Optimal. Leaf size=64 \[ \frac{\sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b \sqrt{d \cos (a+b x)}}-\frac{\csc (a+b x) \sqrt{d \cos (a+b x)}}{b d} \]
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Rubi [A] time = 0.0578789, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2570, 2642, 2641} \[ \frac{\sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b \sqrt{d \cos (a+b x)}}-\frac{\csc (a+b x) \sqrt{d \cos (a+b x)}}{b d} \]
Antiderivative was successfully verified.
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Rule 2570
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{\csc ^2(a+b x)}{\sqrt{d \cos (a+b x)}} \, dx &=-\frac{\sqrt{d \cos (a+b x)} \csc (a+b x)}{b d}+\frac{1}{2} \int \frac{1}{\sqrt{d \cos (a+b x)}} \, dx\\ &=-\frac{\sqrt{d \cos (a+b x)} \csc (a+b x)}{b d}+\frac{\sqrt{\cos (a+b x)} \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{2 \sqrt{d \cos (a+b x)}}\\ &=-\frac{\sqrt{d \cos (a+b x)} \csc (a+b x)}{b d}+\frac{\sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b \sqrt{d \cos (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0836877, size = 47, normalized size = 0.73 \[ \frac{\sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )-\cot (a+b x)}{b \sqrt{d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.27, size = 188, normalized size = 2.9 \begin{align*}{\frac{d}{2\,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}}\sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \left ( 2\, \left ( 2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) ^{3/2}\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \cos \left ( 1/2\,bx+a/2 \right ) -4\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+4\, \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 ( -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}}}{\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}}{\sqrt{d \cos \left (b x + a\right )}}\,{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 \cos \left (b x + a\right )}, 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 )}}{\sqrt{d \cos{\left (a + b x \right )}}}\, 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}}{\sqrt{d \cos \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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