Optimal. Leaf size=463 \[ -\frac{10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{8 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{8 \left (-937 a^2 b^2+480 a^4+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{16 a \left (-279 a^2 b^2+160 a^4+27 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{16 a \left (-439 a^4 b^2+306 a^2 b^4+160 a^6-27 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left (-798 a^4 b^2+435 a^2 b^4+320 a^6-693 b^6\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d} \]
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Rubi [A] time = 1.04769, antiderivative size = 463, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {2895, 3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac{10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{8 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{8 \left (-937 a^2 b^2+480 a^4+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{16 a \left (-279 a^2 b^2+160 a^4+27 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{16 a \left (-439 a^4 b^2+306 a^2 b^4+160 a^6-27 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left (-798 a^4 b^2+435 a^2 b^4+320 a^6-693 b^6\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d} \]
Antiderivative was successfully verified.
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Rule 2895
Rule 3049
Rule 3023
Rule 2753
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \cos ^4(c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)} \, dx &=\frac{20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{4 \int \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left (\frac{1}{4} \left (60 a^2-143 b^2\right )+\frac{1}{2} a b \sin (c+d x)-\frac{5}{4} \left (16 a^2-33 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{143 b^2}\\ &=-\frac{10 \left (16 a^2-33 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{8 \int \sin (c+d x) \sqrt{a+b \sin (c+d x)} \left (-\frac{5}{2} a \left (16 a^2-33 b^2\right )-\frac{1}{2} b \left (5 a^2+33 b^2\right ) \sin (c+d x)+\frac{3}{2} a \left (40 a^2-81 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{1287 b^3}\\ &=\frac{8 a \left (40 a^2-81 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{10 \left (16 a^2-33 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{16 \int \sqrt{a+b \sin (c+d x)} \left (\frac{3}{2} a^2 \left (40 a^2-81 b^2\right )+5 a b \left (2 a^2-3 b^2\right ) \sin (c+d x)-\frac{1}{4} \left (480 a^4-937 a^2 b^2+231 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{9009 b^4}\\ &=-\frac{8 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{8 a \left (40 a^2-81 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{10 \left (16 a^2-33 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{32 \int \sqrt{a+b \sin (c+d x)} \left (-\frac{3}{8} b \left (80 a^4-127 a^2 b^2+231 b^4\right )+\frac{3}{4} a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \sin (c+d x)\right ) \, dx}{45045 b^5}\\ &=\frac{16 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}-\frac{8 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{8 a \left (40 a^2-81 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{10 \left (16 a^2-33 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{64 \int \frac{\frac{3}{16} a b \left (80 a^4-177 a^2 b^2-639 b^4\right )+\frac{3}{16} \left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{135135 b^5}\\ &=\frac{16 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}-\frac{8 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{8 a \left (40 a^2-81 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{10 \left (16 a^2-33 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{\left (4 \left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{45045 b^6}+\frac{\left (8 a \left (160 a^6-439 a^4 b^2+306 a^2 b^4-27 b^6\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{45045 b^6}\\ &=\frac{16 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}-\frac{8 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{8 a \left (40 a^2-81 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{10 \left (16 a^2-33 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{\left (4 \left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 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}{45045 b^6 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (8 a \left (160 a^6-439 a^4 b^2+306 a^2 b^4-27 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}{45045 b^6 \sqrt{a+b \sin (c+d x)}}\\ &=\frac{16 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}-\frac{8 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{8 a \left (40 a^2-81 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{10 \left (16 a^2-33 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{8 \left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 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)}}{45045 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{16 a \left (160 a^6-439 a^4 b^2+306 a^2 b^4-27 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}}}{45045 b^6 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 4.84953, size = 327, normalized size = 0.71 \[ \frac{\sqrt{a+b \sin (c+d x)} \left (-256 a \left (-279 a^3 b^2+279 a^2 b^3-160 a^4 b+160 a^5+27 a b^4-27 b^5\right ) F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )+128 \left (-798 a^4 b^2+435 a^2 b^4+320 a^6-693 b^6\right ) E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )-2 b \cos (c+d x) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \left (13592 a^2 b^3 \sin (c+d x)+1400 a^2 b^3 \sin (3 (c+d x))-1600 \left (2 a^3 b^2-3 a b^4\right ) \cos (2 (c+d x))-21056 a^3 b^2-7680 a^4 b \sin (c+d x)+10240 a^5+630 a b^4 \cos (4 (c+d x))+5898 a b^4-19866 b^5 \sin (c+d x)+5775 b^5 \sin (3 (c+d x))+3465 b^5 \sin (5 (c+d x))\right )\right )}{720720 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.677, size = 1619, normalized size = 3.5 \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 )^{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 (-{\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}, 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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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