Optimal. Leaf size=83 \[ \frac {\cos ^3(c+d x) (a \sec (c+d x)+a)^{n+2}}{3 a^2 d}-\frac {(4-n) (a \sec (c+d x)+a)^{n+2} \, _2F_1(3,n+2;n+3;\sec (c+d x)+1)}{3 a^2 d (n+2)} \]
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Rubi [A] time = 0.07, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3873, 78, 65} \[ \frac {\cos ^3(c+d x) (a \sec (c+d x)+a)^{n+2}}{3 a^2 d}-\frac {(4-n) (a \sec (c+d x)+a)^{n+2} \, _2F_1(3,n+2;n+3;\sec (c+d x)+1)}{3 a^2 d (n+2)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 78
Rule 3873
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^n \sin ^3(c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(-a-a x) (a-a x)^{1+n}}{x^4} \, dx,x,-\sec (c+d x)\right )}{a^2 d}\\ &=\frac {\cos ^3(c+d x) (a+a \sec (c+d x))^{2+n}}{3 a^2 d}+\frac {(4-n) \operatorname {Subst}\left (\int \frac {(a-a x)^{1+n}}{x^3} \, dx,x,-\sec (c+d x)\right )}{3 a d}\\ &=\frac {\cos ^3(c+d x) (a+a \sec (c+d x))^{2+n}}{3 a^2 d}-\frac {(4-n) \, _2F_1(3,2+n;3+n;1+\sec (c+d x)) (a+a \sec (c+d x))^{2+n}}{3 a^2 d (2+n)}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 67, normalized size = 0.81 \[ \frac {(\sec (c+d x)+1)^2 (a (\sec (c+d x)+1))^n \left ((n-4) \, _2F_1(3,n+2;n+3;\sec (c+d x)+1)+(n+2) \cos ^3(c+d x)\right )}{3 d (n+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (d x + c\right )^{2} - 1\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.02, size = 0, normalized size = 0.00 \[ \int \left (a +a \sec \left (d x +c \right )\right )^{n} \left (\sin ^{3}\left (d x +c \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\sin \left (c+d\,x\right )}^3\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
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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