Optimal. Leaf size=404 \[ \frac{2 \sqrt{a+b} \left (-8 a^2+2 a b+21 b^2\right ) \cot (c+d x) \sqrt{-\frac{b (\sec (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sec (c+d x)+1)}{b-a}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{15 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left (8 a^2-21 b^2\right ) \cot (c+d x) \sqrt{-\frac{b (\sec (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{15 b^4 d}-\frac{8 a \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b^2 d}+\frac{2 \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b d}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a d} \]
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Rubi [A] time = 0.763477, antiderivative size = 610, normalized size of antiderivative = 1.51, number of steps used = 11, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {3895, 3784, 3837, 3832, 4004, 3860, 4082, 4005} \[ -\frac{2 \sqrt{a+b} \left (8 a^2-2 a b+9 b^2\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{15 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left (8 a^2+9 b^2\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{15 b^4 d}-\frac{8 a \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b^2 d}+\frac{4 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{b^2 d}+\frac{2 \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b d}+\frac{4 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{b d}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a d} \]
Antiderivative was successfully verified.
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Rule 3895
Rule 3784
Rule 3837
Rule 3832
Rule 4004
Rule 3860
Rule 4082
Rule 4005
Rubi steps
\begin{align*} \int \frac{\tan ^4(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx &=\int \left (\frac{1}{\sqrt{a+b \sec (c+d x)}}-\frac{2 \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}}+\frac{\sec ^4(c+d x)}{\sqrt{a+b \sec (c+d x)}}\right ) \, dx\\ &=-\left (2 \int \frac{\sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx\right )+\int \frac{1}{\sqrt{a+b \sec (c+d x)}} \, dx+\int \frac{\sec ^4(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{2 \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{5 b d}+2 \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx-2 \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx+\frac{\int \frac{\sec (c+d x) \left (2 a+3 b \sec (c+d x)-4 a \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{5 b}\\ &=\frac{4 (a-b) \sqrt{a+b} \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}}}{b^2 d}+\frac{4 \sqrt{a+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}}}{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{8 a \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{15 b^2 d}+\frac{2 \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{5 b d}+\frac{2 \int \frac{\sec (c+d x) \left (a b+\frac{1}{2} \left (8 a^2+9 b^2\right ) \sec (c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{15 b^2}\\ &=\frac{4 (a-b) \sqrt{a+b} \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}}}{b^2 d}+\frac{4 \sqrt{a+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}}}{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{8 a \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{15 b^2 d}+\frac{2 \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{5 b d}+\frac{1}{15} \left (9+\frac{8 a^2}{b^2}\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx-\frac{\left (8 a^2-2 a b+9 b^2\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{15 b^2}\\ &=\frac{4 (a-b) \sqrt{a+b} \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}}}{b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left (8 a^2+9 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}}}{15 b^4 d}+\frac{4 \sqrt{a+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}}}{b d}-\frac{2 \sqrt{a+b} \left (8 a^2-2 a b+9 b^2\right ) \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}}}{15 b^3 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{8 a \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{15 b^2 d}+\frac{2 \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{5 b d}\\ \end{align*}
Mathematica [B] time = 16.8896, size = 839, normalized size = 2.08 \[ \frac{(b+a \cos (c+d x)) \sec (c+d x) \left (-\frac{2 \left (21 b^2-8 a^2\right ) \sin (c+d x)}{15 b^3}+\frac{2 \sec (c+d x) \tan (c+d x)}{5 b}-\frac{8 a \tan (c+d x)}{15 b^2}\right )}{d \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )}} \left (8 a^3 \tan ^5\left (\frac{1}{2} (c+d x)\right )+21 b^3 \tan ^5\left (\frac{1}{2} (c+d x)\right )-21 a b^2 \tan ^5\left (\frac{1}{2} (c+d x)\right )-8 a^2 b \tan ^5\left (\frac{1}{2} (c+d x)\right )-16 a^3 \tan ^3\left (\frac{1}{2} (c+d x)\right )+42 a b^2 \tan ^3\left (\frac{1}{2} (c+d x)\right )+30 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 )+8 a^3 \tan \left (\frac{1}{2} (c+d x)\right )-21 b^3 \tan \left (\frac{1}{2} (c+d x)\right )-21 a b^2 \tan \left (\frac{1}{2} (c+d x)\right )+8 a^2 b \tan \left (\frac{1}{2} (c+d x)\right )+\left (8 a^3+8 b a^2-21 b^2 a-21 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 b \left (4 a^2+b a-18 b^2\right ) \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}}+30 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}}\right )}{15 b^3 d \sqrt{a+b \sec (c+d x)} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )^{3/2} \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}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.566, size = 1780, normalized size = 4.4 \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{\tan \left (d x + c\right )^{4}}{\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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\tan \left (d x + c\right )^{4}}{\sqrt{b \sec \left (d x + c\right ) + a}}, 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{\tan ^{4}{\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{\tan \left (d x + c\right )^{4}}{\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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