 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {4 a^3 b \cos (c+d x)}{d}+\frac {a^4 \sin (c+d x)}{d}+\frac {4 a b^3 \cos (c+d x)}{d}+\frac {4 a b^3 \sec (c+d x)}{d}+\frac {3 b^4 \sin (c+d x)}{2 d}+\frac {b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac {3 b^4 \tanh ^{-1}(\sin (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 206
Rule 288
Rule 321
Rule 2590
Rule 2592
Rule 2637
Rule 2638
Rule 3090
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx &=\int \left (a^4 \cos (c+d x)+4 a^3 b \sin (c+d x)+6 a^2 b^2 \sin (c+d x) \tan (c+d x)+4 a b^3 \sin (c+d x) \tan ^2(c+d x)+b^4 \sin (c+d x) \tan ^3(c+d x)\right ) \, dx\\ &=a^4 \int \cos (c+d x) \, dx+\left (4 a^3 b\right ) \int \sin (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \sin (c+d x) \tan (c+d x) \, dx+\left (4 a b^3\right ) \int \sin (c+d x) \tan ^2(c+d x) \, dx+b^4 \int \sin (c+d x) \tan ^3(c+d x) \, dx\\ &=-\frac {4 a^3 b \cos (c+d x)}{d}+\frac {a^4 \sin (c+d x)}{d}+\frac {\left (6 a^2 b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (4 a b^3\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^4 \operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {4 a^3 b \cos (c+d x)}{d}+\frac {a^4 \sin (c+d x)}{d}-\frac {6 a^2 b^2 \sin (c+d x)}{d}+\frac {b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}+\frac {\left (6 a^2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (4 a b^3\right ) \operatorname {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 b^4\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (c+d x)\right )}{2 d}\\ &=\frac {6 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {4 a^3 b \cos (c+d x)}{d}+\frac {4 a b^3 \cos (c+d x)}{d}+\frac {4 a b^3 \sec (c+d x)}{d}+\frac {a^4 \sin (c+d x)}{d}-\frac {6 a^2 b^2 \sin (c+d x)}{d}+\frac {3 b^4 \sin (c+d x)}{2 d}+\frac {b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac {\left (3 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{2 d}\\ &=\frac {6 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {3 b^4 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {4 a^3 b \cos (c+d x)}{d}+\frac {4 a b^3 \cos (c+d x)}{d}+\frac {4 a b^3 \sec (c+d x)}{d}+\frac {a^4 \sin (c+d x)}{d}-\frac {6 a^2 b^2 \sin (c+d x)}{d}+\frac {3 b^4 \sin (c+d x)}{2 d}+\frac {b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 2.56, size = 268, normalized size = 1.77 \[ \frac {4 a^4 \sin (c+d x)-24 a^2 b^2 \sin (c+d x)-16 a b \left (a^2-b^2\right ) \cos (c+d x)-24 a^2 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 a^2 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+32 a b^3 \sin ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)+16 a b^3+4 b^4 \sin (c+d x)+\frac {b^4}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {b^4}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+6 b^4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-6 b^4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 153, normalized size = 1.01 \[ \frac {16 \, a b^{3} \cos \left (d x + c\right ) - 16 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (4 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (b^{4} + 2 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 206, normalized size = 1.36 \[ \frac {3 \, {\left (4 \, a^{2} b^{2} - b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (4 \, a^{2} b^{2} - b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {4 \, {\left (a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a^{3} b + 4 \, a b^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {2 \, {\left (b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, a b^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 29.43, size = 211, normalized size = 1.40 \[ \frac {a^{4} \sin \left (d x +c \right )}{d}-\frac {4 a^{3} b \cos \left (d x +c \right )}{d}+\frac {6 a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {6 a^{2} b^{2} \sin \left (d x +c \right )}{d}+\frac {4 a \,b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right ) a \,b^{3}}{d}+\frac {8 a \,b^{3} \cos \left (d x +c \right )}{d}+\frac {b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {b^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d}+\frac {3 b^{4} \sin \left (d x +c \right )}{2 d}-\frac {3 b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 142, normalized size = 0.94 \[ -\frac {b^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, \sin \left (d x + c\right )\right )} - 16 \, a b^{3} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} - 12 \, a^{2} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, \sin \left (d x + c\right )\right )} + 16 \, a^{3} b \cos \left (d x + c\right ) - 4 \, a^{4} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.96, size = 221, normalized size = 1.46 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (4\,a^4-24\,a^2\,b^2+2\,b^4\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (2\,a^4-12\,a^2\,b^2+3\,b^4\right )-16\,a\,b^3+8\,a^3\,b-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^4-12\,a^2\,b^2+3\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (16\,a\,b^3-16\,a^3\,b\right )+8\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (3\,b^4-12\,a^2\,b^2\right )}{d} \]
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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