Optimal. Leaf size=144 \[ \frac {8 a^2 (35 A+21 B+19 C) \tan (c+d x)}{105 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a (35 A+21 B+19 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{105 d}+\frac {2 (7 B-2 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 a d} \]
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Rubi [A] time = 0.28, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {4082, 4001, 3793, 3792} \[ \frac {8 a^2 (35 A+21 B+19 C) \tan (c+d x)}{105 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a (35 A+21 B+19 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{105 d}+\frac {2 (7 B-2 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 a d} \]
Antiderivative was successfully verified.
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Rule 3792
Rule 3793
Rule 4001
Rule 4082
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 a d}+\frac {2 \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{2} a (7 A+5 C)+\frac {1}{2} a (7 B-2 C) \sec (c+d x)\right ) \, dx}{7 a}\\ &=\frac {2 (7 B-2 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac {2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 a d}+\frac {1}{35} (35 A+21 B+19 C) \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac {2 a (35 A+21 B+19 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{105 d}+\frac {2 (7 B-2 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac {2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 a d}+\frac {1}{105} (4 a (35 A+21 B+19 C)) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {8 a^2 (35 A+21 B+19 C) \tan (c+d x)}{105 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (35 A+21 B+19 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{105 d}+\frac {2 (7 B-2 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac {2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 a d}\\ \end {align*}
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Mathematica [A] time = 1.65, size = 120, normalized size = 0.83 \[ \frac {a \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \sqrt {a (\sec (c+d x)+1)} ((525 A+462 B+468 C) \cos (c+d x)+2 (35 A+63 B+52 C) \cos (2 (c+d x))+175 A \cos (3 (c+d x))+70 A+126 B \cos (3 (c+d x))+126 B+104 C \cos (3 (c+d x))+164 C)}{210 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 113, normalized size = 0.78 \[ \frac {2 \, {\left ({\left (175 \, A + 126 \, B + 104 \, C\right )} a \cos \left (d x + c\right )^{3} + {\left (35 \, A + 63 \, B + 52 \, C\right )} a \cos \left (d x + c\right )^{2} + 3 \, {\left (7 \, B + 13 \, C\right )} a \cos \left (d x + c\right ) + 15 \, C a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.84, size = 286, normalized size = 1.99 \[ -\frac {4 \, {\left (105 \, \sqrt {2} A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 105 \, \sqrt {2} B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 105 \, \sqrt {2} C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (280 \, \sqrt {2} A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 210 \, \sqrt {2} B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 140 \, \sqrt {2} C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (245 \, \sqrt {2} A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 147 \, \sqrt {2} B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 133 \, \sqrt {2} C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 2 \, {\left (35 \, \sqrt {2} A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 21 \, \sqrt {2} B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 19 \, \sqrt {2} C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{105 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.68, size = 139, normalized size = 0.97 \[ -\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (175 A \left (\cos ^{3}\left (d x +c \right )\right )+126 B \left (\cos ^{3}\left (d x +c \right )\right )+104 C \left (\cos ^{3}\left (d x +c \right )\right )+35 A \left (\cos ^{2}\left (d x +c \right )\right )+63 B \left (\cos ^{2}\left (d x +c \right )\right )+52 C \left (\cos ^{2}\left (d x +c \right )\right )+21 B \cos \left (d x +c \right )+39 C \cos \left (d x +c \right )+15 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a}{105 d \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )} \]
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 = 8.89, size = 585, normalized size = 4.06 \[ \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\,a\,2{}\mathrm {i}}{d}-\frac {a\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (175\,A+126\,B+104\,C\right )\,2{}\mathrm {i}}{105\,d}\right )}{{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1}+\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 ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {a\,\left (3\,A+2\,B\right )\,2{}\mathrm {i}}{7\,d}-\frac {A\,a\,2{}\mathrm {i}}{7\,d}+\frac {a\,\left (5\,A+6\,B+4\,C\right )\,2{}\mathrm {i}}{7\,d}-\frac {a\,\left (7\,A+8\,B+12\,C\right )\,2{}\mathrm {i}}{7\,d}\right )+\frac {a\,\left (3\,A+2\,B\right )\,2{}\mathrm {i}}{7\,d}-\frac {A\,a\,2{}\mathrm {i}}{7\,d}+\frac {a\,\left (5\,A+6\,B+4\,C\right )\,2{}\mathrm {i}}{7\,d}-\frac {a\,\left (7\,A+8\,B+12\,C\right )\,2{}\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 {\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\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {a\,\left (3\,A+2\,B\right )\,2{}\mathrm {i}}{5\,d}+\frac {a\,\left (7\,A-8\,C\right )\,2{}\mathrm {i}}{35\,d}-\frac {a\,\left (2\,A+3\,B+6\,C\right )\,4{}\mathrm {i}}{5\,d}\right )-\frac {A\,a\,2{}\mathrm {i}}{5\,d}+\frac {a\,\left (2\,A+3\,B+2\,C\right )\,4{}\mathrm {i}}{5\,d}-\frac {a\,\left (3\,A+2\,B+8\,C\right )\,2{}\mathrm {i}}{5\,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 {\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\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {a\,\left (3\,A+2\,B\right )\,2{}\mathrm {i}}{3\,d}-\frac {a\,\left (35\,A+28\,B+52\,C\right )\,2{}\mathrm {i}}{105\,d}\right )-\frac {A\,a\,2{}\mathrm {i}}{3\,d}+\frac {a\,\left (3\,A+6\,B+4\,C\right )\,2{}\mathrm {i}}{3\,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 )} \]
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 \[ \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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