Optimal. Leaf size=98 \[ \frac{5 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b d^2 \sqrt{d \cos (a+b x)}}+\frac{5 \sin (a+b x)}{3 b d (d \cos (a+b x))^{3/2}}-\frac{\csc (a+b x)}{b d (d \cos (a+b x))^{3/2}} \]
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Rubi [A] time = 0.0829941, antiderivative size = 98, 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, 2642, 2641} \[ \frac{5 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b d^2 \sqrt{d \cos (a+b x)}}+\frac{5 \sin (a+b x)}{3 b d (d \cos (a+b x))^{3/2}}-\frac{\csc (a+b x)}{b d (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2570
Rule 2636
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{\csc ^2(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx &=-\frac{\csc (a+b x)}{b d (d \cos (a+b x))^{3/2}}+\frac{5}{2} \int \frac{1}{(d \cos (a+b x))^{5/2}} \, dx\\ &=-\frac{\csc (a+b x)}{b d (d \cos (a+b x))^{3/2}}+\frac{5 \sin (a+b x)}{3 b d (d \cos (a+b x))^{3/2}}+\frac{5 \int \frac{1}{\sqrt{d \cos (a+b x)}} \, dx}{6 d^2}\\ &=-\frac{\csc (a+b x)}{b d (d \cos (a+b x))^{3/2}}+\frac{5 \sin (a+b x)}{3 b d (d \cos (a+b x))^{3/2}}+\frac{\left (5 \sqrt{\cos (a+b x)}\right ) \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{6 d^2 \sqrt{d \cos (a+b x)}}\\ &=-\frac{\csc (a+b x)}{b d (d \cos (a+b x))^{3/2}}+\frac{5 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b d^2 \sqrt{d \cos (a+b x)}}+\frac{5 \sin (a+b x)}{3 b d (d \cos (a+b x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.140831, size = 62, normalized size = 0.63 \[ \frac{2 \tan (a+b x)-3 \cot (a+b x)+5 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b d^2 \sqrt{d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.235, size = 190, normalized size = 1.9 \begin{align*}{\frac{1}{6\,db}\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 ( 10\, \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 ) -20\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+20\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-3 \right ) \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \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 ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\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{5}{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^{3} \cos \left (b x + a\right )^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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