Optimal. Leaf size=534 \[ -\frac{2 a^3 \left (-39 A c d+299 A d^2+15 B c^2-75 B c d+280 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^3 \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (-150 c^2 d+15 c^3+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{9009 d^3 f \sqrt{a \sin (e+f x)+a}}-\frac{8 a^2 (5 c-d) (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (-150 c^2 d+15 c^3+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{45045 d^2 f}-\frac{4 a^3 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (-150 c^2 d+15 c^3+799 c d^2-4184 d^3\right )\right ) \cos (e+f x)}{45045 d^3 f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 (-13 A d+5 B c-16 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac{4 a (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (-150 c^2 d+15 c^3+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{15015 d f}-\frac{2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^4}{13 d f} \]
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Rubi [A] time = 1.20306, antiderivative size = 534, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {2976, 2981, 2770, 2761, 2751, 2646} \[ -\frac{2 a^3 \left (-39 A c d+299 A d^2+15 B c^2-75 B c d+280 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^3 \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (-150 c^2 d+15 c^3+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{9009 d^3 f \sqrt{a \sin (e+f x)+a}}-\frac{8 a^2 (5 c-d) (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (-150 c^2 d+15 c^3+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{45045 d^2 f}-\frac{4 a^3 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (-150 c^2 d+15 c^3+799 c d^2-4184 d^3\right )\right ) \cos (e+f x)}{45045 d^3 f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 (-13 A d+5 B c-16 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac{4 a (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (-150 c^2 d+15 c^3+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{15015 d f}-\frac{2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^4}{13 d f} \]
Antiderivative was successfully verified.
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Rule 2976
Rule 2981
Rule 2770
Rule 2761
Rule 2751
Rule 2646
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx &=-\frac{2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^4}{13 d f}+\frac{2 \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \left (\frac{1}{2} a (3 B c+13 A d+8 B d)-\frac{1}{2} a (5 B c-13 A d-16 B d) \sin (e+f x)\right ) \, dx}{13 d}\\ &=\frac{2 a^2 (5 B c-13 A d-16 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac{2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^4}{13 d f}+\frac{4 \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3 \left (\frac{1}{4} a^2 \left (13 A d (c+19 d)-B \left (5 c^2-9 c d-216 d^2\right )\right )+\frac{1}{4} a^2 \left (15 B c^2-39 A c d-75 B c d+299 A d^2+280 B d^2\right ) \sin (e+f x)\right ) \, dx}{143 d^2}\\ &=-\frac{2 a^3 \left (15 B c^2-39 A c d-75 B c d+299 A d^2+280 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 (5 B c-13 A d-16 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac{2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^4}{13 d f}+\frac{\left (a^2 \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right )\right ) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3 \, dx}{1287 d^3}\\ &=-\frac{2 a^3 \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{9009 d^3 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a^3 \left (15 B c^2-39 A c d-75 B c d+299 A d^2+280 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 (5 B c-13 A d-16 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac{2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^4}{13 d f}+\frac{\left (2 a^2 (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right )\right ) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx}{3003 d^3}\\ &=-\frac{4 a (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{15015 d f}-\frac{2 a^3 \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{9009 d^3 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a^3 \left (15 B c^2-39 A c d-75 B c d+299 A d^2+280 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 (5 B c-13 A d-16 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac{2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^4}{13 d f}+\frac{\left (4 a (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right )\right ) \int \sqrt{a+a \sin (e+f x)} \left (\frac{1}{2} a \left (5 c^2+3 d^2\right )+a (5 c-d) d \sin (e+f x)\right ) \, dx}{15015 d^3}\\ &=-\frac{8 a^2 (5 c-d) (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{45045 d^2 f}-\frac{4 a (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{15015 d f}-\frac{2 a^3 \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{9009 d^3 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a^3 \left (15 B c^2-39 A c d-75 B c d+299 A d^2+280 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 (5 B c-13 A d-16 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac{2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^4}{13 d f}+\frac{\left (2 a^2 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right )\right ) \int \sqrt{a+a \sin (e+f x)} \, dx}{45045 d^3}\\ &=-\frac{4 a^3 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x)}{45045 d^3 f \sqrt{a+a \sin (e+f x)}}-\frac{8 a^2 (5 c-d) (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{45045 d^2 f}-\frac{4 a (c+d) \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{15015 d f}-\frac{2 a^3 \left (13 A d \left (3 c^2-38 c d+355 d^2\right )-B \left (15 c^3-150 c^2 d+799 c d^2-4184 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{9009 d^3 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a^3 \left (15 B c^2-39 A c d-75 B c d+299 A d^2+280 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{1287 d^3 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 (5 B c-13 A d-16 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{143 d^2 f}-\frac{2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^4}{13 d f}\\ \end{align*}
Mathematica [C] time = 6.89669, size = 1565, normalized size = 2.93 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.157, size = 374, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( fx+e \right ) \right ){a}^{3} \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( \left ( 4095\,A{d}^{3}+12285\,Bc{d}^{2}+11970\,B{d}^{3} \right ) \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( -19305\,A{c}^{2}d-55770\,Ac{d}^{2}-31265\,A{d}^{3}-6435\,B{c}^{3}-55770\,B{c}^{2}d-93795\,Bc{d}^{2}-44860\,B{d}^{3} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) + \left ( 42042\,A{c}^{3}+167310\,A{c}^{2}d+181038\,Ac{d}^{2}+64090\,A{d}^{3}+55770\,B{c}^{3}+181038\,B{c}^{2}d+192270\,Bc{d}^{2}+66362\,B{d}^{3} \right ) \sin \left ( fx+e \right ) -3465\,B{d}^{3} \left ( \cos \left ( fx+e \right ) \right ) ^{6}+ \left ( 15015\,Ac{d}^{2}+14560\,A{d}^{3}+15015\,B{c}^{2}d+43680\,Bc{d}^{2}+28700\,B{d}^{3} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( -9009\,A{c}^{3}-77220\,A{c}^{2}d-123981\,Ac{d}^{2}-56810\,A{d}^{3}-25740\,B{c}^{3}-123981\,B{c}^{2}d-170430\,Bc{d}^{2}-72109\,B{d}^{3} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+138138\,A{c}^{3}+373230\,A{c}^{2}d+359502\,Ac{d}^{2}+116090\,A{d}^{3}+124410\,B{c}^{3}+359502\,B{c}^{2}d+348270\,Bc{d}^{2}+113818\,B{d}^{3} \right ) }{45045\,f\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \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{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}{\left (d \sin \left (f x + e\right ) + c\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22997, size = 2237, normalized size = 4.19 \begin{align*} \text{result too large to display} \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(-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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