Optimal. Leaf size=128 \[ \frac {2 b^7 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b^5 \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {14 b^3 \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}}-\frac {14 b^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}} \]
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Rubi [A] time = 0.09, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 2636, 2640, 2639} \[ \frac {2 b^7 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b^5 \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {14 b^3 \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}}-\frac {14 b^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2636
Rule 2639
Rule 2640
Rubi steps
\begin {align*} \int (b \cos (c+d x))^{5/2} \sec ^8(c+d x) \, dx &=b^8 \int \frac {1}{(b \cos (c+d x))^{11/2}} \, dx\\ &=\frac {2 b^7 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {1}{9} \left (7 b^6\right ) \int \frac {1}{(b \cos (c+d x))^{7/2}} \, dx\\ &=\frac {2 b^7 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b^5 \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {1}{15} \left (7 b^4\right ) \int \frac {1}{(b \cos (c+d x))^{3/2}} \, dx\\ &=\frac {2 b^7 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b^5 \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {14 b^3 \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}}-\frac {1}{15} \left (7 b^2\right ) \int \sqrt {b \cos (c+d x)} \, dx\\ &=\frac {2 b^7 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b^5 \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {14 b^3 \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}}-\frac {\left (7 b^2 \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 \sqrt {\cos (c+d x)}}\\ &=-\frac {14 b^2 \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {2 b^7 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b^5 \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {14 b^3 \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 79, normalized size = 0.62 \[ \frac {\sec ^7(c+d x) (b \cos (c+d x))^{5/2} \left (150 \sin (c+d x)+91 \sin (3 (c+d x))+21 \sin (5 (c+d x))-336 \cos ^{\frac {9}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{360 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \cos \left (d x + c\right )} b^{2} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{8}, x\right ) \]
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 \[ \int \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}} \sec \left (d x + c\right )^{8}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.23, size = 414, normalized size = 3.23 \[ -\frac {2 \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, b^{3} \left (-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}{144 b \left (-\frac {1}{2}+\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {7 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}{180 b \left (-\frac {1}{2}+\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {14 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{15 \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}+\frac {7 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}-\frac {7 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \left (\EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-\EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{15 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d} \]
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 \[ \int \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}} \sec \left (d x + c\right )^{8}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^8} \,d x \]
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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