Optimal. Leaf size=146 \[ \frac {832 a^3 \tan (c+d x)}{315 d \sqrt {a \sec (c+d x)+a}}+\frac {208 a^2 \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{315 d}+\frac {2 \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}-\frac {4 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}+\frac {26 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d} \]
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Rubi [A] time = 0.23, antiderivative size = 146, 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 = {3800, 4001, 3793, 3792} \[ \frac {208 a^2 \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{315 d}+\frac {832 a^3 \tan (c+d x)}{315 d \sqrt {a \sec (c+d x)+a}}+\frac {2 \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}-\frac {4 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}+\frac {26 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d} \]
Antiderivative was successfully verified.
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Rule 3792
Rule 3793
Rule 3800
Rule 4001
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \, dx &=\frac {2 (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {2 \int \sec (c+d x) \left (\frac {7 a}{2}-a \sec (c+d x)\right ) (a+a \sec (c+d x))^{5/2} \, dx}{9 a}\\ &=-\frac {4 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {13}{21} \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \, dx\\ &=\frac {26 a (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}-\frac {4 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {1}{105} (104 a) \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac {208 a^2 \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {26 a (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}-\frac {4 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {1}{315} \left (416 a^2\right ) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {832 a^3 \tan (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {208 a^2 \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {26 a (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}-\frac {4 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 70, normalized size = 0.48 \[ \frac {2 a^3 \tan (c+d x) \left (35 \sec ^4(c+d x)+130 \sec ^3(c+d x)+219 \sec ^2(c+d x)+292 \sec (c+d x)+584\right )}{315 d \sqrt {a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 108, normalized size = 0.74 \[ \frac {2 \, {\left (584 \, a^{2} \cos \left (d x + c\right )^{4} + 292 \, a^{2} \cos \left (d x + c\right )^{3} + 219 \, a^{2} \cos \left (d x + c\right )^{2} + 130 \, a^{2} \cos \left (d x + c\right ) + 35 \, a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 5.91, size = 180, normalized size = 1.23 \[ \frac {8 \, {\left (315 \, \sqrt {2} a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (630 \, \sqrt {2} a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 13 \, {\left (63 \, \sqrt {2} a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 4 \, {\left (2 \, \sqrt {2} a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \sqrt {2} a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{315 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.87, size = 95, normalized size = 0.65 \[ -\frac {2 \left (584 \left (\cos ^{5}\left (d x +c \right )\right )-292 \left (\cos ^{4}\left (d x +c \right )\right )-73 \left (\cos ^{3}\left (d x +c \right )\right )-89 \left (\cos ^{2}\left (d x +c \right )\right )-95 \cos \left (d x +c \right )-35\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{2}}{315 d \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )} \]
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 = 8.18, size = 456, normalized size = 3.12 \[ \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\,32{}\mathrm {i}}{9\,d}-\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,32{}\mathrm {i}}{9\,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 )}^4}-\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\,96{}\mathrm {i}}{7\,d}-\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,32{}\mathrm {i}}{63\,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 {\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\,8{}\mathrm {i}}{3\,d}-\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,584{}\mathrm {i}}{315\,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\,56{}\mathrm {i}}{5\,d}+\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,904{}\mathrm {i}}{105\,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 )}^2}-\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}}}\,1168{}\mathrm {i}}{315\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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