Optimal. Leaf size=226 \[ \frac {8 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 a (16 A+18 B+21 C) \sin (c+d x)}{105 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {16 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {2 a (A+9 B) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]
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Rubi [A] time = 0.52, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3043, 2980, 2772, 2771} \[ \frac {8 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 a (16 A+18 B+21 C) \sin (c+d x)}{105 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {16 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {2 a (A+9 B) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2771
Rule 2772
Rule 2980
Rule 3043
Rubi steps
\begin {align*} \int \frac {\sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx &=\frac {2 A \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \int \frac {\sqrt {a+a \cos (c+d x)} \left (\frac {1}{2} a (A+9 B)+\frac {3}{2} a (2 A+3 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{9 a}\\ &=\frac {2 a (A+9 B) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 A \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {1}{21} (16 A+18 B+21 C) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 a (A+9 B) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a (16 A+18 B+21 C) \sin (c+d x)}{105 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 A \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {1}{105} (4 (16 A+18 B+21 C)) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a (A+9 B) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a (16 A+18 B+21 C) \sin (c+d x)}{105 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {8 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 A \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {1}{315} (8 (16 A+18 B+21 C)) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a (A+9 B) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a (16 A+18 B+21 C) \sin (c+d x)}{105 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {8 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {16 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {2 A \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 0.88, size = 155, normalized size = 0.69 \[ \frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} (2 (88 A+99 B+63 C) \cos (c+d x)+11 (16 A+18 B+21 C) \cos (2 (c+d x))+32 A \cos (3 (c+d x))+32 A \cos (4 (c+d x))+214 A+36 B \cos (3 (c+d x))+36 B \cos (4 (c+d x))+162 B+42 C \cos (3 (c+d x))+42 C \cos (4 (c+d x))+189 C)}{315 d \cos ^{\frac {9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 130, normalized size = 0.58 \[ \frac {2 \, {\left (8 \, {\left (16 \, A + 18 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (16 \, A + 18 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (16 \, A + 18 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (8 \, A + 9 \, B\right )} \cos \left (d x + c\right ) + 35 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.40, size = 163, normalized size = 0.72 \[ -\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (128 A \left (\cos ^{4}\left (d x +c \right )\right )+144 B \left (\cos ^{4}\left (d x +c \right )\right )+168 C \left (\cos ^{4}\left (d x +c \right )\right )+64 A \left (\cos ^{3}\left (d x +c \right )\right )+72 B \left (\cos ^{3}\left (d x +c \right )\right )+84 C \left (\cos ^{3}\left (d x +c \right )\right )+48 A \left (\cos ^{2}\left (d x +c \right )\right )+54 B \left (\cos ^{2}\left (d x +c \right )\right )+63 C \left (\cos ^{2}\left (d x +c \right )\right )+40 A \cos \left (d x +c \right )+45 B \cos \left (d x +c \right )+35 A \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{315 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.89, size = 848, normalized size = 3.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.70, size = 629, normalized size = 2.78 \[ \frac {\sqrt {a+a\,\left (\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}\right )}\,\left (\frac {\left (256\,A+288\,B+336\,C\right )\,1{}\mathrm {i}}{315\,d}-\frac {C\,{\mathrm {e}}^{c\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,8{}\mathrm {i}}{3\,d}+\frac {C\,{\mathrm {e}}^{c\,6{}\mathrm {i}+d\,x\,6{}\mathrm {i}}\,8{}\mathrm {i}}{3\,d}-\frac {{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}\,\left (256\,A+288\,B+336\,C\right )\,1{}\mathrm {i}}{315\,d}+\frac {{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,\left (1152\,A+1296\,B+1512\,C\right )\,1{}\mathrm {i}}{315\,d}-\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\left (1152\,A+1296\,B+1512\,C\right )\,1{}\mathrm {i}}{315\,d}+\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\left (2016\,A+1008\,B+2016\,C\right )\,1{}\mathrm {i}}{315\,d}-\frac {{\mathrm {e}}^{c\,5{}\mathrm {i}+d\,x\,5{}\mathrm {i}}\,\left (2016\,A+1008\,B+2016\,C\right )\,1{}\mathrm {i}}{315\,d}\right )}{\sqrt {\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}}+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\sqrt {\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}}+4\,{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,\sqrt {\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}}+4\,{\mathrm {e}}^{c\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,\sqrt {\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}}+6\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sqrt {\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}}+6\,{\mathrm {e}}^{c\,5{}\mathrm {i}+d\,x\,5{}\mathrm {i}}\,\sqrt {\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}}+4\,{\mathrm {e}}^{c\,6{}\mathrm {i}+d\,x\,6{}\mathrm {i}}\,\sqrt {\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}}+4\,{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\sqrt {\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}}+{\mathrm {e}}^{c\,8{}\mathrm {i}+d\,x\,8{}\mathrm {i}}\,\sqrt {\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}}+{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}\,\sqrt {\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}}} \]
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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