Optimal. Leaf size=161 \[ \frac {2 \left (3 a^2+b^2\right ) \log (\tan (c+d x))}{b^5 d}+\frac {2 \left (3 a^2+b^2\right ) \log (a \cot (c+d x)+b)}{b^5 d}-\frac {\left (3 a^2-b^2\right ) \left (a^2+b^2\right )}{a^2 b^4 d (a \cot (c+d x)+b)}-\frac {\left (a^2+b^2\right )^2}{2 a^2 b^3 d (a \cot (c+d x)+b)^2}-\frac {3 a \tan (c+d x)}{b^4 d}+\frac {\tan ^2(c+d x)}{2 b^3 d} \]
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Rubi [A] time = 0.17, antiderivative size = 161, 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} \[ -\frac {\left (a^2+b^2\right )^2}{2 a^2 b^3 d (a \cot (c+d x)+b)^2}-\frac {\left (3 a^2-b^2\right ) \left (a^2+b^2\right )}{a^2 b^4 d (a \cot (c+d x)+b)}+\frac {2 \left (3 a^2+b^2\right ) \log (\tan (c+d x))}{b^5 d}+\frac {2 \left (3 a^2+b^2\right ) \log (a \cot (c+d x)+b)}{b^5 d}-\frac {3 a \tan (c+d x)}{b^4 d}+\frac {\tan ^2(c+d x)}{2 b^3 d} \]
Antiderivative was successfully verified.
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Rule 894
Rule 3088
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^3 (b+a x)^3} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{b^3 x^3}-\frac {3 a}{b^4 x^2}+\frac {2 \left (3 a^2+b^2\right )}{b^5 x}-\frac {\left (a^2+b^2\right )^2}{a b^3 (b+a x)^3}+\frac {-3 a^4-2 a^2 b^2+b^4}{a b^4 (b+a x)^2}-\frac {2 a \left (3 a^2+b^2\right )}{b^5 (b+a x)}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {\left (a^2+b^2\right )^2}{2 a^2 b^3 d (b+a \cot (c+d x))^2}-\frac {\left (3 a^2-b^2\right ) \left (a^2+b^2\right )}{a^2 b^4 d (b+a \cot (c+d x))}+\frac {2 \left (3 a^2+b^2\right ) \log (b+a \cot (c+d x))}{b^5 d}+\frac {2 \left (3 a^2+b^2\right ) \log (\tan (c+d x))}{b^5 d}-\frac {3 a \tan (c+d x)}{b^4 d}+\frac {\tan ^2(c+d x)}{2 b^3 d}\\ \end {align*}
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Mathematica [A] time = 3.00, size = 140, normalized size = 0.87 \[ \frac {-2 a \left (-\frac {a^2+b^2}{a+b \tan (c+d x)}-2 a \log (a+b \tan (c+d x))+b \tan (c+d x)\right )+2 \left (a^2+b^2\right ) \left (\frac {3 a^2+4 a b \tan (c+d x)-b^2}{2 (a+b \tan (c+d x))^2}+\log (a+b \tan (c+d x))\right )+\frac {b^4 \sec ^4(c+d x)}{2 (a+b \tan (c+d x))^2}}{b^5 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 354, normalized size = 2.20 \[ \frac {24 \, a^{2} b^{2} \cos \left (d x + c\right )^{4} + b^{4} - 2 \, {\left (9 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left ({\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + {\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 2 \, {\left ({\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + {\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) - 4 \, {\left (a b^{3} \cos \left (d x + c\right ) + 3 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (2 \, a b^{6} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + b^{7} d \cos \left (d x + c\right )^{2} + {\left (a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 140, normalized size = 0.87 \[ \frac {\frac {4 \, {\left (3 \, a^{2} + b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b^{5}} + \frac {b^{3} \tan \left (d x + c\right )^{2} - 6 \, a b^{2} \tan \left (d x + c\right )}{b^{6}} - \frac {18 \, a^{2} b^{2} \tan \left (d x + c\right )^{2} + 6 \, b^{4} \tan \left (d x + c\right )^{2} + 28 \, a^{3} b \tan \left (d x + c\right ) + 4 \, a b^{3} \tan \left (d x + c\right ) + 11 \, a^{4} + b^{4}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} b^{5}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 184, normalized size = 1.14 \[ \frac {\tan ^{2}\left (d x +c \right )}{2 b^{3} d}-\frac {3 a \tan \left (d x +c \right )}{b^{4} d}+\frac {4 a^{3}}{d \,b^{5} \left (a +b \tan \left (d x +c \right )\right )}+\frac {4 a}{d \,b^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {6 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{2}}{d \,b^{5}}+\frac {2 \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,b^{3}}-\frac {a^{4}}{2 d \,b^{5} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {a^{2}}{d \,b^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {1}{2 b d \left (a +b \tan \left (d x +c \right )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 652, normalized size = 4.05 \[ -\frac {2 \, {\left (\frac {\frac {{\left (6 \, a^{5} + 2 \, a^{3} b^{2} - a b^{4}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {{\left (18 \, a^{4} b + 6 \, a^{2} b^{3} - b^{5}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {{\left (18 \, a^{5} - 2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {2 \, {\left (18 \, a^{4} b + 8 \, a^{2} b^{3} - b^{5}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {{\left (18 \, a^{5} - 2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {{\left (18 \, a^{4} b + 6 \, a^{2} b^{3} - b^{5}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {{\left (6 \, a^{5} + 2 \, a^{3} b^{2} - a b^{4}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4} b^{4} + \frac {4 \, a^{3} b^{5} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {12 \, a^{3} b^{5} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {12 \, a^{3} b^{5} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {4 \, a^{3} b^{5} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {a^{4} b^{4} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {4 \, {\left (a^{4} b^{4} - a^{2} b^{6}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, {\left (3 \, a^{4} b^{4} - 4 \, a^{2} b^{6}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4 \, {\left (a^{4} b^{4} - a^{2} b^{6}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {{\left (3 \, a^{2} + b^{2}\right )} \log \left (-a - \frac {2 \, b \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{b^{5}} + \frac {{\left (3 \, a^{2} + b^{2}\right )} \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{b^{5}} + \frac {{\left (3 \, a^{2} + b^{2}\right )} \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{b^{5}}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.74, size = 1204, normalized size = 7.48 \[ \text {result too large to display} \]
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 \[ \int \frac {\sec ^{3}{\left (c + d x \right )}}{\left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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