Optimal. Leaf size=186 \[ \frac{a \left (a^2 (3-n)-3 b^2 (n+1)\right ) \sin ^{n+1}(c+d x) \, _2F_1\left (2,\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right )}{4 d (n+1)}+\frac{b \left (3 a^2 (2-n)-b^2 (n+2)\right ) \sin ^{n+2}(c+d x) \, _2F_1\left (2,\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right )}{4 d (n+2)}+\frac{\sec ^4(c+d x) \sin ^{n+1}(c+d x) \left (b \left (3 a^2+b^2\right ) \sin (c+d x)+a \left (a^2+3 b^2\right )\right )}{4 d} \]
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Rubi [A] time = 0.296152, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {2837, 1806, 808, 364} \[ \frac{a \left (a^2 (3-n)-3 b^2 (n+1)\right ) \sin ^{n+1}(c+d x) \, _2F_1\left (2,\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right )}{4 d (n+1)}+\frac{b \left (3 a^2 (2-n)-b^2 (n+2)\right ) \sin ^{n+2}(c+d x) \, _2F_1\left (2,\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right )}{4 d (n+2)}+\frac{\sec ^4(c+d x) \sin ^{n+1}(c+d x) \left (b \left (3 a^2+b^2\right ) \sin (c+d x)+a \left (a^2+3 b^2\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 1806
Rule 808
Rule 364
Rubi steps
\begin{align*} \int \sec ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac{b^5 \operatorname{Subst}\left (\int \frac{\left (\frac{x}{b}\right )^n (a+x)^3}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{\sec ^4(c+d x) \sin ^{1+n}(c+d x) \left (a \left (a^2+3 b^2\right )+b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 d}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\left (\frac{x}{b}\right )^n \left (-a \left (a^2 (3-n)-3 b^2 (1+n)\right )-\left (3 a^2 (2-n)-b^2 (2+n)\right ) x\right )}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=\frac{\sec ^4(c+d x) \sin ^{1+n}(c+d x) \left (a \left (a^2+3 b^2\right )+b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 d}+\frac{\left (a b^3 \left (a^2 (3-n)-3 b^2 (1+n)\right )\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{x}{b}\right )^n}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d}+\frac{\left (b^4 \left (3 a^2 (2-n)-b^2 (2+n)\right )\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{x}{b}\right )^{1+n}}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=\frac{a \left (a^2 (3-n)-3 b^2 (1+n)\right ) \, _2F_1\left (2,\frac{1+n}{2};\frac{3+n}{2};\sin ^2(c+d x)\right ) \sin ^{1+n}(c+d x)}{4 d (1+n)}+\frac{b \left (3 a^2 (2-n)-b^2 (2+n)\right ) \, _2F_1\left (2,\frac{2+n}{2};\frac{4+n}{2};\sin ^2(c+d x)\right ) \sin ^{2+n}(c+d x)}{4 d (2+n)}+\frac{\sec ^4(c+d x) \sin ^{1+n}(c+d x) \left (a \left (a^2+3 b^2\right )+b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 d}\\ \end{align*}
Mathematica [A] time = 0.161711, size = 158, normalized size = 0.85 \[ \frac{\sin ^{n+1}(c+d x) \left (6 a (a+b) (a-b) \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right )+2 (a-b)^3 \, _2F_1(3,n+1;n+2;-\sin (c+d x))+3 (a+b) (a-b)^2 \, _2F_1(2,n+1;n+2;-\sin (c+d x))+3 (a+b)^2 (a-b) \, _2F_1(2,n+1;n+2;\sin (c+d x))+2 (a+b)^3 \, _2F_1(3,n+1;n+2;\sin (c+d x))\right )}{16 d (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.816, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{n} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right ) + a\right )}^{3} \sin \left (d x + c\right )^{n} \sec \left (d x + c\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left ({\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sec \left (d x + c\right )^{5} \sin \left (d x + c\right ) +{\left (3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2}\right )} \sec \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right ) + a\right )}^{3} \sin \left (d x + c\right )^{n} \sec \left (d x + c\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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