Optimal. Leaf size=261 \[ -\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{35 b^4 d}+\frac{8 \left (-37 a^2 b^2+32 a^4+5 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{35 b^5 d \sqrt{a+b \sin (c+d x)}}-\frac{8 a \left (32 a^2-29 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{35 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt{a+b \sin (c+d x)}} \]
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Rubi [A] time = 0.424217, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2863, 2865, 2752, 2663, 2661, 2655, 2653} \[ -\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{35 b^4 d}+\frac{8 \left (-37 a^2 b^2+32 a^4+5 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{35 b^5 d \sqrt{a+b \sin (c+d x)}}-\frac{8 a \left (32 a^2-29 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{35 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2863
Rule 2865
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=\frac{2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{12 \int \frac{\cos ^2(c+d x) \left (-\frac{b}{2}-4 a \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{7 b^2}\\ &=\frac{2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{35 b^4 d}-\frac{16 \int \frac{\frac{1}{4} b \left (8 a^2-5 b^2\right )+\frac{1}{4} a \left (32 a^2-29 b^2\right ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{35 b^4}\\ &=\frac{2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{35 b^4 d}-\frac{\left (4 a \left (32 a^2-29 b^2\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{35 b^5}+\frac{\left (4 \left (32 a^4-37 a^2 b^2+5 b^4\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{35 b^5}\\ &=\frac{2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{35 b^4 d}-\frac{\left (4 a \left (32 a^2-29 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}{35 b^5 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (4 \left (32 a^4-37 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}{35 b^5 \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{8 a \left (32 a^2-29 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)}}{35 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{8 \left (32 a^4-37 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}}}{35 b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{35 b^4 d}\\ \end{align*}
Mathematica [A] time = 3.91684, size = 222, normalized size = 0.85 \[ \frac{b \cos (c+d x) \left (\left (45 b^3-64 a^2 b\right ) \sin (c+d x)-256 a^3-16 a b^2 \cos (2 (c+d x))+216 a b^2+5 b^3 \sin (3 (c+d x))\right )-16 \left (-37 a^2 b^2+32 a^4+5 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )+16 a \left (32 a^2 b+32 a^3-29 a b^2-29 b^3\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{70 b^5 d \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.546, size = 943, normalized size = 3.6 \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} \sin \left (d x + c\right )}{{\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} \sin \left (d x + c\right )}{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} \sin \left (d x + c\right )}{{\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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