Optimal. Leaf size=361 \[ -\frac{\cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{d \sqrt{a+b}}+\frac{b^2 \tan (c+d x)}{d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}-\frac{\cot (c+d x)}{d \sqrt{a+b \sec (c+d x)}}+\frac{\cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{d \sqrt{a+b}}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a d} \]
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Rubi [A] time = 0.420432, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {3896, 3784, 3875, 3833, 21, 3829, 3832, 4004} \[ \frac{b^2 \tan (c+d x)}{d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}-\frac{\cot (c+d x)}{d \sqrt{a+b \sec (c+d x)}}-\frac{\cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{d \sqrt{a+b}}+\frac{\cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{d \sqrt{a+b}}+\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a d} \]
Antiderivative was successfully verified.
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Rule 3896
Rule 3784
Rule 3875
Rule 3833
Rule 21
Rule 3829
Rule 3832
Rule 4004
Rubi steps
\begin{align*} \int \frac{\cot ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx &=\int \left (-\frac{1}{\sqrt{a+b \sec (c+d x)}}+\frac{\csc ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}}\right ) \, dx\\ &=-\int \frac{1}{\sqrt{a+b \sec (c+d x)}} \, dx+\int \frac{\csc ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{2 \sqrt{a+b} \cot (c+d x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{a d}-\frac{\cot (c+d x)}{d \sqrt{a+b \sec (c+d x)}}-\frac{1}{2} b \int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx\\ &=\frac{2 \sqrt{a+b} \cot (c+d x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{a d}-\frac{\cot (c+d x)}{d \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}+\frac{b \int \frac{\sec (c+d x) \left (-\frac{a}{2}-\frac{1}{2} b \sec (c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{a^2-b^2}\\ &=\frac{2 \sqrt{a+b} \cot (c+d x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{a d}-\frac{\cot (c+d x)}{d \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}-\frac{b \int \sec (c+d x) \sqrt{a+b \sec (c+d x)} \, dx}{2 \left (a^2-b^2\right )}\\ &=\frac{2 \sqrt{a+b} \cot (c+d x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{a d}-\frac{\cot (c+d x)}{d \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}-\frac{b \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{2 (a+b)}-\frac{b^2 \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{2 \left (a^2-b^2\right )}\\ &=\frac{\cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{\sqrt{a+b} d}-\frac{\cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{\sqrt{a+b} d}+\frac{2 \sqrt{a+b} \cot (c+d x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{a d}-\frac{\cot (c+d x)}{d \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 18.735, size = 1198, normalized size = 3.32 \[ \frac{(b+a \cos (c+d x)) \sec (c+d x) \left (\frac{(a \cos (c+d x)-b) \csc (c+d x)}{b^2-a^2}+\frac{b \sin (c+d x)}{b^2-a^2}\right )}{d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \left (-b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left (\frac{1}{2} (c+d x)\right )+a b \sqrt{\frac{b-a}{a+b}} \tan ^5\left (\frac{1}{2} (c+d x)\right )-2 a b \sqrt{\frac{b-a}{a+b}} \tan ^3\left (\frac{1}{2} (c+d x)\right )+4 i a^2 \Pi \left (-\frac{a+b}{a-b};i \sinh ^{-1}\left (\sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}} \tan ^2\left (\frac{1}{2} (c+d x)\right )-4 i b^2 \Pi \left (-\frac{a+b}{a-b};i \sinh ^{-1}\left (\sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}} \tan ^2\left (\frac{1}{2} (c+d x)\right )+b^2 \sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )+a b \sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )-i (a-b) b E\left (i \sinh ^{-1}\left (\sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right ) \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}}-i \left (2 a^2-b a-b^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right ),\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right ) \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}}+4 i a^2 \Pi \left (-\frac{a+b}{a-b};i \sinh ^{-1}\left (\sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}}-4 i b^2 \Pi \left (-\frac{a+b}{a-b};i \sinh ^{-1}\left (\sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a+b}{a-b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}}\right )}{\sqrt{\frac{b-a}{a+b}} \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)} \sqrt{\frac{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )}} \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}} \left (\tan ^4\left (\frac{1}{2} (c+d x)\right )-1\right )} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.335, size = 1408, normalized size = 3.9 \begin{align*} \text{result too large to display} \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{\cot \left (d x + c\right )^{2}}{\sqrt{b \sec \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{2}{\left (c + d x \right )}}{\sqrt{a + b \sec{\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{\cot \left (d x + c\right )^{2}}{\sqrt{b \sec \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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