Optimal. Leaf size=332 \[ -\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{693 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+32 a^4+45 b^4\right )}{3465 b^4 d}-\frac{8 \left (-101 a^4 b^2+114 a^2 b^4+32 a^6-45 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3465 b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{8 a \left (-93 a^2 b^2+32 a^4+93 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3465 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{11 d} \]
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Rubi [A] time = 0.633782, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2862, 2865, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{693 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+32 a^4+45 b^4\right )}{3465 b^4 d}-\frac{8 \left (-101 a^4 b^2+114 a^2 b^4+32 a^6-45 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3465 b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{8 a \left (-93 a^2 b^2+32 a^4+93 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3465 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{11 d} \]
Antiderivative was successfully verified.
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Rule 2862
Rule 2865
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \cos ^4(c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)} \, dx &=-\frac{2 \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{11 d}+\frac{2}{11} \int \frac{\cos ^4(c+d x) \left (\frac{b}{2}+\frac{1}{2} a \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx\\ &=-\frac{2 \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{11 d}-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (8 a^2-9 b^2-7 a b \sin (c+d x)\right )}{693 b^2 d}+\frac{8 \int \frac{\cos ^2(c+d x) \left (-\frac{1}{4} b \left (a^2-9 b^2\right )-2 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{231 b^2}\\ &=-\frac{2 \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{11 d}-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (8 a^2-9 b^2-7 a b \sin (c+d x)\right )}{693 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (32 a^4-69 a^2 b^2+45 b^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{3465 b^4 d}+\frac{32 \int \frac{\frac{1}{8} b \left (8 a^4-21 a^2 b^2+45 b^4\right )+\frac{1}{8} a \left (32 a^4-93 a^2 b^2+93 b^4\right ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{3465 b^4}\\ &=-\frac{2 \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{11 d}-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (8 a^2-9 b^2-7 a b \sin (c+d x)\right )}{693 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (32 a^4-69 a^2 b^2+45 b^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{3465 b^4 d}+\frac{\left (4 a \left (32 a^4-93 a^2 b^2+93 b^4\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{3465 b^5}-\frac{\left (4 \left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{3465 b^5}\\ &=-\frac{2 \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{11 d}-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (8 a^2-9 b^2-7 a b \sin (c+d x)\right )}{693 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (32 a^4-69 a^2 b^2+45 b^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{3465 b^4 d}+\frac{\left (4 a \left (32 a^4-93 a^2 b^2+93 b^4\right ) \sqrt{a+b \sin (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}} \, dx}{3465 b^5 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (4 \left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\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}{3465 b^5 \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{2 \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{11 d}+\frac{8 a \left (32 a^4-93 a^2 b^2+93 b^4\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)}}{3465 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{8 \left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\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}}}{3465 b^5 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (8 a^2-9 b^2-7 a b \sin (c+d x)\right )}{693 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (32 a^4-69 a^2 b^2+45 b^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{3465 b^4 d}\\ \end{align*}
Mathematica [A] time = 3.90255, size = 326, normalized size = 0.98 \[ \frac{b \cos (c+d x) \left (-692 a^2 b^3 \sin (c+d x)-20 a^2 b^3 \sin (3 (c+d x))+16 \left (4 a^3 b^2-183 a b^4\right ) \cos (2 (c+d x))-2912 a^3 b^2+256 a^4 b \sin (c+d x)+1024 a^5-700 a b^4 \cos (4 (c+d x))+748 a b^4+990 b^5 \sin (c+d x)-765 b^5 \sin (3 (c+d x))-315 b^5 \sin (5 (c+d x))\right )+64 \left (-101 a^4 b^2+114 a^2 b^4+32 a^6-45 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )-64 a \left (-93 a^3 b^2-93 a^2 b^3+32 a^4 b+32 a^5+93 a b^4+93 b^5\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{27720 b^5 d \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.645, size = 1356, normalized size = 4.1 \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 \sqrt{b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\,{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 (\sqrt{b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ), 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 \sqrt{b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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