Optimal. Leaf size=86 \[ \frac {8 a^2 \tan (c+d x)}{5 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{5 d}+\frac {2 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d} \]
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Rubi [A] time = 0.12, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3798, 3793, 3792} \[ \frac {8 a^2 \tan (c+d x)}{5 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{5 d}+\frac {2 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 3792
Rule 3793
Rule 3798
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx &=\frac {2 (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 d}+\frac {3}{5} \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac {2 a \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{5 d}+\frac {2 (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 d}+\frac {1}{5} (4 a) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {8 a^2 \tan (c+d x)}{5 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{5 d}+\frac {2 (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 48, normalized size = 0.56 \[ \frac {2 a^2 \tan (c+d x) \left (\sec ^2(c+d x)+3 \sec (c+d x)+6\right )}{5 d \sqrt {a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 74, normalized size = 0.86 \[ \frac {2 \, {\left (6 \, a \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) + a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{5 \, {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 7.77, size = 121, normalized size = 1.41 \[ \frac {4 \, {\left (5 \, \sqrt {2} a^{4} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + {\left (2 \, \sqrt {2} a^{4} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5 \, \sqrt {2} a^{4} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{5 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.84, size = 73, normalized size = 0.85 \[ -\frac {2 \left (6 \left (\cos ^{3}\left (d x +c \right )\right )-3 \left (\cos ^{2}\left (d x +c \right )\right )-2 \cos \left (d x +c \right )-1\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a}{5 d \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.36, size = 116, normalized size = 1.35 \[ \frac {4\,a\,\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,5{}\mathrm {i}-{\mathrm {e}}^{c\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,5{}\mathrm {i}-{\mathrm {e}}^{c\,5{}\mathrm {i}+d\,x\,5{}\mathrm {i}}\,3{}\mathrm {i}+3{}\mathrm {i}\right )}{5\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2} \]
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 \[ \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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