Optimal. Leaf size=39 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \csc ^2(c+d x)}}\right )}{\sqrt{a} d} \]
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Rubi [A] time = 0.0360174, antiderivative size = 40, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {4128, 377, 203} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right )}{\sqrt{a} d} \]
Antiderivative was successfully verified.
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Rule 4128
Rule 377
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b \csc ^2(c+d x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{a+b+b x^2}} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,\frac{\cot (c+d x)}{\sqrt{a+b+b \cot ^2(c+d x)}}\right )}{d}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b+b \cot ^2(c+d x)}}\right )}{\sqrt{a} d}\\ \end{align*}
Mathematica [B] time = 0.144836, size = 98, normalized size = 2.51 \[ -\frac{\csc (c+d x) \sqrt{a \cos (2 (c+d x))-a-2 b} \log \left (\sqrt{a \cos (2 (c+d x))-a-2 b}+\sqrt{2} \sqrt{a} \cos (c+d x)\right )}{\sqrt{2} \sqrt{a} d \sqrt{a+b \csc ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.371, size = 182, normalized size = 4.7 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{d \left ( -1+\cos \left ( dx+c \right ) \right ) }\sqrt{-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}}\ln \left ( 4\,\cos \left ( dx+c \right ) \sqrt{-a}\sqrt{-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}}-4\,a\cos \left ( dx+c \right ) +4\,\sqrt{-a}\sqrt{-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}} \right ){\frac{1}{\sqrt{-a}}}{\frac{1}{\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}-1}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.685864, size = 995, normalized size = 25.51 \begin{align*} \left [-\frac{\sqrt{-a} \log \left (128 \, a^{4} \cos \left (d x + c\right )^{8} - 256 \,{\left (a^{4} + a^{3} b\right )} \cos \left (d x + c\right )^{6} + 160 \,{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \,{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (d x + c\right )^{2} + 8 \,{\left (16 \, a^{3} \cos \left (d x + c\right )^{7} - 24 \,{\left (a^{3} + a^{2} b\right )} \cos \left (d x + c\right )^{5} + 10 \,{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )^{3} -{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right )^{2} - a - b}{\cos \left (d x + c\right )^{2} - 1}} \sin \left (d x + c\right )\right )}{8 \, a d}, \frac{\arctan \left (\frac{{\left (8 \, a^{2} \cos \left (d x + c\right )^{4} - 8 \,{\left (a^{2} + a b\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right )^{2} - a - b}{\cos \left (d x + c\right )^{2} - 1}} \sin \left (d x + c\right )}{4 \,{\left (2 \, a^{3} \cos \left (d x + c\right )^{5} - 3 \,{\left (a^{3} + a^{2} b\right )} \cos \left (d x + c\right )^{3} +{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )\right )}}\right )}{4 \, \sqrt{a} d}\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{1}{\sqrt{a + b \csc ^{2}{\left (c + d 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{1}{\sqrt{b \csc \left (d x + c\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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