Optimal. Leaf size=64 \[ -\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^2 b \log (\cos (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.10, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3872, 2833, 12, 43} \[ -\frac {3 a^2 b \log (\cos (c+d x))}{d}-\frac {a^3 \cos (c+d x)}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rule 3872
Rubi steps
\begin {align*} \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx &=-\int (-b-a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {a^3 (-b+x)^3}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \frac {(-b+x)^3}{x^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \left (1-\frac {b^3}{x^3}+\frac {3 b^2}{x^2}-\frac {3 b}{x}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^2 b \log (\cos (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 56, normalized size = 0.88 \[ \frac {b \left (-6 a^2 \log (\cos (c+d x))+6 a b \sec (c+d x)+b^2 \sec ^2(c+d x)\right )-2 a^3 \cos (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 67, normalized size = 1.05 \[ -\frac {2 \, a^{3} \cos \left (d x + c\right )^{3} + 6 \, a^{2} b \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 6 \, a b^{2} \cos \left (d x + c\right ) - b^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.56, size = 66, normalized size = 1.03 \[ -\frac {a^{3} \cos \left (d x + c\right )}{d} - \frac {3 \, a^{2} b \log \left (\frac {{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac {6 \, a b^{2} \cos \left (d x + c\right ) + b^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 65, normalized size = 1.02 \[ \frac {b^{3} \left (\sec ^{2}\left (d x +c \right )\right )}{2 d}+\frac {3 a \,b^{2} \sec \left (d x +c \right )}{d}+\frac {3 a^{2} b \ln \left (\sec \left (d x +c \right )\right )}{d}-\frac {a^{3}}{d \sec \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 57, normalized size = 0.89 \[ -\frac {2 \, a^{3} \cos \left (d x + c\right ) + 6 \, a^{2} b \log \left (\cos \left (d x + c\right )\right ) - \frac {6 \, a b^{2}}{\cos \left (d x + c\right )} - \frac {b^{3}}{\cos \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 57, normalized size = 0.89 \[ -\frac {a^3\,\cos \left (c+d\,x\right )-\frac {\frac {b^3}{2}+3\,a\,\cos \left (c+d\,x\right )\,b^2}{{\cos \left (c+d\,x\right )}^2}+3\,a^2\,b\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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 \[ \int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \sin {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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