Optimal. Leaf size=103 \[ \frac {b^2 \left (3 a^2-b^2\right ) \tan (c+d x)}{d}-\frac {4 a b \left (a^2-b^2\right ) \log (\cos (c+d x))}{d}+x \left (a^4-6 a^2 b^2+b^4\right )+\frac {b (a+b \tan (c+d x))^3}{3 d}+\frac {a b (a+b \tan (c+d x))^2}{d} \]
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Rubi [A] time = 0.16, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3086, 3482, 3528, 3525, 3475} \[ \frac {b^2 \left (3 a^2-b^2\right ) \tan (c+d x)}{d}-\frac {4 a b \left (a^2-b^2\right ) \log (\cos (c+d x))}{d}+x \left (-6 a^2 b^2+a^4+b^4\right )+\frac {b (a+b \tan (c+d x))^3}{3 d}+\frac {a b (a+b \tan (c+d x))^2}{d} \]
Antiderivative was successfully verified.
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Rule 3086
Rule 3475
Rule 3482
Rule 3525
Rule 3528
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx &=\int (a+b \tan (c+d x))^4 \, dx\\ &=\frac {b (a+b \tan (c+d x))^3}{3 d}+\int (a+b \tan (c+d x))^2 \left (a^2-b^2+2 a b \tan (c+d x)\right ) \, dx\\ &=\frac {a b (a+b \tan (c+d x))^2}{d}+\frac {b (a+b \tan (c+d x))^3}{3 d}+\int (a+b \tan (c+d x)) \left (a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\left (a^4-6 a^2 b^2+b^4\right ) x+\frac {b^2 \left (3 a^2-b^2\right ) \tan (c+d x)}{d}+\frac {a b (a+b \tan (c+d x))^2}{d}+\frac {b (a+b \tan (c+d x))^3}{3 d}+\left (4 a b \left (a^2-b^2\right )\right ) \int \tan (c+d x) \, dx\\ &=\left (a^4-6 a^2 b^2+b^4\right ) x-\frac {4 a b \left (a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac {b^2 \left (3 a^2-b^2\right ) \tan (c+d x)}{d}+\frac {a b (a+b \tan (c+d x))^2}{d}+\frac {b (a+b \tan (c+d x))^3}{3 d}\\ \end {align*}
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Mathematica [C] time = 0.40, size = 105, normalized size = 1.02 \[ \frac {-6 b^2 \left (b^2-6 a^2\right ) \tan (c+d x)+12 a b^3 \tan ^2(c+d x)+3 i (a-i b)^4 \log (\tan (c+d x)+i)-3 i (a+i b)^4 \log (-\tan (c+d x)+i)+2 b^4 \tan ^3(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 119, normalized size = 1.16 \[ \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} d x \cos \left (d x + c\right )^{3} + 6 \, a b^{3} \cos \left (d x + c\right ) - 12 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\cos \left (d x + c\right )\right ) + {\left (b^{4} + 2 \, {\left (9 \, a^{2} b^{2} - 2 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 104, normalized size = 1.01 \[ \frac {b^{4} \tan \left (d x + c\right )^{3} + 6 \, a b^{3} \tan \left (d x + c\right )^{2} + 18 \, a^{2} b^{2} \tan \left (d x + c\right ) - 3 \, b^{4} \tan \left (d x + c\right ) + 3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )} + 6 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.92, size = 145, normalized size = 1.41 \[ a^{4} x +\frac {a^{4} c}{d}-\frac {4 a^{3} b \ln \left (\cos \left (d x +c \right )\right )}{d}-6 a^{2} b^{2} x +\frac {6 a^{2} b^{2} \tan \left (d x +c \right )}{d}-\frac {6 a^{2} b^{2} c}{d}+\frac {2 a \,b^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {4 a \,b^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {b^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {b^{4} \tan \left (d x +c \right )}{d}+b^{4} x +\frac {b^{4} c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 116, normalized size = 1.13 \[ \frac {3 \, {\left (d x + c\right )} a^{4} - 18 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{2} b^{2} + {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} b^{4} - 6 \, a b^{3} {\left (\frac {1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )} - 6 \, a^{3} b \log \left (-\sin \left (d x + c\right )^{2} + 1\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.79, size = 546, normalized size = 5.30 \[ \frac {\frac {3\,a^4\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}-\frac {b^4\,\sin \left (3\,c+3\,d\,x\right )}{3}+\frac {3\,b^4\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}-\frac {a\,b^3\,\cos \left (3\,c+3\,d\,x\right )}{2}+\frac {3\,a^2\,b^2\,\sin \left (c+d\,x\right )}{2}+\frac {a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )}{2}+\frac {b^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )}{2}+\frac {3\,a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{2}+\frac {a\,b^3\,\cos \left (c+d\,x\right )}{2}+3\,a\,b^3\,\ln \left (-\frac {\cos \left (c+d\,x\right )}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )\,\cos \left (c+d\,x\right )-3\,a^3\,b\,\ln \left (-\frac {\cos \left (c+d\,x\right )}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )\,\cos \left (c+d\,x\right )-3\,a^2\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )-3\,a\,b^3\,\cos \left (c+d\,x\right )\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )+3\,a^3\,b\,\cos \left (c+d\,x\right )\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )+a\,b^3\,\ln \left (-\frac {\cos \left (c+d\,x\right )}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )\,\cos \left (3\,c+3\,d\,x\right )-a^3\,b\,\ln \left (-\frac {\cos \left (c+d\,x\right )}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )\,\cos \left (3\,c+3\,d\,x\right )-a\,b^3\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )\,\cos \left (3\,c+3\,d\,x\right )+a^3\,b\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )\,\cos \left (3\,c+3\,d\,x\right )-9\,a^2\,b^2\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d\,\left (\frac {3\,\cos \left (c+d\,x\right )}{4}+\frac {\cos \left (3\,c+3\,d\,x\right )}{4}\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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