Optimal. Leaf size=89 \[ \frac {a \sin ^{n+1}(c+d x) \, _2F_1\left (3,\frac {n+1}{2};\frac {n+3}{2};\sin ^2(c+d x)\right )}{d (n+1)}+\frac {b \sin ^{n+2}(c+d x) \, _2F_1\left (3,\frac {n+2}{2};\frac {n+4}{2};\sin ^2(c+d x)\right )}{d (n+2)} \]
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Rubi [A] time = 0.11, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2837, 808, 364} \[ \frac {a \sin ^{n+1}(c+d x) \, _2F_1\left (3,\frac {n+1}{2};\frac {n+3}{2};\sin ^2(c+d x)\right )}{d (n+1)}+\frac {b \sin ^{n+2}(c+d x) \, _2F_1\left (3,\frac {n+2}{2};\frac {n+4}{2};\sin ^2(c+d x)\right )}{d (n+2)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 808
Rule 2837
Rubi steps
\begin {align*} \int \sec ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x)) \, dx &=\frac {b^5 \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n (a+x)}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\left (a b^5\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}+\frac {b^6 \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^{1+n}}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {a \, _2F_1\left (3,\frac {1+n}{2};\frac {3+n}{2};\sin ^2(c+d x)\right ) \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {b \, _2F_1\left (3,\frac {2+n}{2};\frac {4+n}{2};\sin ^2(c+d x)\right ) \sin ^{2+n}(c+d x)}{d (2+n)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 89, normalized size = 1.00 \[ \frac {\sin ^{n+1}(c+d x) \left (a (n+2) \, _2F_1\left (3,\frac {n+1}{2};\frac {n+3}{2};\sin ^2(c+d x)\right )+b (n+1) \sin (c+d x) \, _2F_1\left (3,\frac {n+2}{2};\frac {n+4}{2};\sin ^2(c+d x)\right )\right )}{d (n+1) (n+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (d x + c\right )^{5} \sin \left (d x + c\right ) + a \sec \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )^{n} \sec \left (d x + c\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.70, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{5}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )^{n} \sec \left (d x + c\right )^{5}\,{d x} \]
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 \[ \int \frac {{\sin \left (c+d\,x\right )}^n\,\left (a+b\,\sin \left (c+d\,x\right )\right )}{{\cos \left (c+d\,x\right )}^5} \,d x \]
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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