Optimal. Leaf size=384 \[ -\frac {2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}-\frac {128 a \left (8 a^2-9 b^2\right ) \left (4 a^2-3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{99 b^8 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {32 \left (128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{99 b^8 d \sqrt {a+b \sin (c+d x)}}-\frac {28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {40 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-3 b^2-28 a b \sin (c+d x)\right )}{99 b^5 d}-\frac {16 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (128 a^4-144 a^2 b^2+15 b^4-3 a b \left (32 a^2-31 b^2\right ) \sin (c+d x)\right )}{99 b^7 d} \]
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Rubi [A]
time = 0.49, 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 = {2772, 2942,
2944, 2831, 2742, 2740, 2734, 2732} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2772
Rule 2831
Rule 2942
Rule 2944
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 = 1.74, size = 356, normalized size = 0.93 \begin {gather*} \frac {256 (a+b) \left (b \left (32 a^4 b-51 a^2 b^3+15 b^5\right ) F\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )+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+\pi -2 d x)|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )\right )\right ) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2}+\frac {1}{2} b \cos (c+d x) \left (-32768 a^6+55296 a^4 b^2-18144 a^2 b^4-2574 b^6+\left (2048 a^4 b^2-3648 a^2 b^4+1383 b^6\right ) \cos (2 (c+d x))+\left (-96 a^2 b^4+126 b^6\right ) \cos (4 (c+d x))+9 b^6 \cos (6 (c+d x))-40960 a^5 b \sin (c+d x)+74112 a^3 b^3 \sin (c+d x)-30920 a b^5 \sin (c+d x)-384 a^3 b^3 \sin (3 (c+d x))+596 a b^5 \sin (3 (c+d x))+28 a b^5 \sin (5 (c+d x))\right )}{792 b^8 d (a+b \sin (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2252\) vs.
\(2(422)=844\).
time = 3.03, size = 2253, normalized size = 5.87
method | result | size |
default | \(\text {Expression too large to display}\) | \(2253\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.41, size = 1043, normalized size = 2.72 \begin {gather*} \frac {2 \, {\left (8 \, {\left (\sqrt {2} {\left (256 \, a^{6} b^{2} - 576 \, a^{4} b^{4} + 369 \, a^{2} b^{6} - 45 \, b^{8}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} {\left (256 \, a^{7} b - 576 \, a^{5} b^{3} + 369 \, a^{3} b^{5} - 45 \, a b^{7}\right )} \sin \left (d x + c\right ) - \sqrt {2} {\left (256 \, a^{8} - 320 \, a^{6} b^{2} - 207 \, a^{4} b^{4} + 324 \, a^{2} b^{6} - 45 \, b^{8}\right )}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + 8 \, {\left (\sqrt {2} {\left (256 \, a^{6} b^{2} - 576 \, a^{4} b^{4} + 369 \, a^{2} b^{6} - 45 \, b^{8}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} {\left (256 \, a^{7} b - 576 \, a^{5} b^{3} + 369 \, a^{3} b^{5} - 45 \, a b^{7}\right )} \sin \left (d x + c\right ) - \sqrt {2} {\left (256 \, a^{8} - 320 \, a^{6} b^{2} - 207 \, a^{4} b^{4} + 324 \, a^{2} b^{6} - 45 \, b^{8}\right )}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) + 96 \, {\left (\sqrt {2} {\left (32 i \, a^{5} b^{3} - 60 i \, a^{3} b^{5} + 27 i \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (-32 i \, a^{6} b^{2} + 60 i \, a^{4} b^{4} - 27 i \, a^{2} b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-32 i \, a^{7} b + 28 i \, a^{5} b^{3} + 33 i \, a^{3} b^{5} - 27 i \, a b^{7}\right )}\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) + 96 \, {\left (\sqrt {2} {\left (-32 i \, a^{5} b^{3} + 60 i \, a^{3} b^{5} - 27 i \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (32 i \, a^{6} b^{2} - 60 i \, a^{4} b^{4} + 27 i \, a^{2} b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (32 i \, a^{7} b - 28 i \, a^{5} b^{3} - 33 i \, a^{3} b^{5} + 27 i \, a b^{7}\right )}\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) - 3 \, {\left (9 \, b^{8} \cos \left (d x + c\right )^{7} - 6 \, {\left (4 \, a^{2} b^{6} - 3 \, b^{8}\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (32 \, a^{4} b^{4} - 51 \, a^{2} b^{6} + 15 \, b^{8}\right )} \cos \left (d x + c\right )^{3} - 8 \, {\left (128 \, a^{6} b^{2} - 208 \, a^{4} b^{4} + 57 \, a^{2} b^{6} + 15 \, b^{8}\right )} \cos \left (d x + c\right ) + 2 \, {\left (7 \, a b^{7} \cos \left (d x + c\right )^{5} - 8 \, {\left (3 \, a^{3} b^{5} - 4 \, a b^{7}\right )} \cos \left (d x + c\right )^{3} - 4 \, {\left (160 \, a^{5} b^{3} - 291 \, a^{3} b^{5} + 123 \, a b^{7}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{297 \, {\left (b^{11} d \cos \left (d x + c\right )^{2} - 2 \, a b^{10} d \sin \left (d x + c\right ) - {\left (a^{2} b^{9} + b^{11}\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^8}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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