Optimal. Leaf size=94 \[ -\frac {3 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{b d^2 \sqrt {\cos (a+b x)}}+\frac {3 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {\csc (a+b x)}{b d \sqrt {d \cos (a+b x)}} \]
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Rubi [A] time = 0.08, antiderivative size = 94, 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 = {2570, 2636, 2640, 2639} \[ -\frac {3 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{b d^2 \sqrt {\cos (a+b x)}}+\frac {3 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {\csc (a+b x)}{b d \sqrt {d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2570
Rule 2636
Rule 2639
Rule 2640
Rubi steps
\begin {align*} \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx &=-\frac {\csc (a+b x)}{b d \sqrt {d \cos (a+b x)}}+\frac {3}{2} \int \frac {1}{(d \cos (a+b x))^{3/2}} \, dx\\ &=-\frac {\csc (a+b x)}{b d \sqrt {d \cos (a+b x)}}+\frac {3 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {3 \int \sqrt {d \cos (a+b x)} \, dx}{2 d^2}\\ &=-\frac {\csc (a+b x)}{b d \sqrt {d \cos (a+b x)}}+\frac {3 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {\left (3 \sqrt {d \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{2 d^2 \sqrt {\cos (a+b x)}}\\ &=-\frac {\csc (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {3 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b d^2 \sqrt {\cos (a+b x)}}+\frac {3 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 65, normalized size = 0.69 \[ \frac {2 \sin (a+b x)-\cos (a+b x) \cot (a+b x)-3 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b d \sqrt {d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \cos \left (b x + a\right )} \csc \left (b x + a\right )^{2}}{d^{2} \cos \left (b x + a\right )^{2}}, x\right ) \]
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 \[ \int \frac {\csc \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 209, normalized size = 2.22 \[ -\frac {\sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d \right )^{\frac {3}{2}} \left (6 \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+12 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1\right )}{2 d^{3} \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{5} \left (2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )^{2} \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b} \]
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 \[ \int \frac {\csc \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\sin \left (a+b\,x\right )}^2\,{\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}{\left (a + b x \right )}}{\left (d \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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