Optimal. Leaf size=396 \[ \frac{(a+3 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{a^2 d \sqrt{a+b}}+\frac{b \left (a^2-3 b^2\right ) \tan (c+d x)}{a^2 d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{\left (a^2-3 b^2\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a^2 b d \sqrt{a+b}}+\frac{3 b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a^3 d}+\frac{\sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}} \]
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Rubi [A] time = 0.497515, antiderivative size = 396, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3846, 4061, 4058, 3921, 3784, 3832, 4004} \[ \frac{b \left (a^2-3 b^2\right ) \tan (c+d x)}{a^2 d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{\left (a^2-3 b^2\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a^2 b d \sqrt{a+b}}+\frac{(a+3 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a^2 d \sqrt{a+b}}+\frac{3 b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a^3 d}+\frac{\sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3846
Rule 4061
Rule 4058
Rule 3921
Rule 3784
Rule 3832
Rule 4004
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx &=\frac{\sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}+\frac{\int \frac{-\frac{3 b}{2}+\frac{1}{2} b \sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx}{a}\\ &=\frac{\sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}+\frac{b \left (a^2-3 b^2\right ) \tan (c+d x)}{a^2 \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}-\frac{2 \int \frac{\frac{3}{4} b \left (a^2-b^2\right )-\frac{1}{2} a b^2 \sec (c+d x)+\frac{1}{4} b \left (a^2-3 b^2\right ) \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{a^2 \left (a^2-b^2\right )}\\ &=\frac{\sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}+\frac{b \left (a^2-3 b^2\right ) \tan (c+d x)}{a^2 \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}-\frac{2 \int \frac{\frac{3}{4} b \left (a^2-b^2\right )+\left (-\frac{a b^2}{2}-\frac{1}{4} b \left (a^2-3 b^2\right )\right ) \sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{a^2 \left (a^2-b^2\right )}-\frac{\left (b \left (a^2-3 b^2\right )\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{2 a^2 \left (a^2-b^2\right )}\\ &=\frac{\left (a^2-3 b^2\right ) \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}}}{a^2 b \sqrt{a+b} d}+\frac{\sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}+\frac{b \left (a^2-3 b^2\right ) \tan (c+d x)}{a^2 \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}-\frac{(3 b) \int \frac{1}{\sqrt{a+b \sec (c+d x)}} \, dx}{2 a^2}+\frac{(b (a+3 b)) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{2 a^2 (a+b)}\\ &=\frac{\left (a^2-3 b^2\right ) \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}}}{a^2 b \sqrt{a+b} d}+\frac{(a+3 b) \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}}}{a^2 \sqrt{a+b} d}+\frac{3 b \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^3 d}+\frac{\sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}+\frac{b \left (a^2-3 b^2\right ) \tan (c+d x)}{a^2 \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}\\ \end{align*}
Mathematica [B] time = 15.4708, size = 1077, normalized size = 2.72 \[ \frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \left (-\frac{2 \sin (c+d x) b^3}{a^2 \left (a^2-b^2\right ) (b+a \cos (c+d x))}-\frac{2 \sin (c+d x) b^2}{a^2 \left (b^2-a^2\right )}\right )}{d (a+b \sec (c+d x))^{3/2}}-\frac{(b+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \sqrt{\frac{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 (a^3 \tan ^5\left (\frac{1}{2} (c+d x)\right )+3 b^3 \tan ^5\left (\frac{1}{2} (c+d x)\right )-3 a b^2 \tan ^5\left (\frac{1}{2} (c+d x)\right )-a^2 b \tan ^5\left (\frac{1}{2} (c+d x)\right )-2 a^3 \tan ^3\left (\frac{1}{2} (c+d x)\right )+6 a b^2 \tan ^3\left (\frac{1}{2} (c+d x)\right )-6 b^3 \Pi \left (-1;-\sin ^{-1}\left (\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 )+6 a^2 b \Pi \left (-1;-\sin ^{-1}\left (\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 )+a^3 \tan \left (\frac{1}{2} (c+d x)\right )-3 b^3 \tan \left (\frac{1}{2} (c+d x)\right )-3 a b^2 \tan \left (\frac{1}{2} (c+d x)\right )+a^2 b \tan \left (\frac{1}{2} (c+d x)\right )+\left (a^3+b a^2-3 b^2 a-3 b^3\right ) E\left (\sin ^{-1}\left (\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}}+2 a b (a+b) \text{EllipticF}\left (\sin ^{-1}\left (\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}}-6 b^3 \Pi \left (-1;-\sin ^{-1}\left (\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}}+6 a^2 b \Pi \left (-1;-\sin ^{-1}\left (\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 )}{a^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2} \sqrt{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1} \left (a \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )-1\right )-b \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.27, size = 1662, normalized size = 4.2 \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{\cos \left (d x + c\right )}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{3}{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{b \sec \left (d x + c\right ) + a} \cos \left (d x + c\right )}{b^{2} \sec \left (d x + c\right )^{2} + 2 \, a b \sec \left (d x + c\right ) + a^{2}}, 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{\cos{\left (c + d x \right )}}{\left (a + b \sec{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, 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{\cos \left (d x + c\right )}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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