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 = 0.0804821, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, 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 2640
Rule 2639
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*}
Mathematica [A] time = 0.171588, 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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Maple [A] time = 0.319, size = 209, normalized size = 2.2 \begin{align*} -{\frac{1}{2\,{d}^{3}b}\sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \left ( -2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}d+ \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}d \right ) ^{{\frac{3}{2}}} \left ( 6\,\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \cos \left ( 1/2\,bx+a/2 \right ) +12\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}-12\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1 \right ) \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-5} \left ( 2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) ^{-2}{\frac{1}{\sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{2}{\left (a + b x \right )}}{\left (d \cos{\left (a + b x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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