Optimal. Leaf size=96 \[ -\frac {21 d^4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{5 b \sqrt {\cos (a+b x)}}-\frac {7 d^3 \sin (a+b x) (d \cos (a+b x))^{3/2}}{5 b}-\frac {d \csc (a+b x) (d \cos (a+b x))^{7/2}}{b} \]
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Rubi [A] time = 0.08, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2567, 2635, 2640, 2639} \[ -\frac {7 d^3 \sin (a+b x) (d \cos (a+b x))^{3/2}}{5 b}-\frac {21 d^4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{5 b \sqrt {\cos (a+b x)}}-\frac {d \csc (a+b x) (d \cos (a+b x))^{7/2}}{b} \]
Antiderivative was successfully verified.
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Rule 2567
Rule 2635
Rule 2639
Rule 2640
Rubi steps
\begin {align*} \int (d \cos (a+b x))^{9/2} \csc ^2(a+b x) \, dx &=-\frac {d (d \cos (a+b x))^{7/2} \csc (a+b x)}{b}-\frac {1}{2} \left (7 d^2\right ) \int (d \cos (a+b x))^{5/2} \, dx\\ &=-\frac {d (d \cos (a+b x))^{7/2} \csc (a+b x)}{b}-\frac {7 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}-\frac {1}{10} \left (21 d^4\right ) \int \sqrt {d \cos (a+b x)} \, dx\\ &=-\frac {d (d \cos (a+b x))^{7/2} \csc (a+b x)}{b}-\frac {7 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}-\frac {\left (21 d^4 \sqrt {d \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{10 \sqrt {\cos (a+b x)}}\\ &=-\frac {d (d \cos (a+b x))^{7/2} \csc (a+b x)}{b}-\frac {21 d^4 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b \sqrt {\cos (a+b x)}}-\frac {7 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 74, normalized size = 0.77 \[ -\frac {d^4 \sqrt {d \cos (a+b x)} \left (21 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+\sqrt {\cos (a+b x)} (\sin (2 (a+b x))+5 \cot (a+b x))\right )}{5 b \sqrt {\cos (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {d \cos \left (b x + a\right )} d^{4} \cos \left (b x + a\right )^{4} \csc \left (b x + a\right )^{2}, x\right ) \]
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 \[ \int \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \csc \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.27, size = 229, normalized size = 2.39 \[ \frac {\sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, d^{6} \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (-64 \left (\sin ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+160 \left (\sin ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+42 \left (2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )^{\frac {3}{2}} \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-104 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-4 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+22 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-5\right )}{10 \left (-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d \right )^{\frac {3}{2}} \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b} \]
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 \[ \int \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \csc \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^{9/2}}{{\sin \left (a+b\,x\right )}^2} \,d x \]
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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