Optimal. Leaf size=232 \[ -\frac {4 a \left (5 a^2+3 b^2\right ) \log (\tan (c+d x))}{b^7 d}-\frac {4 a \left (5 a^2+3 b^2\right ) \log (a \cot (c+d x)+b)}{b^7 d}+\frac {\left (10 a^2+3 b^2\right ) \tan (c+d x)}{b^6 d}+\frac {\left (a^2+b^2\right )^3}{3 a^3 b^4 d (a \cot (c+d x)+b)^3}+\frac {10 a^6+9 a^4 b^2+b^6}{a^3 b^6 d (a \cot (c+d x)+b)}+\frac {2 a^6+3 a^4 b^2-b^6}{a^3 b^5 d (a \cot (c+d x)+b)^2}-\frac {2 a \tan ^2(c+d x)}{b^5 d}+\frac {\tan ^3(c+d x)}{3 b^4 d} \]
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Rubi [A] time = 0.25, antiderivative size = 232, 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} \[ \frac {\left (10 a^2+3 b^2\right ) \tan (c+d x)}{b^6 d}+\frac {\left (a^2+b^2\right )^3}{3 a^3 b^4 d (a \cot (c+d x)+b)^3}+\frac {9 a^4 b^2+10 a^6+b^6}{a^3 b^6 d (a \cot (c+d x)+b)}+\frac {3 a^4 b^2+2 a^6-b^6}{a^3 b^5 d (a \cot (c+d x)+b)^2}-\frac {4 a \left (5 a^2+3 b^2\right ) \log (\tan (c+d x))}{b^7 d}-\frac {4 a \left (5 a^2+3 b^2\right ) \log (a \cot (c+d x)+b)}{b^7 d}-\frac {2 a \tan ^2(c+d x)}{b^5 d}+\frac {\tan ^3(c+d x)}{3 b^4 d} \]
Antiderivative was successfully verified.
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Rule 894
Rule 3088
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^4 (b+a x)^4} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{b^4 x^4}-\frac {4 a}{b^5 x^3}+\frac {10 a^2+3 b^2}{b^6 x^2}-\frac {4 \left (5 a^3+3 a b^2\right )}{b^7 x}+\frac {\left (a^2+b^2\right )^3}{a^2 b^4 (b+a x)^4}+\frac {2 \left (2 a^6+3 a^4 b^2-b^6\right )}{a^2 b^5 (b+a x)^3}+\frac {10 a^6+9 a^4 b^2+b^6}{a^2 b^6 (b+a x)^2}+\frac {4 \left (5 a^4+3 a^2 b^2\right )}{b^7 (b+a x)}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {\left (a^2+b^2\right )^3}{3 a^3 b^4 d (b+a \cot (c+d x))^3}+\frac {2 a^6+3 a^4 b^2-b^6}{a^3 b^5 d (b+a \cot (c+d x))^2}+\frac {10 a^6+9 a^4 b^2+b^6}{a^3 b^6 d (b+a \cot (c+d x))}-\frac {4 a \left (5 a^2+3 b^2\right ) \log (b+a \cot (c+d x))}{b^7 d}-\frac {4 a \left (5 a^2+3 b^2\right ) \log (\tan (c+d x))}{b^7 d}+\frac {\left (10 a^2+3 b^2\right ) \tan (c+d x)}{b^6 d}-\frac {2 a \tan ^2(c+d x)}{b^5 d}+\frac {\tan ^3(c+d x)}{3 b^4 d}\\ \end {align*}
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Mathematica [A] time = 2.01, size = 295, normalized size = 1.27 \[ \frac {3 b^4 \sec ^4(c+d x) \left (a^2-a b \tan (c+d x)+2 b^2\right )-2 \left (37 a^6+36 a^4 b^2-6 a^2 b^4 \tan ^4(c+d x)+3 a^2 b^4+6 a b^3 \tan ^3(c+d x) \left (\left (5 a^2+3 b^2\right ) \log (a+b \tan (c+d x))-3 a^2\right )+6 a^4 \left (5 a^2+3 b^2\right ) \log (a+b \tan (c+d x))+6 b^2 \tan ^2(c+d x) \left (6 a^4+3 a^2 \left (5 a^2+3 b^2\right ) \log (a+b \tan (c+d x))+11 a^2 b^2+2 b^4\right )+3 a b \tan (c+d x) \left (27 a^4+6 a^2 \left (5 a^2+3 b^2\right ) \log (a+b \tan (c+d x))+30 a^2 b^2+b^4\right )+4 b^6\right )+b^6 \sec ^6(c+d x)}{3 b^7 d (a+b \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 553, normalized size = 2.38 \[ -\frac {4 \, {\left (45 \, a^{4} b^{2} - 3 \, a^{2} b^{4} - 4 \, b^{6}\right )} \cos \left (d x + c\right )^{6} - b^{6} - 6 \, {\left (25 \, a^{4} b^{2} - 5 \, a^{2} b^{4} - 4 \, b^{6}\right )} \cos \left (d x + c\right )^{4} - 3 \, {\left (5 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left ({\left (5 \, a^{6} - 12 \, a^{4} b^{2} - 9 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (5 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{4} + {\left ({\left (15 \, a^{5} b + 4 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} + {\left (5 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 6 \, {\left ({\left (5 \, a^{6} - 12 \, a^{4} b^{2} - 9 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (5 