Optimal. Leaf size=466 \[ -\frac{2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{33 a b^3 d}-\frac{20 \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{231 b^4 d}+\frac{8 a \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^5 d}-\frac{8 \left (-592 a^2 b^2+640 a^4+15 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^6 d}+\frac{8 \left (-1664 a^4 b^2+369 a^2 b^4+1280 a^6+15 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{1155 b^7 d \sqrt{a+b \sin (c+d x)}}-\frac{8 a \left (-1344 a^2 b^2+1280 a^4+123 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{1155 b^7 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{11 b^2 d} \]
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Rubi [A] time = 1.21402, antiderivative size = 466, 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 = {2892, 3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{33 a b^3 d}-\frac{20 \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{231 b^4 d}+\frac{8 a \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^5 d}-\frac{8 \left (-592 a^2 b^2+640 a^4+15 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^6 d}+\frac{8 \left (-1664 a^4 b^2+369 a^2 b^4+1280 a^6+15 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{1155 b^7 d \sqrt{a+b \sin (c+d x)}}-\frac{8 a \left (-1344 a^2 b^2+1280 a^4+123 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{1155 b^7 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{11 b^2 d} \]
Antiderivative was successfully verified.
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Rule 2892
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))^{3/2}} \, dx &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{11 b^2 d}+\frac{4 \int \frac{\sin ^3(c+d x) \left (\frac{1}{4} \left (96 a^2-77 b^2\right )-\frac{1}{2} a b \sin (c+d x)-\frac{3}{4} \left (40 a^2-33 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{11 a b^2}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{33 a b^3 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{11 b^2 d}+\frac{8 \int \frac{\sin ^2(c+d x) \left (-\frac{9}{4} a \left (40 a^2-33 b^2\right )+3 a^2 b \sin (c+d x)+\frac{15}{4} a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{99 a b^3}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{20 \left (32 a^2-27 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{231 b^4 d}+\frac{2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{33 a b^3 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{11 b^2 d}+\frac{16 \int \frac{\sin (c+d x) \left (\frac{15}{2} a^2 \left (32 a^2-27 b^2\right )-\frac{3}{4} a b \left (20 a^2-9 b^2\right ) \sin (c+d x)-\frac{3}{4} a^2 \left (480 a^2-419 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{693 a b^4}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{8 a \left (480 a^2-419 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^5 d}-\frac{20 \left (32 a^2-27 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{231 b^4 d}+\frac{2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{33 a b^3 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{11 b^2 d}+\frac{32 \int \frac{-\frac{3}{4} a^3 \left (480 a^2-419 b^2\right )+\frac{3}{8} a^2 b \left (160 a^2-93 b^2\right ) \sin (c+d x)+\frac{9}{8} a \left (640 a^4-592 a^2 b^2+15 b^4\right ) \sin ^2(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{3465 a b^5}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^6 d}+\frac{8 a \left (480 a^2-419 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^5 d}-\frac{20 \left (32 a^2-27 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{231 b^4 d}+\frac{2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{33 a b^3 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{11 b^2 d}+\frac{64 \int \frac{-\frac{9}{16} a b \left (320 a^4-246 a^2 b^2-15 b^4\right )-\frac{9}{16} a^2 \left (1280 a^4-1344 a^2 b^2+123 b^4\right ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{10395 a b^6}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^6 d}+\frac{8 a \left (480 a^2-419 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^5 d}-\frac{20 \left (32 a^2-27 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{231 b^4 d}+\frac{2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{33 a b^3 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{11 b^2 d}-\frac{\left (4 a \left (1280 a^4-1344 a^2 b^2+123 b^4\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{1155 b^7}+\frac{\left (4 \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{1155 b^7}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^6 d}+\frac{8 a \left (480 a^2-419 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^5 d}-\frac{20 \left (32 a^2-27 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{231 b^4 d}+\frac{2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{33 a b^3 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{11 b^2 d}-\frac{\left (4 a \left (1280 a^4-1344 a^2 b^2+123 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}{1155 b^7 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (4 \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 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}{1155 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)}{a b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^6 d}+\frac{8 a \left (480 a^2-419 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^5 d}-\frac{20 \left (32 a^2-27 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{231 b^4 d}+\frac{2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{33 a b^3 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) \sqrt{a+b \sin (c+d x)}}{11 b^2 d}-\frac{8 a \left (1280 a^4-1344 a^2 b^2+123 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)}}{1155 b^7 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{8 \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 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}}}{1155 b^7 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 6.57906, size = 326, normalized size = 0.7 \[ \frac{b \cos (c+d x) \left (8672 a^2 b^3 \sin (c+d x)+800 a^2 b^3 \sin (3 (c+d x))-16 \left (160 a^3 b^2-93 a b^4\right ) \cos (2 (c+d x))+40448 a^3 b^2-10240 a^4 b \sin (c+d x)-40960 a^5+280 a b^4 \cos (4 (c+d x))-2728 a b^4+330 b^5 \sin (c+d x)-255 b^5 \sin (3 (c+d x))-105 b^5 \sin (5 (c+d x))\right )-64 \left (-1664 a^4 b^2+369 a^2 b^4+1280 a^6+15 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )+64 a \left (-1344 a^3 b^2-1344 a^2 b^3+1280 a^4 b+1280 a^5+123 a b^4+123 b^5\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{9240 b^7 d \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.581, size = 1356, normalized size = 2.9 \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{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{{\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 )}{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 )^{3}}{{\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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