Optimal. Leaf size=116 \[ \frac {64 a^3 \tan (c+d x)}{21 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a^2 \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{21 d}+\frac {2 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac {2 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d} \]
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Rubi [A] time = 0.16, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3798, 3793, 3792} \[ \frac {64 a^3 \tan (c+d x)}{21 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a^2 \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{21 d}+\frac {2 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac {2 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d} \]
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))^{5/2} \, dx &=\frac {2 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {5}{7} \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \, dx\\ &=\frac {2 a (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac {2 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {1}{7} (8 a) \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac {16 a^2 \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{21 d}+\frac {2 a (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac {2 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {1}{21} \left (32 a^2\right ) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {64 a^3 \tan (c+d x)}{21 d \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{21 d}+\frac {2 a (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac {2 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 60, normalized size = 0.52 \[ \frac {2 a^3 \tan (c+d x) \left (3 \sec ^3(c+d x)+12 \sec ^2(c+d x)+23 \sec (c+d x)+46\right )}{21 d \sqrt {a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 95, normalized size = 0.82 \[ \frac {2 \, {\left (46 \, a^{2} \cos \left (d x + c\right )^{3} + 23 \, a^{2} \cos \left (d x + c\right )^{2} + 12 \, a^{2} \cos \left (d x + c\right ) + 3 \, a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{21 \, {\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 5.11, size = 151, normalized size = 1.30 \[ -\frac {8 \, {\left (21 \, \sqrt {2} a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (35 \, \sqrt {2} a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 4 \, {\left (2 \, \sqrt {2} a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7 \, \sqrt {2} a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{21 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.82, size = 85, normalized size = 0.73 \[ -\frac {2 \left (46 \left (\cos ^{4}\left (d x +c \right )\right )-23 \left (\cos ^{3}\left (d x +c \right )\right )-11 \left (\cos ^{2}\left (d x +c \right )\right )-9 \cos \left (d x +c \right )-3\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{2}}{21 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}} \]
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 = 4.58, size = 349, normalized size = 3.01 \[ \frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a^2\,20{}\mathrm {i}}{3\,d}-\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,4{}\mathrm {i}}{21\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a^2\,16{}\mathrm {i}}{7\,d}+\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,16{}\mathrm {i}}{7\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,92{}\mathrm {i}}{21\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )}+\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,48{}\mathrm {i}}{7\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2} \]
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 \[ \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}} \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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