Optimal. Leaf size=100 \[ \frac {2 (a+b \sec (c+d x))^{5/2}}{5 b^2 d}-\frac {2 a (a+b \sec (c+d x))^{3/2}}{3 b^2 d}-\frac {2 \sqrt {a+b \sec (c+d x)}}{d}+\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{d} \]
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Rubi [A] time = 0.11, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3885, 898, 1261, 207} \[ \frac {2 (a+b \sec (c+d x))^{5/2}}{5 b^2 d}-\frac {2 a (a+b \sec (c+d x))^{3/2}}{3 b^2 d}-\frac {2 \sqrt {a+b \sec (c+d x)}}{d}+\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 898
Rule 1261
Rule 3885
Rubi steps
\begin {align*} \int \sqrt {a+b \sec (c+d x)} \tan ^3(c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+x} \left (b^2-x^2\right )}{x} \, dx,x,b \sec (c+d x)\right )}{b^2 d}\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {x^2 \left (-a^2+b^2+2 a x^2-x^4\right )}{-a+x^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{b^2 d}\\ &=-\frac {2 \operatorname {Subst}\left (\int \left (b^2+a x^2-x^4+\frac {a b^2}{-a+x^2}\right ) \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{b^2 d}\\ &=-\frac {2 \sqrt {a+b \sec (c+d x)}}{d}-\frac {2 a (a+b \sec (c+d x))^{3/2}}{3 b^2 d}+\frac {2 (a+b \sec (c+d x))^{5/2}}{5 b^2 d}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{-a+x^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{d}\\ &=\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {2 \sqrt {a+b \sec (c+d x)}}{d}-\frac {2 a (a+b \sec (c+d x))^{3/2}}{3 b^2 d}+\frac {2 (a+b \sec (c+d x))^{5/2}}{5 b^2 d}\\ \end {align*}
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Mathematica [A] time = 6.22, size = 194, normalized size = 1.94 \[ \frac {\sqrt {a+b \sec (c+d x)} \left (-\frac {2 \left (2 a^2+15 b^2\right )}{15 b^2}+\frac {2 a \sec (c+d x)}{15 b}+\frac {2}{5} \sec ^2(c+d x)\right )}{d}+\frac {\sin ^2(c+d x) \sqrt {a \cos (c+d x)} \sqrt {a+b \sec (c+d x)} \left (\log \left (\frac {\sqrt {a \cos (c+d x)+b}}{\sqrt {a \cos (c+d x)}}+1\right )-\log \left (1-\frac {\sqrt {a \cos (c+d x)+b}}{\sqrt {a \cos (c+d x)}}\right )\right )}{d \left (1-\cos ^2(c+d x)\right ) \sqrt {a \cos (c+d x)+b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.45, size = 311, normalized size = 3.11 \[ \left [\frac {15 \, \sqrt {a} b^{2} \cos \left (d x + c\right )^{2} \log \left (-8 \, a^{2} \cos \left (d x + c\right )^{2} - 8 \, a b \cos \left (d x + c\right ) - b^{2} - 4 \, {\left (2 \, a \cos \left (d x + c\right )^{2} + b \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}}\right ) + 4 \, {\left (a b \cos \left (d x + c\right ) - {\left (2 \, a^{2} + 15 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, b^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}}}{30 \, b^{2} d \cos \left (d x + c\right )^{2}}, -\frac {15 \, \sqrt {-a} b^{2} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + b}\right ) \cos \left (d x + c\right )^{2} - 2 \, {\left (a b \cos \left (d x + c\right ) - {\left (2 \, a^{2} + 15 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, b^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}}}{15 \, b^{2} d \cos \left (d x + c\right )^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.17, size = 539, normalized size = 5.39 \[ -\frac {2 \, {\left (\frac {15 \, a \arctan \left (-\frac {\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b} + \sqrt {a - b}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} - \frac {2 \, {\left (15 \, {\left (\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}\right )}^{4} a - 30 \, {\left (\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}\right )}^{3} {\left (a + 2 \, b\right )} \sqrt {a - b} + 20 \, {\left (\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}\right )}^{2} {\left (4 \, a b - 3 \, b^{2}\right )} - 15 \, a^{3} - 10 \, a^{2} b - 35 \, a b^{2} + 12 \, b^{3} + 10 \, {\left (\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}\right )} {\left (3 \, a^{2} - a b + 6 \, b^{2}\right )} \sqrt {a - b}\right )}}{{\left (\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b} - \sqrt {a - b}\right )}^{5}}\right )} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.72, size = 2342, normalized size = 23.42 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 108, normalized size = 1.08 \[ -\frac {15 \, \sqrt {a} \log \left (\frac {\sqrt {a + \frac {b}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {b}{\cos \left (d x + c\right )}} + \sqrt {a}}\right ) + 30 \, \sqrt {a + \frac {b}{\cos \left (d x + c\right )}} - \frac {6 \, {\left (a + \frac {b}{\cos \left (d x + c\right )}\right )}^{\frac {5}{2}}}{b^{2}} + \frac {10 \, {\left (a + \frac {b}{\cos \left (d x + c\right )}\right )}^{\frac {3}{2}} a}{b^{2}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^3\,\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \sec {\left (c + d x \right )}} \tan ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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