Optimal. Leaf size=471 \[ -\frac{10 \left (8 a^2-11 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{429 b^3 d}+\frac{4 a \left (160 a^2-223 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3003 b^4 d}-\frac{8 \left (-683 a^2 b^2+480 a^4+77 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^5 d}+\frac{64 a \left (-118 a^2 b^2+80 a^4+17 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^6 d}-\frac{8 a \left (-2368 a^4 b^2+875 a^2 b^4+1280 a^6+213 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{15015 b^7 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left (-2048 a^4 b^2+453 a^2 b^4+1280 a^6+231 b^6\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{15015 b^7 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{24 a \sin ^4(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{143 b^2 d}-\frac{2 \sin ^5(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{13 b d} \]
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Rubi [A] time = 1.18478, antiderivative size = 471, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {2895, 3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac{10 \left (8 a^2-11 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{429 b^3 d}+\frac{4 a \left (160 a^2-223 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3003 b^4 d}-\frac{8 \left (-683 a^2 b^2+480 a^4+77 b^4\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^5 d}+\frac{64 a \left (-118 a^2 b^2+80 a^4+17 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^6 d}-\frac{8 a \left (-2368 a^4 b^2+875 a^2 b^4+1280 a^6+213 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{15015 b^7 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left (-2048 a^4 b^2+453 a^2 b^4+1280 a^6+231 b^6\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{15015 b^7 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{24 a \sin ^4(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{143 b^2 d}-\frac{2 \sin ^5(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{13 b d} \]
Antiderivative was successfully verified.
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Rule 2895
Rule 3049
Rule 3023
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx &=\frac{24 a \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{13 b d}-\frac{4 \int \frac{\sin ^3(c+d x) \left (\frac{1}{4} \left (96 a^2-143 b^2\right )-\frac{1}{2} a b \sin (c+d x)-\frac{15}{4} \left (8 a^2-11 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{143 b^2}\\ &=-\frac{10 \left (8 a^2-11 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{429 b^3 d}+\frac{24 a \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{13 b d}-\frac{8 \int \frac{\sin ^2(c+d x) \left (-\frac{45}{4} a \left (8 a^2-11 b^2\right )+\frac{3}{2} b \left (2 a^2-11 b^2\right ) \sin (c+d x)+\frac{3}{4} a \left (160 a^2-223 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{1287 b^3}\\ &=\frac{4 a \left (160 a^2-223 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{3003 b^4 d}-\frac{10 \left (8 a^2-11 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{429 b^3 d}+\frac{24 a \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{13 b d}-\frac{16 \int \frac{\sin (c+d x) \left (\frac{3}{2} a^2 \left (160 a^2-223 b^2\right )-15 a b \left (a^2-b^2\right ) \sin (c+d x)-\frac{3}{4} \left (480 a^4-683 a^2 b^2+77 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{9009 b^4}\\ &=-\frac{8 \left (480 a^4-683 a^2 b^2+77 b^4\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^5 d}+\frac{4 a \left (160 a^2-223 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{3003 b^4 d}-\frac{10 \left (8 a^2-11 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{429 b^3 d}+\frac{24 a \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{13 b d}-\frac{32 \int \frac{-\frac{3}{4} a \left (480 a^4-683 a^2 b^2+77 b^4\right )+\frac{3}{8} b \left (160 a^4-181 a^2 b^2-231 b^4\right ) \sin (c+d x)+9 a \left (80 a^4-118 a^2 b^2+17 b^4\right ) \sin ^2(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{45045 b^5}\\ &=\frac{64 a \left (80 a^4-118 a^2 b^2+17 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^6 d}-\frac{8 \left (480 a^4-683 a^2 b^2+77 b^4\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^5 d}+\frac{4 a \left (160 a^2-223 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{3003 b^4 d}-\frac{10 \left (8 a^2-11 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{429 b^3 d}+\frac{24 a \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{13 b d}-\frac{64 \int \frac{-\frac{9}{8} a b \left (160 a^4-211 a^2 b^2+9 b^4\right )-\frac{9}{16} \left (1280 a^6-2048 a^4 b^2+453 a^2 b^4+231 b^6\right ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{135135 b^6}\\ &=\frac{64 a \left (80 a^4-118 a^2 b^2+17 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^6 d}-\frac{8 \left (480 a^4-683 a^2 b^2+77 b^4\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^5 d}+\frac{4 a \left (160 a^2-223 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{3003 b^4 d}-\frac{10 \left (8 a^2-11 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{429 b^3 d}+\frac{24 a \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{13 b d}-\frac{\left (4 a \left (1280 a^6-2368 a^4 b^2+875 a^2 b^4+213 b^6\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{15015 b^7}+\frac{\left (4 \left (1280 a^6-2048 a^4 b^2+453 a^2 b^4+231 b^6\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{15015 b^7}\\ &=\frac{64 a \left (80 a^4-118 a^2 b^2+17 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^6 d}-\frac{8 \left (480 a^4-683 a^2 b^2+77 b^4\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^5 d}+\frac{4 a \left (160 a^2-223 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{3003 b^4 d}-\frac{10 \left (8 a^2-11 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{429 b^3 d}+\frac{24 a \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{13 b d}+\frac{\left (4 \left (1280 a^6-2048 a^4 b^2+453 a^2 b^4+231 b^6\right ) \sqrt{a+b \sin (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}} \, dx}{15015 b^7 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (4 a \left (1280 a^6-2368 a^4 b^2+875 a^2 b^4+213 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}{15015 b^7 \sqrt{a+b \sin (c+d x)}}\\ &=\frac{64 a \left (80 a^4-118 a^2 b^2+17 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^6 d}-\frac{8 \left (480 a^4-683 a^2 b^2+77 b^4\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^5 d}+\frac{4 a \left (160 a^2-223 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{3003 b^4 d}-\frac{10 \left (8 a^2-11 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{429 b^3 d}+\frac{24 a \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{13 b d}+\frac{8 \left (1280 a^6-2048 a^4 b^2+453 a^2 b^4+231 b^6\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)}}{15015 b^7 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{8 a \left (1280 a^6-2368 a^4 b^2+875 a^2 b^4+213 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}}}{15015 b^7 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 5.42158, size = 382, normalized size = 0.81 \[ \frac{3 b \cos (c+d x) \left (-28608 a^3 b^3 \sin (c+d x)-1600 a^3 b^3 \sin (3 (c+d x))+\left (-5792 a^2 b^4+5120 a^4 b^2-8547 b^6\right ) \cos (2 (c+d x))-70 \left (8 a^2 b^4-11 b^6\right ) \cos (4 (c+d x))-125952 a^4 b^2+23760 a^2 b^4+20480 a^5 b \sin (c+d x)+81920 a^6+2332 a b^5 \sin (c+d x)+1390 a b^5 \sin (3 (c+d x))+210 a b^5 \sin (5 (c+d x))+1155 b^6 \cos (6 (c+d x))+6622 b^6\right )+384 a \left (-2368 a^4 b^2+875 a^2 b^4+1280 a^6+213 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )-384 \left (-2048 a^5 b^2-2048 a^4 b^3+453 a^3 b^4+453 a^2 b^5+1280 a^6 b+1280 a^7+231 a b^6+231 b^7\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{720720 b^7 d \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.55, size = 1619, normalized size = 3.4 \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{\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{3}}{\sqrt{b \sin \left (d x + c\right ) + a}}\,{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 (\cos \left (d x + c\right )^{6} - \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\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 \frac{\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{3}}{\sqrt{b \sin \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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