Optimal. Leaf size=54 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2 \sqrt {a \sec (c+d x)+a}}{a^2 d} \]
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Rubi [A] time = 0.07, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3880, 80, 63, 207} \[ \frac {2 \sqrt {a \sec (c+d x)+a}}{a^2 d}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{3/2} d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 207
Rule 3880
Rubi steps
\begin {align*} \int \frac {\tan ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {-a+a x}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {2 \sqrt {a+a \sec (c+d x)}}{a^2 d}-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac {2 \sqrt {a+a \sec (c+d x)}}{a^2 d}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{a^2 d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2 \sqrt {a+a \sec (c+d x)}}{a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 56, normalized size = 1.04 \[ \frac {2 \left (\sec (c+d x)+\sqrt {\sec (c+d x)+1} \tanh ^{-1}\left (\sqrt {\sec (c+d x)+1}\right )+1\right )}{a d \sqrt {a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 191, normalized size = 3.54 \[ \left [\frac {\sqrt {a} \log \left (-8 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 4 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{2 \, a^{2} d}, -\frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) - 2 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{a^{2} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.20, size = 99, normalized size = 1.83 \[ \frac {2 \, {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {\sqrt {2}}{\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}\right )}}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.05, size = 81, normalized size = 1.50 \[ -\frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \sqrt {2}-2\right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 71, normalized size = 1.31 \[ -\frac {\frac {\log \left (\frac {\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {2 \, \sqrt {a + \frac {a}{\cos \left (d x + c\right )}}}{a^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\tan ^{3}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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