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{4} + {\left ({\left (15 \, a^{5} b + 4 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} + {\left (5 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) + {\left (3 \, a b^{5} \cos \left (d x + c\right ) - 4 \, {\left (15 \, a^{5} b - 41 \, a^{3} b^{3} - 12 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} - 2 \, {\left (55 \, a^{3} b^{3} + 21 \, a b^{5}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{3 \, {\left (3 \, a b^{9} d \cos \left (d x + c\right )^{4} + {\left (a^{3} b^{7} - 3 \, a b^{9}\right )} d \cos \left (d x + c\right )^{6} + {\left (b^{10} d \cos \left (d x + c\right )^{3} + {\left (3 \, a^{2} b^{8} - b^{10}\right )} d \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 249, normalized size = 1.07 \[ -\frac {\frac {12 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b^{7}} - \frac {110 \, a^{3} b^{3} \tan \left (d x + c\right )^{3} + 66 \, a b^{5} \tan \left (d x + c\right )^{3} + 285 \, a^{4} b^{2} \tan \left (d x + c\right )^{2} + 144 \, a^{2} b^{4} \tan \left (d x + c\right )^{2} - 9 \, b^{6} \tan \left (d x + c\right )^{2} + 249 \, a^{5} b \tan \left (d x + c\right ) + 108 \, a^{3} b^{3} \tan \left (d x + c\right ) - 9 \, a b^{5} \tan \left (d x + c\right ) + 73 \, a^{6} + 27 \, a^{4} b^{2} - 3 \, a^{2} b^{4} - b^{6}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} b^{7}} - \frac {b^{8} \tan \left (d x + c\right )^{3} - 6 \, a b^{7} \tan \left (d x + c\right )^{2} + 30 \, a^{2} b^{6} \tan \left (d x + c\right ) + 9 \, b^{8} \tan \left (d x + c\right )}{b^{12}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 330, normalized size = 1.42 \[ \frac {\tan ^{3}\left (d x +c \right )}{3 b^{4} d}-\frac {2 a \left (\tan ^{2}\left (d x +c \right )\right )}{b^{5} d}+\frac {10 a^{2} \tan \left (d x +c \right )}{d \,b^{6}}+\frac {3 \tan \left (d x +c \right )}{b^{4} d}-\frac {20 a^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,b^{7}}-\frac {12 a \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,b^{5}}+\frac {3 a^{5}}{d \,b^{7} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {6 a^{3}}{d \,b^{5} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {3 a}{d \,b^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {a^{6}}{3 d \,b^{7} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {a^{4}}{d \,b^{5} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {a^{2}}{d \,b^{3} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {1}{3 d b \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {15 a^{4}}{d \,b^{7} \left (a +b \tan \left (d x +c \right )\right )}-\frac {18 a^{2}}{d \,b^{5} \left (a +b \tan \left (d x +c \right )\right )}-\frac {3}{d \,b^{3} \left (a +b \tan \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 217, normalized size = 0.94 \[ -\frac {\frac {37 \, a^{6} + 39 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6} + 9 \, {\left (5 \, a^{4} b^{2} + 6 \, a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )^{2} + 9 \, {\left (9 \, a^{5} b + 10 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (d x + c\right )}{b^{10} \tan \left (d x + c\right )^{3} + 3 \, a b^{9} \tan \left (d x + c\right )^{2} + 3 \, a^{2} b^{8} \tan \left (d x + c\right ) + a^{3} b^{7}} - \frac {b^{2} \tan \left (d x + c\right )^{3} - 6 \, a b \tan \left (d x + c\right )^{2} + 3 \, {\left (10 \, a^{2} + 3 \, b^{2}\right )} \tan \left (d x + c\right )}{b^{6}} + \frac {12 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{b^{7}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.97, size = 1599, normalized size = 6.89 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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