Optimal. Leaf size=469 \[ \frac{2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}-\frac{10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{9 a^2 b^3 d}+\frac{4 \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{63 a b^4 d}-\frac{8 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^5 d}+\frac{128 a \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^6 d}-\frac{8 a \left (-1088 a^2 b^2+1280 a^4+123 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{315 b^7 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left (-768 a^2 b^2+1280 a^4+21 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{315 b^7 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}} \]
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Rubi [A] time = 1.18088, antiderivative size = 469, 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 = {2891, 3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}-\frac{10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{9 a^2 b^3 d}+\frac{4 \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{63 a b^4 d}-\frac{8 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^5 d}+\frac{128 a \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^6 d}-\frac{8 a \left (-1088 a^2 b^2+1280 a^4+123 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{315 b^7 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left (-768 a^2 b^2+1280 a^4+21 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{315 b^7 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2891
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)}{(a+b \sin (c+d x))^{5/2}} \, dx &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac{2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{4 \int \frac{\sin ^3(c+d x) \left (\frac{1}{4} \left (96 a^2-35 b^2\right )-\frac{1}{2} a b \sin (c+d x)-\frac{15}{4} \left (8 a^2-3 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{3 a^2 b^2}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac{2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{9 a^2 b^3 d}-\frac{8 \int \frac{\sin ^2(c+d x) \left (-\frac{45}{4} a \left (8 a^2-3 b^2\right )+3 a^2 b \sin (c+d x)+\frac{3}{4} a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{27 a^2 b^3}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac{2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{4 \left (160 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{63 a b^4 d}-\frac{10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{9 a^2 b^3 d}-\frac{16 \int \frac{\sin (c+d x) \left (\frac{3}{2} a^2 \left (160 a^2-63 b^2\right )-15 a^3 b \sin (c+d x)-\frac{3}{4} a^2 \left (480 a^2-203 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{189 a^2 b^4}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac{2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left (480 a^2-203 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^5 d}+\frac{4 \left (160 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{63 a b^4 d}-\frac{10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{9 a^2 b^3 d}-\frac{32 \int \frac{-\frac{3}{4} a^3 \left (480 a^2-203 b^2\right )+\frac{3}{8} a^2 b \left (160 a^2-21 b^2\right ) \sin (c+d x)+18 a^3 \left (40 a^2-19 b^2\right ) \sin ^2(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{945 a^2 b^5}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac{2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{128 a \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^6 d}-\frac{8 \left (480 a^2-203 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^5 d}+\frac{4 \left (160 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{63 a b^4 d}-\frac{10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{9 a^2 b^3 d}-\frac{64 \int \frac{-\frac{9}{8} a^3 b \left (160 a^2-51 b^2\right )-\frac{9}{16} a^2 \left (1280 a^4-768 a^2 b^2+21 b^4\right ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{2835 a^2 b^6}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac{2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{128 a \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^6 d}-\frac{8 \left (480 a^2-203 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^5 d}+\frac{4 \left (160 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{63 a b^4 d}-\frac{10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{9 a^2 b^3 d}+\frac{\left (4 \left (1280 a^4-768 a^2 b^2+21 b^4\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{315 b^7}-\frac{\left (4 a \left (1280 a^4-1088 a^2 b^2+123 b^4\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{315 b^7}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac{2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{128 a \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^6 d}-\frac{8 \left (480 a^2-203 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^5 d}+\frac{4 \left (160 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{63 a b^4 d}-\frac{10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{9 a^2 b^3 d}+\frac{\left (4 \left (1280 a^4-768 a^2 b^2+21 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}{315 b^7 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (4 a \left (1280 a^4-1088 a^2 b^2+123 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}{315 b^7 \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac{2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{128 a \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^6 d}-\frac{8 \left (480 a^2-203 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^5 d}+\frac{4 \left (160 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{63 a b^4 d}-\frac{10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{9 a^2 b^3 d}+\frac{8 \left (1280 a^4-768 a^2 b^2+21 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)}}{315 b^7 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{8 a \left (1280 a^4-1088 a^2 b^2+123 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}}}{315 b^7 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}
Mathematica [B] time = 9.29774, size = 1044, normalized size = 2.23 \[ \frac{315 \left (\frac{\left (\left (a^2+3 b^2\right ) E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )+a (b-a) F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )\right ) \left (\frac{a+b \sin (c+d x)}{a+b}\right )^{3/2}}{(a-b)^2 b}-\frac{\cos (c+d x) \left (2 a \left (a^2+b^2\right )+b \left (a^2+3 b^2\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^2}\right )+\frac{315 \left (\frac{\left (\left (32 a^4-57 b^2 a^2+21 b^4\right ) E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )+a \left (-32 a^3+32 b a^2+33 b^2 a-33 b^3\right ) F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )\right ) \left (\frac{a+b \sin (c+d x)}{a+b}\right )^{3/2}}{(a-b)^2}-\frac{b \left (4 a \left (8 a^4-13 b^2 a^2+3 b^4\right ) \cos (c+d x)+b \left (20 a^4-33 b^2 a^2+9 b^4\right ) \sin (2 (c+d x))\right )}{2 \left (a^2-b^2\right )^2}\right )}{b^3}-\frac{21 \left (\frac{\left (\left (-2048 a^6+4192 b^2 a^4-2355 b^4 a^2+231 b^6\right ) E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )+a \left (2048 a^5-2048 b a^4-2656 b^2 a^3+2656 b^3 a^2+603 b^4 a-603 b^5\right ) F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )\right ) \left (\frac{a+b \sin (c+d x)}{a+b}\right )^{3/2}}{(a-b)^2}+\frac{b \cos (c+d x) \left (-64 a b^2 \cos (2 (c+d x)) \left (a^2-b^2\right )^2+b \left (1280 a^6-2536 b^2 a^4+1347 b^4 a^2-111 b^6\right ) \sin (c+d x)+2 \left (512 a^7-952 b^2 a^5+423 b^4 a^3+7 b^6 a+6 b^3 \left (a^2-b^2\right )^2 \sin (3 (c+d x))\right )\right )}{\left (a^2-b^2\right )^2}\right )}{b^5}-\frac{5 (a+b \sin (c+d x)) \left (\frac{\sqrt{\frac{a+b \sin (c+d x)}{a+b}} \left (\left (65536 a^8-161792 b^2 a^6+129664 b^4 a^4-35109 b^6 a^2+1617 b^8\right ) \left ((a+b) E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )\right )-4 b \left (-4096 b a^7+8960 b^3 a^5-5884 b^5 a^3+1041 b^7 a\right ) F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )\right )}{(a-b)^2 (a+b)^2}+b (a+b \sin (c+d x)) \left (-56 \sin (4 (c+d x)) b^3+416 a \cos (3 (c+d x)) b^2-8 \left (35 b^2-276 a^2\right ) \sin (2 (c+d x)) b-128 a \left (88 a^2-27 b^2\right ) \cos (c+d x)-\frac{21 \left (1088 a^8-2576 b^2 a^6+1960 b^4 a^4-497 b^6 a^2+21 b^8\right ) \cos (c+d x)}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))}+\frac{21 a \left (64 a^6-112 b^2 a^4+56 b^4 a^2-7 b^6\right ) \cos (c+d x)}{\left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\right )\right )}{b^7}}{10080 d (a+b \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.77, size = 2033, normalized size = 4.3 \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}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{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{{\left (\cos \left (d x + c\right )^{6} - \cos \left (d x + c\right )^{4}\right )} \sqrt{b \sin \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} +{\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \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 \frac{\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{3}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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