Optimal. Leaf size=247 \[ \frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (3 a^2+24 a b \sin (c+d x)+5 b^2\right )}{105 b d}-\frac{4 \left (2 a^2 b^2+3 a^4-5 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{105 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{4 a \left (3 a^2+29 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{105 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 b \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{7 d} \]
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Rubi [A] time = 0.462135, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {2692, 2865, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (3 a^2+24 a b \sin (c+d x)+5 b^2\right )}{105 b d}-\frac{4 \left (2 a^2 b^2+3 a^4-5 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{105 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{4 a \left (3 a^2+29 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{105 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 b \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2692
Rule 2865
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx &=-\frac{2 b \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{7 d}+\frac{2}{7} \int \frac{\cos ^2(c+d x) \left (\frac{7 a^2}{2}+\frac{b^2}{2}+4 a b \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx\\ &=-\frac{2 b \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{7 d}+\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (3 a^2+5 b^2+24 a b \sin (c+d x)\right )}{105 b d}+\frac{8 \int \frac{\frac{1}{4} b^2 \left (27 a^2+5 b^2\right )+\frac{1}{4} a b \left (3 a^2+29 b^2\right ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{105 b^2}\\ &=-\frac{2 b \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{7 d}+\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (3 a^2+5 b^2+24 a b \sin (c+d x)\right )}{105 b d}+\frac{1}{105} \left (2 a \left (29+\frac{3 a^2}{b^2}\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx-\frac{\left (2 \left (3 a^4+2 a^2 b^2-5 b^4\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{105 b^2}\\ &=-\frac{2 b \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{7 d}+\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (3 a^2+5 b^2+24 a b \sin (c+d x)\right )}{105 b d}+\frac{\left (2 a \left (29+\frac{3 a^2}{b^2}\right ) \sqrt{a+b \sin (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}} \, dx}{105 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (2 \left (3 a^4+2 a^2 b^2-5 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}}} \, dx}{105 b^2 \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{2 b \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{7 d}+\frac{4 a \left (29+\frac{3 a^2}{b^2}\right ) E\left (\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{a+b \sin (c+d x)}}{105 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{4 \left (3 a^4+2 a^2 b^2-5 b^4\right ) F\left (\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}{105 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (3 a^2+5 b^2+24 a b \sin (c+d x)\right )}{105 b d}\\ \end{align*}
Mathematica [A] time = 1.01005, size = 222, normalized size = 0.9 \[ \frac{b \cos (c+d x) \left (b \left (108 a^2+5 b^2\right ) \sin (c+d x)+12 a^3-78 a b^2 \cos (2 (c+d x))+38 a b^2-15 b^3 \sin (3 (c+d x))\right )+8 \left (2 a^2 b^2+3 a^4-5 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )-8 a \left (3 a^2 b+3 a^3+29 a b^2+29 b^3\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{210 b^2 d \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.432, size = 943, normalized size = 3.8 \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{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{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 ({\left (b \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + a \cos \left (d x + c\right )^{2}\right )} \sqrt{b \sin \left (d x + c\right ) + a}, 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 )}^{\frac{3}{2}} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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