Optimal. Leaf size=411 \[ \frac{2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}-\frac{2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 a^2 b^3 d}+\frac{8 \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 a b^4 d}-\frac{8 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 b^5 d}+\frac{8 \left (-46 a^2 b^2+64 a^4+3 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{21 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 a \left (32 a^2-15 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{21 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.91817, antiderivative size = 411, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{2 \left (11 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}-\frac{2 \left (80 a^2-21 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 a^2 b^3 d}+\frac{8 \left (24 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 a b^4 d}-\frac{8 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 b^5 d}+\frac{8 \left (-46 a^2 b^2+64 a^4+3 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{21 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 a \left (32 a^2-15 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{21 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}} \]
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 ^2(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 ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac{2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{4 \int \frac{\sin ^2(c+d x) \left (\frac{15}{4} \left (4 a^2-b^2\right )-\frac{1}{2} a b \sin (c+d x)-\frac{1}{4} \left (80 a^2-21 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 ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac{2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left (80 a^2-21 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{21 a^2 b^3 d}-\frac{8 \int \frac{\sin (c+d x) \left (-\frac{1}{2} a \left (80 a^2-21 b^2\right )+\frac{5}{2} a^2 b \sin (c+d x)+\frac{5}{2} a \left (24 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{21 a^2 b^3}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac{2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left (24 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{21 a b^4 d}-\frac{2 \left (80 a^2-21 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{21 a^2 b^3 d}-\frac{16 \int \frac{\frac{5}{2} a^2 \left (24 a^2-7 b^2\right )-10 a^3 b \sin (c+d x)-\frac{15}{4} a^2 \left (32 a^2-11 b^2\right ) \sin ^2(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{105 a^2 b^4}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac{2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 b^5 d}+\frac{8 \left (24 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{21 a b^4 d}-\frac{2 \left (80 a^2-21 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{21 a^2 b^3 d}-\frac{32 \int \frac{\frac{15}{8} a^2 b \left (16 a^2-3 b^2\right )+\frac{15}{4} a^3 \left (32 a^2-15 b^2\right ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{315 a^2 b^5}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac{2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 b^5 d}+\frac{8 \left (24 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{21 a b^4 d}-\frac{2 \left (80 a^2-21 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{21 a^2 b^3 d}-\frac{\left (8 a \left (32 a^2-15 b^2\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{21 b^6}+\frac{\left (4 \left (64 a^4-46 a^2 b^2+3 b^4\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{21 b^6}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac{2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 b^5 d}+\frac{8 \left (24 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{21 a b^4 d}-\frac{2 \left (80 a^2-21 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{21 a^2 b^3 d}-\frac{\left (8 a \left (32 a^2-15 b^2\right ) \sqrt{a+b \sin (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}} \, dx}{21 b^6 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (4 \left (64 a^4-46 a^2 b^2+3 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}{21 b^6 \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac{2 \left (11 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left (32 a^2-11 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 b^5 d}+\frac{8 \left (24 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)}}{21 a b^4 d}-\frac{2 \left (80 a^2-21 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{21 a^2 b^3 d}-\frac{16 a \left (32 a^2-15 b^2\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)}}{21 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{8 \left (64 a^4-46 a^2 b^2+3 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}}}{21 b^6 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 7.60269, size = 257, normalized size = 0.63 \[ \frac{-\frac{1}{2} b \cos (c+d x) \left (-8 \left (8 a^2 b^2-3 b^4\right ) \cos (2 (c+d x))-288 a^2 b^2+1280 a^3 b \sin (c+d x)+1024 a^4-516 a b^3 \sin (c+d x)+12 a b^3 \sin (3 (c+d x))+3 b^4 \cos (4 (c+d x))-27 b^4\right )-16 \left (-46 a^2 b^2+64 a^4+3 b^4\right ) (a+b) \left (\frac{a+b \sin (c+d x)}{a+b}\right )^{3/2} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )+32 a \left (32 a^2-15 b^2\right ) (a+b)^2 \left (\frac{a+b \sin (c+d x)}{a+b}\right )^{3/2} E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{42 b^6 d (a+b \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.618, size = 1642, normalized size = 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 )^{2}}{{\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}}{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 )^{2}}{{\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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