Optimal. Leaf size=398 \[ \frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{3003 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15015 b^3 d}+\frac {32 a \left (5 a^6-45 a^4 b^2-53 a^2 b^4+93 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{15015 b^4 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{15015 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d} \]
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Rubi [A] time = 0.94, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2692, 2862, 2865, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)+a \left (5 a^2+59 b^2\right )\right )}{3003 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (-40 a^2 b^2+5 a^4-93 b^4\right )-3 b \left (430 a^2 b^2+5 a^4+77 b^4\right ) \sin (c+d x)\right )}{15015 b^3 d}+\frac {32 a \left (-45 a^4 b^2-53 a^2 b^4+5 a^6+93 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{15015 b^4 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \left (-175 a^4 b^2-1662 a^2 b^4+20 a^6-231 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{15015 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2692
Rule 2752
Rule 2862
Rule 2865
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}+\frac {2}{13} \int \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \left (\frac {13 a^2}{2}+\frac {3 b^2}{2}+8 a b \sin (c+d x)\right ) \, dx\\ &=-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}+\frac {4}{143} \int \frac {\cos ^4(c+d x) \left (\frac {1}{4} a \left (143 a^2+49 b^2\right )+\frac {3}{4} b \left (53 a^2+11 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx\\ &=-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (5 a^2+59 b^2\right )+7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b d}+\frac {16 \int \frac {\cos ^2(c+d x) \left (3 a b^2 \left (47 a^2+17 b^2\right )+\frac {3}{8} b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3003 b^2}\\ &=-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (5 a^2+59 b^2\right )+7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15015 b^3 d}+\frac {64 \int \frac {-\frac {3}{16} a b^2 \left (5 a^4-1450 a^2 b^2-603 b^4\right )-\frac {3}{16} b \left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{45045 b^4}\\ &=-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (5 a^2+59 b^2\right )+7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15015 b^3 d}-\frac {\left (4 \left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{15015 b^4}+\frac {\left (16 a \left (5 a^6-45 a^4 b^2-53 a^2 b^4+93 b^6\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{15015 b^4}\\ &=-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (5 a^2+59 b^2\right )+7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15015 b^3 d}-\frac {\left (4 \left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 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}{15015 b^4 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (16 a \left (5 a^6-45 a^4 b^2-53 a^2 b^4+93 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}{15015 b^4 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {32 a b \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac {8 \left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 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)}}{15015 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {32 a \left (5 a^6-45 a^4 b^2-53 a^2 b^4+93 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}}}{15015 b^4 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (5 a^2+59 b^2\right )+7 b \left (53 a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b d}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (5 a^4-40 a^2 b^2-93 b^4\right )-3 b \left (5 a^4+430 a^2 b^2+77 b^4\right ) \sin (c+d x)\right )}{15015 b^3 d}\\ \end {align*}
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Mathematica [A] time = 1.30, size = 321, normalized size = 0.81 \[ \frac {128 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \left (b \left (5 a^5 b-1450 a^3 b^3-603 a b^5\right ) F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+\left (20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right ) \left ((a+b) E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )\right )\right )-b (a+b \sin (c+d x)) \left (-10 a b^2 \left (20 a^2-2599 b^2\right ) \cos (3 (c+d x))+140 b^3 \left (22 b^2-53 a^2\right ) \sin (4 (c+d x))-b \left (480 a^4+56120 a^2 b^2+4697 b^4\right ) \sin (2 (c+d x))+4 a \left (320 a^4-2710 a^2 b^2+6453 b^4\right ) \cos (c+d x)+5670 a b^4 \cos (5 (c+d x))+1155 b^5 \sin (6 (c+d x))\right )}{240240 b^4 d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.13, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (b^{2} \cos \left (d x + c\right )^{6} - 2 \, a b \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{4}\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.96, size = 1619, normalized size = 4.07 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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