Optimal. Leaf size=394 \[ -\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15015 b^4 d}-\frac {8 a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{15015 b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{15015 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d} \]
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Rubi [A] time = 0.86, antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2862, 2865, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (-113 a^2 b^2+32 a^4+177 b^4\right )-3 b \left (-27 a^2 b^2+8 a^4-77 b^4\right ) \sin (c+d x)\right )}{15015 b^4 d}-\frac {8 a \left (-145 a^4 b^2+290 a^2 b^4+32 a^6-177 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{15015 b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \left (-137 a^4 b^2+258 a^2 b^4+32 a^6+231 b^6\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{15015 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2862
Rule 2865
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx &=-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}+\frac {2}{13} \int \cos ^4(c+d x) \left (\frac {3 b}{2}+\frac {3}{2} a \sin (c+d x)\right ) \sqrt {a+b \sin (c+d x)} \, dx\\ &=-\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 \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 (9 a b+\frac {3}{4} \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx\\ &=-\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 \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 (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b^2 d}+\frac {16 \int \frac {\cos ^2(c+d x) \left (-\frac {3}{8} a b \left (a^2-97 b^2\right )-\frac {3}{8} \left (8 a^4-27 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 {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 \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 (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15015 b^4 d}+\frac {64 \int \frac {\frac {3}{2} a b \left (a^4-4 a^2 b^2+51 b^4\right )+\frac {3}{16} \left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{45045 b^4}\\ &=-\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 \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 (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15015 b^4 d}-\frac {\left (4 a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{15015 b^5}+\frac {\left (4 \left (32 a^6-137 a^4 b^2+258 a^2 b^4+231 b^6\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{15015 b^5}\\ &=-\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 \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 (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15015 b^4 d}+\frac {\left (4 \left (32 a^6-137 a^4 b^2+258 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^5 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (4 a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 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^5 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {6 a \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{143 d}-\frac {2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}+\frac {8 \left (32 a^6-137 a^4 b^2+258 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^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 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^5 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a \left (2 a^2-5 b^2\right )-7 b \left (a^2+11 b^2\right ) \sin (c+d x)\right )}{3003 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^4-113 a^2 b^2+177 b^4\right )-3 b \left (8 a^4-27 a^2 b^2-77 b^4\right ) \sin (c+d x)\right )}{15015 b^4 d}\\ \end {align*}
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Mathematica [A] time = 12.07, size = 382, normalized size = 0.97 \[ \frac {384 a \left (32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )-3 b \cos (c+d x) \left (-2048 a^6-512 a^5 b \sin (c+d x)+8640 a^4 b^2+2088 a^3 b^3 \sin (c+d x)+40 a^3 b^3 \sin (3 (c+d x))+1980 a^2 b^4+70 \left (86 a^2 b^4-11 b^6\right ) \cos (4 (c+d x))+\left (-128 a^4 b^2+24512 a^2 b^4+8547 b^6\right ) \cos (2 (c+d x))-19492 a b^5 \sin (c+d x)+11870 a b^5 \sin (3 (c+d x))+5250 a b^5 \sin (5 (c+d x))-1155 b^6 \cos (6 (c+d x))-6622 b^6\right )-384 \left (32 a^7+32 a^6 b-137 a^5 b^2-137 a^4 b^3+258 a^3 b^4+258 a^2 b^5+231 a b^6+231 b^7\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{720720 b^5 d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.17, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (b \cos \left (d x + c\right )^{6} - a \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - b \cos \left (d x + c\right )^{4}\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.96, size = 1619, normalized size = 4.11 \[ \text {result too large to display} \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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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