Optimal. Leaf size=89 \[ -\frac {b \cos ^2(c+d x)}{2 a^2 d}+\frac {b \left (a^2-b^2\right ) \log (a \cos (c+d x)+b)}{a^4 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x)}{a^3 d}+\frac {\cos ^3(c+d x)}{3 a d} \]
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Rubi [A] time = 0.16, antiderivative size = 89, 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, 772} \[ -\frac {\left (a^2-b^2\right ) \cos (c+d x)}{a^3 d}+\frac {b \left (a^2-b^2\right ) \log (a \cos (c+d x)+b)}{a^4 d}-\frac {b \cos ^2(c+d x)}{2 a^2 d}+\frac {\cos ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 772
Rule 2837
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sin ^3(c+d x)}{a+b \sec (c+d x)} \, dx &=-\int \frac {\cos (c+d x) \sin ^3(c+d x)}{-b-a \cos (c+d x)} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x \left (a^2-x^2\right )}{a (-b+x)} \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x \left (a^2-x^2\right )}{-b+x} \, dx,x,-a \cos (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2 \left (1-\frac {b^2}{a^2}\right )+\frac {-a^2 b+b^3}{b-x}-b x-x^2\right ) \, dx,x,-a \cos (c+d x)\right )}{a^4 d}\\ &=-\frac {\left (a^2-b^2\right ) \cos (c+d x)}{a^3 d}-\frac {b \cos ^2(c+d x)}{2 a^2 d}+\frac {\cos ^3(c+d x)}{3 a d}+\frac {b \left (a^2-b^2\right ) \log (b+a \cos (c+d x))}{a^4 d}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 89, normalized size = 1.00 \[ \frac {\left (12 a b^2-9 a^3\right ) \cos (c+d x)+a^3 \cos (3 (c+d x))-3 a^2 b \cos (2 (c+d x))+12 a^2 b \log (a \cos (c+d x)+b)-12 b^3 \log (a \cos (c+d x)+b)}{12 a^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 78, normalized size = 0.88 \[ \frac {2 \, a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{2} b \cos \left (d x + c\right )^{2} - 6 \, {\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right ) + 6 \, {\left (a^{2} b - b^{3}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{6 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 102, normalized size = 1.15 \[ \frac {{\left (a^{2} b - b^{3}\right )} \log \left ({\left | -a \cos \left (d x + c\right ) - b \right |}\right )}{a^{4} d} + \frac {2 \, a^{2} d^{2} \cos \left (d x + c\right )^{3} - 3 \, a b d^{2} \cos \left (d x + c\right )^{2} - 6 \, a^{2} d^{2} \cos \left (d x + c\right ) + 6 \, b^{2} d^{2} \cos \left (d x + c\right )}{6 \, a^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 106, normalized size = 1.19 \[ \frac {\cos ^{3}\left (d x +c \right )}{3 d a}-\frac {b \left (\cos ^{2}\left (d x +c \right )\right )}{2 a^{2} d}-\frac {\cos \left (d x +c \right )}{d a}+\frac {\cos \left (d x +c \right ) b^{2}}{d \,a^{3}}+\frac {b \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{2} d}-\frac {b^{3} \ln \left (b +a \cos \left (d x +c \right )\right )}{d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 80, normalized size = 0.90 \[ \frac {\frac {2 \, a^{2} \cos \left (d x + c\right )^{3} - 3 \, a b \cos \left (d x + c\right )^{2} - 6 \, {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )}{a^{3}} + \frac {6 \, {\left (a^{2} b - b^{3}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{4}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 79, normalized size = 0.89 \[ -\frac {\cos \left (c+d\,x\right )\,\left (\frac {1}{a}-\frac {b^2}{a^3}\right )-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a}+\frac {b\,{\cos \left (c+d\,x\right )}^2}{2\,a^2}-\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (a^2\,b-b^3\right )}{a^4}}{d} \]
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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