Optimal. Leaf size=384 \[ -\frac {128 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{99 b^8 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{99 b^5 d}-\frac {16 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)-144 a^2 b^2+15 b^4\right )}{99 b^7 d}+\frac {32 \left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{99 b^8 d \sqrt {a+b \sin (c+d x)}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}} \]
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Rubi [A] time = 0.76, antiderivative size = 384, 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 = {2693, 2863, 2865, 2752, 2663, 2661, 2655, 2653} \[ \frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-28 a b \sin (c+d x)-3 b^2\right )}{99 b^5 d}-\frac {16 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)-144 a^2 b^2+128 a^4+15 b^4\right )}{99 b^7 d}+\frac {32 \left (-272 a^4 b^2+159 a^2 b^4+128 a^6-15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{99 b^8 d \sqrt {a+b \sin (c+d x)}}-\frac {128 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{99 b^8 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2693
Rule 2752
Rule 2863
Rule 2865
Rubi steps
\begin {align*} \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx &=-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {14 \int \frac {\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx}{3 b}\\ &=-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {280 \int \frac {\cos ^4(c+d x) \left (-\frac {b}{2}-6 a \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{33 b^3}\\ &=-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-3 b^2-28 a b \sin (c+d x)\right )}{99 b^5 d}+\frac {160 \int \frac {\cos ^2(c+d x) \left (\frac {3}{4} b \left (4 a^2-3 b^2\right )+\frac {3}{4} a \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{99 b^5}\\ &=-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-3 b^2-28 a b \sin (c+d x)\right )}{99 b^5 d}-\frac {16 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-144 a^2 b^2+15 b^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{99 b^7 d}+\frac {128 \int \frac {-\frac {3}{8} b \left (32 a^4-51 a^2 b^2+15 b^4\right )-\frac {3}{2} a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{297 b^7}\\ &=-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-3 b^2-28 a b \sin (c+d x)\right )}{99 b^5 d}-\frac {16 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-144 a^2 b^2+15 b^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{99 b^7 d}-\frac {\left (64 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{99 b^8}+\frac {\left (16 \left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{99 b^8}\\ &=-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-3 b^2-28 a b \sin (c+d x)\right )}{99 b^5 d}-\frac {16 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-144 a^2 b^2+15 b^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{99 b^7 d}-\frac {\left (64 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 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}{99 b^8 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (16 \left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 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}{99 b^8 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {128 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 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)}}{99 b^8 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {32 \left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 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}}}{99 b^8 d \sqrt {a+b \sin (c+d x)}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-3 b^2-28 a b \sin (c+d x)\right )}{99 b^5 d}-\frac {16 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-144 a^2 b^2+15 b^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{99 b^7 d}\\ \end {align*}
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Mathematica [A] time = 1.79, size = 356, normalized size = 0.93 \[ \frac {256 (a+b) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2} \left (4 \left (32 a^5-60 a^3 b^2+27 a b^4\right ) \left ((a+b) E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )\right )+b \left (32 a^4 b-51 a^2 b^3+15 b^5\right ) F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )\right )+\frac {1}{2} b \cos (c+d x) \left (-32768 a^6-40960 a^5 b \sin (c+d x)+55296 a^4 b^2+74112 a^3 b^3 \sin (c+d x)-384 a^3 b^3 \sin (3 (c+d x))-18144 a^2 b^4+\left (126 b^6-96 a^2 b^4\right ) \cos (4 (c+d x))+\left (2048 a^4 b^2-3648 a^2 b^4+1383 b^6\right ) \cos (2 (c+d x))-30920 a b^5 \sin (c+d x)+596 a b^5 \sin (3 (c+d x))+28 a b^5 \sin (5 (c+d x))+9 b^6 \cos (6 (c+d x))-2574 b^6\right )}{792 b^8 d (a+b \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{8}}{3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} + {\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )^{8}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.89, size = 2253, normalized size = 5.87 \[ \text {result too large to display} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{8}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^8}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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