Optimal. Leaf size=122 \[ -\frac{b^2 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{b \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d} \]
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Rubi [A] time = 0.307234, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524, Rules used = {3518, 3110, 3768, 3770, 2621, 321, 207, 2622, 3104, 3074, 206} \[ -\frac{b^2 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{b \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 3518
Rule 3110
Rule 3768
Rule 3770
Rule 2621
Rule 321
Rule 207
Rule 2622
Rule 3104
Rule 3074
Rule 206
Rubi steps
\begin{align*} \int \frac{\csc ^3(c+d x)}{a+b \tan (c+d x)} \, dx &=\int \frac{\cot (c+d x) \csc ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx\\ &=\int \left (\frac{\csc ^3(c+d x)}{a}-\frac{b \csc ^2(c+d x) \sec (c+d x)}{a^2}+\frac{b^2 \csc (c+d x) \sec ^2(c+d x)}{a^3}-\frac{b^3 \sec ^2(c+d x)}{a^3 (a \cos (c+d x)+b \sin (c+d x))}\right ) \, dx\\ &=\frac{\int \csc ^3(c+d x) \, dx}{a}-\frac{b \int \csc ^2(c+d x) \sec (c+d x) \, dx}{a^2}+\frac{b^2 \int \csc (c+d x) \sec ^2(c+d x) \, dx}{a^3}-\frac{b^3 \int \frac{\sec ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{a^3}\\ &=-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}-\frac{b^2 \sec (c+d x)}{a^3 d}+\frac{\int \csc (c+d x) \, dx}{2 a}+\frac{b \int \sec (c+d x) \, dx}{a^2}-\frac{\left (b \left (a^2+b^2\right )\right ) \int \frac{1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{a^3}+\frac{b \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{a^2 d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{a^2 d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^3 d}+\frac{\left (b \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{b^2 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{b \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}\\ \end{align*}
Mathematica [A] time = 0.826587, size = 179, normalized size = 1.47 \[ \frac{-16 b \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )-b}{\sqrt{a^2+b^2}}\right )+a^2 \left (-\csc ^2\left (\frac{1}{2} (c+d x)\right )\right )+a^2 \sec ^2\left (\frac{1}{2} (c+d x)\right )+4 a^2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-4 a^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+4 a b \tan \left (\frac{1}{2} (c+d x)\right )+4 a b \cot \left (\frac{1}{2} (c+d x)\right )+8 b^2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-8 b^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{8 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 162, normalized size = 1.3 \begin{align*}{\frac{1}{8\,ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}+{\frac{b}{2\,{a}^{2}d}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-2\,{\frac{b\sqrt{{a}^{2}+{b}^{2}}}{d{a}^{3}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tan \left ( 1/2\,dx+c/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }-{\frac{1}{8\,ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}+{\frac{1}{2\,ad}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{{b}^{2}}{d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{b}{2\,{a}^{2}d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-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 = 2.68281, size = 644, normalized size = 5.28 \begin{align*} \frac{2 \, a^{2} \cos \left (d x + c\right ) - 4 \, a b \sin \left (d x + c\right ) + 2 \,{\left (b \cos \left (d x + c\right )^{2} - b\right )} \sqrt{a^{2} + b^{2}} \log \left (\frac{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt{a^{2} + b^{2}}{\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) -{\left ({\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} - 2 \, b^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) +{\left ({\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} - 2 \, b^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{4 \,{\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} 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{\csc ^{3}{\left (c + d x \right )}}{a + b \tan{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35144, size = 282, normalized size = 2.31 \begin{align*} \frac{\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 4 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{2}} + \frac{4 \,{\left (a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac{8 \,{\left (a^{2} b + b^{3}\right )} \log \left (\frac{{\left | 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{\sqrt{a^{2} + b^{2}} a^{3}} - \frac{6 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 12 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 4 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{2}}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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