Optimal. Leaf size=188 \[ \frac{2 b \left (5 a^2-4 b^2\right ) \sqrt{e \cos (c+d x)}}{21 d e^5}+\frac{2 \left (\left (5 a^2-4 b^2\right ) \sin (c+d x)+a b\right ) (a+b \sin (c+d x))}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac{2 a \left (5 a^2-6 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d e^4 \sqrt{e \cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}} \]
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Rubi [A] time = 0.267374, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2691, 2861, 2669, 2642, 2641} \[ \frac{2 b \left (5 a^2-4 b^2\right ) \sqrt{e \cos (c+d x)}}{21 d e^5}+\frac{2 \left (\left (5 a^2-4 b^2\right ) \sin (c+d x)+a b\right ) (a+b \sin (c+d x))}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac{2 a \left (5 a^2-6 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d e^4 \sqrt{e \cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2691
Rule 2861
Rule 2669
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{9/2}} \, dx &=\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}-\frac{2 \int \frac{(a+b \sin (c+d x)) \left (-\frac{5 a^2}{2}+2 b^2-\frac{1}{2} a b \sin (c+d x)\right )}{(e \cos (c+d x))^{5/2}} \, dx}{7 e^2}\\ &=\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}+\frac{2 (a+b \sin (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac{4 \int \frac{\frac{1}{4} a \left (5 a^2-6 b^2\right )-\frac{1}{4} b \left (5 a^2-4 b^2\right ) \sin (c+d x)}{\sqrt{e \cos (c+d x)}} \, dx}{21 e^4}\\ &=\frac{2 b \left (5 a^2-4 b^2\right ) \sqrt{e \cos (c+d x)}}{21 d e^5}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}+\frac{2 (a+b \sin (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac{\left (a \left (5 a^2-6 b^2\right )\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{21 e^4}\\ &=\frac{2 b \left (5 a^2-4 b^2\right ) \sqrt{e \cos (c+d x)}}{21 d e^5}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}+\frac{2 (a+b \sin (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac{\left (a \left (5 a^2-6 b^2\right ) \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 e^4 \sqrt{e \cos (c+d x)}}\\ &=\frac{2 b \left (5 a^2-4 b^2\right ) \sqrt{e \cos (c+d x)}}{21 d e^5}+\frac{2 a \left (5 a^2-6 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d e^4 \sqrt{e \cos (c+d x)}}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}+\frac{2 (a+b \sin (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{21 d e^3 (e \cos (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.638152, size = 140, normalized size = 0.74 \[ \frac{\sec ^4(c+d x) \sqrt{e \cos (c+d x)} \left (4 a \left (5 a^2-6 b^2\right ) \cos ^{\frac{7}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+36 a^2 b+17 a^3 \sin (c+d x)+5 a^3 \sin (3 (c+d x))+30 a b^2 \sin (c+d x)-6 a b^2 \sin (3 (c+d x))-14 b^3 \cos (2 (c+d x))-2 b^3\right )}{42 d e^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.631, size = 750, normalized size = 4. \begin{align*} \text{result too large to display} \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 \frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac{9}{2}}}\,{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 (-\frac{{\left (3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} +{\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt{e \cos \left (d x + c\right )}}{e^{5} \cos \left (d x + c\right )^{5}}, 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 \frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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