Optimal. Leaf size=119 \[ -\frac {b \cos ^2(c+d x)}{a^3 d}+\frac {\cos ^3(c+d x)}{3 a^2 d}+\frac {b^2 \left (a^2-b^2\right )}{a^5 d (a \cos (c+d x)+b)}+\frac {2 b \left (a^2-2 b^2\right ) \log (a \cos (c+d x)+b)}{a^5 d}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{a^4 d} \]
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Rubi [A] time = 0.23, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3872, 2837, 12, 894} \[ -\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{a^4 d}+\frac {b^2 \left (a^2-b^2\right )}{a^5 d (a \cos (c+d x)+b)}+\frac {2 b \left (a^2-2 b^2\right ) \log (a \cos (c+d x)+b)}{a^5 d}-\frac {b \cos ^2(c+d x)}{a^3 d}+\frac {\cos ^3(c+d x)}{3 a^2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 894
Rule 2837
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sin ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(-b-a \cos (c+d x))^2} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a^2-x^2\right )}{a^2 (-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a^2-x^2\right )}{(-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2 \left (1-\frac {3 b^2}{a^2}\right )-\frac {b^2 \left (-a^2+b^2\right )}{(b-x)^2}+\frac {2 b \left (-a^2+2 b^2\right )}{b-x}-2 b x-x^2\right ) \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{a^4 d}-\frac {b \cos ^2(c+d x)}{a^3 d}+\frac {\cos ^3(c+d x)}{3 a^2 d}+\frac {b^2 \left (a^2-b^2\right )}{a^5 d (b+a \cos (c+d x))}+\frac {2 b \left (a^2-2 b^2\right ) \log (b+a \cos (c+d x))}{a^5 d}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 167, normalized size = 1.40 \[ \frac {a^4 \cos (4 (c+d x))-9 a^4-4 a^3 b \cos (3 (c+d x))+48 a^2 b^2 \log (a \cos (c+d x)+b)+24 a b \cos (c+d x) \left (2 \left (a^2-2 b^2\right ) \log (a \cos (c+d x)+b)-a^2+3 b^2\right )+60 a^2 b^2-8 \left (a^4-3 a^2 b^2\right ) \cos (2 (c+d x))-96 b^4 \log (a \cos (c+d x)+b)-24 b^4}{24 a^5 d (a \cos (c+d x)+b)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 150, normalized size = 1.26 \[ \frac {2 \, a^{4} \cos \left (d x + c\right )^{4} - 4 \, a^{3} b \cos \left (d x + c\right )^{3} + 9 \, a^{2} b^{2} - 6 \, b^{4} - 6 \, {\left (a^{4} - 2 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (a^{3} b - 6 \, a b^{3}\right )} \cos \left (d x + c\right ) + 12 \, {\left (a^{2} b^{2} - 2 \, b^{4} + {\left (a^{3} b - 2 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{6 \, {\left (a^{6} d \cos \left (d x + c\right ) + a^{5} b d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 139, normalized size = 1.17 \[ \frac {2 \, {\left (a^{2} b - 2 \, b^{3}\right )} \log \left ({\left | -a \cos \left (d x + c\right ) - b \right |}\right )}{a^{5} d} + \frac {a^{2} b^{2} - b^{4}}{{\left (a \cos \left (d x + c\right ) + b\right )} a^{5} d} + \frac {a^{4} d^{5} \cos \left (d x + c\right )^{3} - 3 \, a^{3} b d^{5} \cos \left (d x + c\right )^{2} - 3 \, a^{4} d^{5} \cos \left (d x + c\right ) + 9 \, a^{2} b^{2} d^{5} \cos \left (d x + c\right )}{3 \, a^{6} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 153, normalized size = 1.29 \[ \frac {\cos ^{3}\left (d x +c \right )}{3 a^{2} d}-\frac {b \left (\cos ^{2}\left (d x +c \right )\right )}{a^{3} d}-\frac {\cos \left (d x +c \right )}{a^{2} d}+\frac {3 \cos \left (d x +c \right ) b^{2}}{d \,a^{4}}+\frac {2 b \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{3} d}-\frac {4 b^{3} \ln \left (b +a \cos \left (d x +c \right )\right )}{d \,a^{5}}+\frac {b^{2}}{a^{3} d \left (b +a \cos \left (d x +c \right )\right )}-\frac {b^{4}}{d \,a^{5} \left (b +a \cos \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 112, normalized size = 0.94 \[ \frac {\frac {3 \, {\left (a^{2} b^{2} - b^{4}\right )}}{a^{6} \cos \left (d x + c\right ) + a^{5} b} + \frac {a^{2} \cos \left (d x + c\right )^{3} - 3 \, a b \cos \left (d x + c\right )^{2} - 3 \, {\left (a^{2} - 3 \, b^{2}\right )} \cos \left (d x + c\right )}{a^{4}} + \frac {6 \, {\left (a^{2} b - 2 \, b^{3}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{5}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 113, normalized size = 0.95 \[ -\frac {\cos \left (c+d\,x\right )\,\left (\frac {1}{a^2}-\frac {3\,b^2}{a^4}\right )-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a^2}+\frac {b\,{\cos \left (c+d\,x\right )}^2}{a^3}-\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (2\,a^2\,b-4\,b^3\right )}{a^5}+\frac {b^4-a^2\,b^2}{a\,\left (\cos \left (c+d\,x\right )\,a^5+b\,a^4\right )}}{d} \]
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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