Optimal. Leaf size=534 \[ \frac{d \left (-5 a^2 d^2+6 a b c d+b^2 \left (-\left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b^2 f \left (a^2-b^2\right )}-\frac{\left (2 a^2 b^2 d^2 \left (c^2+8 d^2\right )+24 a^3 b c d^3-15 a^4 d^4-12 a b^3 c d \left (c^2+3 d^2\right )+b^4 \left (16 c^2 d^2+3 c^4+2 d^4\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 b^4 f \left (a^2-b^2\right ) \sqrt{c+d \sin (e+f x)}}+\frac{\left (29 a^2 b c d^2-15 a^3 d^3-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 b^3 f \left (a^2-b^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}+\frac{\left (5 a^2 d+2 a b c-7 b^2 d\right ) (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{b^4 f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}} \]
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Rubi [A] time = 2.04004, antiderivative size = 534, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.37, Rules used = {2792, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac{d \left (-5 a^2 d^2+6 a b c d+b^2 \left (-\left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b^2 f \left (a^2-b^2\right )}-\frac{\left (2 a^2 b^2 d^2 \left (c^2+8 d^2\right )+24 a^3 b c d^3-15 a^4 d^4-12 a b^3 c d \left (c^2+3 d^2\right )+b^4 \left (16 c^2 d^2+3 c^4+2 d^4\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 b^4 f \left (a^2-b^2\right ) \sqrt{c+d \sin (e+f x)}}+\frac{\left (29 a^2 b c d^2-15 a^3 d^3-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 b^3 f \left (a^2-b^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}+\frac{\left (5 a^2 d+2 a b c-7 b^2 d\right ) (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{b^4 f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2792
Rule 3049
Rule 3059
Rule 2655
Rule 2653
Rule 3002
Rule 2663
Rule 2661
Rule 2807
Rule 2805
Rubi steps
\begin{align*} \int \frac{(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^2} \, dx &=\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac{\int \frac{\sqrt{c+d \sin (e+f x)} \left (\frac{1}{2} \left (7 b^2 c^2 d+3 a^2 d^3-2 a b c \left (c^2+4 d^2\right )\right )-d \left (a^2 c d-3 b^2 c d+a b \left (c^2+d^2\right )\right ) \sin (e+f x)+\frac{1}{2} d \left (6 a b c d-5 a^2 d^2-b^2 \left (3 c^2-2 d^2\right )\right ) \sin ^2(e+f x)\right )}{a+b \sin (e+f x)} \, dx}{b \left (a^2-b^2\right )}\\ &=\frac{d \left (6 a b c d-5 a^2 d^2-b^2 \left (3 c^2-2 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b^2 \left (a^2-b^2\right ) f}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac{2 \int \frac{\frac{1}{4} \left (21 b^3 c^3 d+15 a^2 b c d^3-5 a^3 d^4-a b^2 \left (6 c^4+27 c^2 d^2-2 d^4\right )\right )+\frac{1}{2} d \left (5 a^3 c d^2+b^3 d \left (18 c^2+d^2\right )-a b^2 c \left (3 c^2+14 d^2\right )-a^2 b \left (9 c^2 d-2 d^3\right )\right ) \sin (e+f x)-\frac{1}{4} d \left (29 a^2 b c d^2-15 a^3 d^3+b^3 \left (3 c^3-20 c d^2\right )-a b^2 \left (9 c^2 d-12 d^3\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{3 b^2 \left (a^2-b^2\right )}\\ &=\frac{d \left (6 a b c d-5 a^2 d^2-b^2 \left (3 c^2-2 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b^2 \left (a^2-b^2\right ) f}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac{2 \int \frac{-\frac{1}{4} d \left (21 b^4 c^3 d-15 a^4 c d^3-9 a^2 b^2 c d \left (c^2-3 d^2\right )+a^3 b \left (29 c^2 d^2-5 d^4\right )-a b^3 \left (3 c^4+47 c^2 d^2-2 d^4\right )\right )-\frac{1}{4} d \left (24 a^3 b c d^3-15 a^4 d^4-12 a b^3 c d \left (c^2+3 d^2\right )+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{3 b^3 \left (a^2-b^2\right ) d}+\frac{\left (29 a^2 b c d^2-15 a^3 d^3+b^3 \left (3 c^3-20 c d^2\right )-a b^2 \left (9 c^2 d-12 d^3\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{6 b^3 \left (a^2-b^2\right )}\\ &=\frac{d \left (6 a b c d-5 a^2 d^2-b^2 \left (3 c^2-2 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b^2 \left (a^2-b^2\right ) f}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac{\left ((b c-a d)^3 \left (2 a b c+5 a^2 d-7 b^2 d\right )\right ) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{2 b^4 \left (a^2-b^2\right )}-\frac{\left (24 a^3 b c d^3-15 a^4 d^4-12 a b^3 c d \left (c^2+3 d^2\right )+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{6 b^4 \left (a^2-b^2\right )}+\frac{\left (\left (29 a^2 b c d^2-15 a^3 d^3+b^3 \left (3 c^3-20 c d^2\right )-a b^2 \left (9 c^2 d-12 d^3\right )\right ) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{6 b^3 \left (a^2-b^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}\\ &=\frac{d \left (6 a b c d-5 a^2 d^2-b^2 \left (3 c^2-2 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b^2 \left (a^2-b^2\right ) f}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac{\left (29 a^2 b c d^2-15 a^3 d^3+b^3 \left (3 c^3-20 c d^2\right )-a b^2 \left (9 c^2 d-12 d^3\right )\right ) E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{3 b^3 \left (a^2-b^2\right ) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left ((b c-a d)^3 \left (2 a b c+5 a^2 d-7 b^2 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{2 b^4 \left (a^2-b^2\right ) \sqrt{c+d \sin (e+f x)}}-\frac{\left (\left (24 a^3 b c d^3-15 a^4 d^4-12 a b^3 c d \left (c^2+3 d^2\right )+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{6 b^4 \left (a^2-b^2\right ) \sqrt{c+d \sin (e+f x)}}\\ &=\frac{d \left (6 a b c d-5 a^2 d^2-b^2 \left (3 c^2-2 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b^2 \left (a^2-b^2\right ) f}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac{\left (29 a^2 b c d^2-15 a^3 d^3+b^3 \left (3 c^3-20 c d^2\right )-a b^2 \left (9 c^2 d-12 d^3\right )\right ) E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{3 b^3 \left (a^2-b^2\right ) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left (24 a^3 b c d^3-15 a^4 d^4-12 a b^3 c d \left (c^2+3 d^2\right )+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{3 b^4 \left (a^2-b^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{(b c-a d)^3 \left (2 a b c+5 a^2 d-7 b^2 d\right ) \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{(a-b) b^4 (a+b)^2 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 8.1094, size = 1109, normalized size = 2.08 \[ \frac{\sqrt{c+d \sin (e+f x)} \left (\frac{-b^3 \cos (e+f x) c^3+3 a b^2 d \cos (e+f x) c^2-3 a^2 b d^2 \cos (e+f x) c+a^3 d^3 \cos (e+f x)}{b^2 \left (b^2-a^2\right ) (a+b \sin (e+f x))}-\frac{2 d^3 \cos (e+f x)}{3 b^2}\right )}{f}-\frac{-\frac{2 \left (-12 a b^2 c^4+39 b^3 d c^3-45 a b^2 d^2 c^2+20 b^3 d^3 c+a^2 b d^3 c+5 a^3 d^4-8 a b^2 d^4\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left (4 b^3 d^4+8 a^2 b d^4+20 a^3 c d^3-56 a b^2 c d^3+72 b^3 c^2 d^2-36 a^2 b c^2 d^2-12 a b^2 c^3 d\right ) \cos (e+f x) \left ((b c-a d) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )+a d \Pi \left (\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )\right ) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left (-15 a^3 d^4+12 a b^2 d^4-20 b^3 c d^3+29 a^2 b c d^3-9 a b^2 c^2 d^2+3 b^3 c^3 d\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (c-d) (b c-a d) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )+d \left (\left (2 a^2-b^2\right ) d \Pi \left (\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )-2 (a+b) (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )\right )\right ) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left (-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right ) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{12 (a-b) b^2 (a+b) f} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 6.068, size = 1886, 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 \frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{7}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{7}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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