Optimal. Leaf size=229 \[ -\frac{32 a \left (a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{5 b^4 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left (4 a^2-3 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{5 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{5 b^3 d}-\frac{2 \cos ^3(c+d x)}{b d \sqrt{a+b \sin (c+d x)}} \]
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Rubi [A] time = 0.332658, antiderivative size = 229, 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 = {2693, 2865, 2752, 2663, 2661, 2655, 2653} \[ -\frac{32 a \left (a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{5 b^4 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left (4 a^2-3 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{5 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{5 b^3 d}-\frac{2 \cos ^3(c+d x)}{b d \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2693
Rule 2865
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=-\frac{2 \cos ^3(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}-\frac{6 \int \frac{\cos ^2(c+d x) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{b}\\ &=-\frac{2 \cos ^3(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}+\frac{4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{5 b^3 d}-\frac{8 \int \frac{-\frac{a b}{2}-\frac{1}{2} \left (4 a^2-3 b^2\right ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{5 b^3}\\ &=-\frac{2 \cos ^3(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}+\frac{4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{5 b^3 d}+\frac{\left (4 \left (4 a^2-3 b^2\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{5 b^4}-\frac{\left (16 a \left (a^2-b^2\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{5 b^4}\\ &=-\frac{2 \cos ^3(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}+\frac{4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{5 b^3 d}+\frac{\left (4 \left (4 a^2-3 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}{5 b^4 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (16 a \left (a^2-b^2\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}{5 b^4 \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{2 \cos ^3(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}+\frac{4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{5 b^3 d}+\frac{8 \left (4 a^2-3 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)}}{5 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{32 a \left (a^2-b^2\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}}}{5 b^4 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.05006, size = 187, normalized size = 0.82 \[ \frac{b \cos (c+d x) \left (16 a^2+4 a b \sin (c+d x)+b^2 \cos (2 (c+d x))-11 b^2\right )+32 a \left (a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )-8 \left (4 a^2 b+4 a^3-3 a b^2-3 b^3\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{5 b^4 d \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.471, size = 797, normalized size = 3.5 \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{\cos \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{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{\sqrt{b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}, 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{\cos \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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