Optimal. Leaf size=183 \[ -\frac{\sec ^2(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{m+1}}{2 d \left (a^2-b^2\right )}-\frac{(a-b (1-m)) (a+b \sin (c+d x))^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{a+b \sin (c+d x)}{a-b}\right )}{4 d (m+1) (a-b)^2}+\frac{(a-b m+b) (a+b \sin (c+d x))^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{a+b \sin (c+d x)}{a+b}\right )}{4 d (m+1) (a+b)^2} \]
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Rubi [A] time = 0.226047, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2668, 741, 831, 68} \[ -\frac{\sec ^2(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{m+1}}{2 d \left (a^2-b^2\right )}-\frac{(a-b (1-m)) (a+b \sin (c+d x))^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{a+b \sin (c+d x)}{a-b}\right )}{4 d (m+1) (a-b)^2}+\frac{(a-b m+b) (a+b \sin (c+d x))^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{a+b \sin (c+d x)}{a+b}\right )}{4 d (m+1) (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 741
Rule 831
Rule 68
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+b \sin (c+d x))^m \, dx &=\frac{b^3 \operatorname{Subst}\left (\int \frac{(a+x)^m}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{\sec ^2(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{2 \left (a^2-b^2\right ) d}+\frac{b \operatorname{Subst}\left (\int \frac{(a+x)^m \left (a^2-b^2 (1-m)-a m x\right )}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{2 \left (a^2-b^2\right ) d}\\ &=-\frac{\sec ^2(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{2 \left (a^2-b^2\right ) d}+\frac{b \operatorname{Subst}\left (\int \left (\frac{\left (b \left (a^2-b^2 (1-m)\right )-a b^2 m\right ) (a+x)^m}{2 b^2 (b-x)}+\frac{\left (b \left (a^2-b^2 (1-m)\right )+a b^2 m\right ) (a+x)^m}{2 b^2 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{2 \left (a^2-b^2\right ) d}\\ &=-\frac{\sec ^2(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{2 \left (a^2-b^2\right ) d}+\frac{((a+b) (a-b (1-m))) \operatorname{Subst}\left (\int \frac{(a+x)^m}{b+x} \, dx,x,b \sin (c+d x)\right )}{4 \left (a^2-b^2\right ) d}+\frac{(a+b-b m) \operatorname{Subst}\left (\int \frac{(a+x)^m}{b-x} \, dx,x,b \sin (c+d x)\right )}{4 (a+b) d}\\ &=-\frac{(a-b (1-m)) \, _2F_1\left (1,1+m;2+m;\frac{a+b \sin (c+d x)}{a-b}\right ) (a+b \sin (c+d x))^{1+m}}{4 (a-b)^2 d (1+m)}+\frac{(a+b-b m) \, _2F_1\left (1,1+m;2+m;\frac{a+b \sin (c+d x)}{a+b}\right ) (a+b \sin (c+d x))^{1+m}}{4 (a+b)^2 d (1+m)}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{2 \left (a^2-b^2\right ) d}\\ \end{align*}
Mathematica [A] time = 0.566822, size = 157, normalized size = 0.86 \[ \frac{(a+b \sin (c+d x))^{m+1} \left (\frac{b \left ((a+b)^2 (a+b (m-1)) \, _2F_1\left (1,m+1;m+2;\frac{a+b \sin (c+d x)}{a-b}\right )-(a-b)^2 (a-b m+b) \, _2F_1\left (1,m+1;m+2;\frac{a+b \sin (c+d x)}{a+b}\right )\right )}{(m+1) (a-b) (a+b)}+2 b \sec ^2(c+d x) (b-a \sin (c+d x))\right )}{4 b d \left (b^2-a^2\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.319, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{3} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{m}\, 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 )}^{m} \sec \left (d x + c\right )^{3}\,{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 (b \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{3}, 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 )}^{m} \sec \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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