Optimal. Leaf size=228 \[ \frac {2 a^3 (292 A+345 B) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 a^3 (124 A+135 B) \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {4 a^3 (292 A+345 B) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (4 A+3 B) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]
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Rubi [A] time = 0.70, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2975, 2980, 2772, 2771} \[ \frac {2 a^3 (292 A+345 B) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 a^3 (124 A+135 B) \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (4 A+3 B) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 a^3 (292 A+345 B) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {2 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2771
Rule 2772
Rule 2975
Rule 2980
Rubi steps
\begin {align*} \int \frac {(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)} \, dx &=\frac {2 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2}{9} \int \frac {(a+a \cos (c+d x))^{3/2} \left (\frac {3}{2} a (4 A+3 B)+\frac {1}{2} a (4 A+9 B) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (4 A+3 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {4}{63} \int \frac {\sqrt {a+a \cos (c+d x)} \left (\frac {1}{4} a^2 (124 A+135 B)+\frac {1}{4} a^2 (76 A+99 B) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 a^3 (124 A+135 B) \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (4 A+3 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {1}{105} \left (a^2 (292 A+345 B)\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a^3 (124 A+135 B) \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (292 A+345 B) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (4 A+3 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {1}{315} \left (2 a^2 (292 A+345 B)\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a^3 (124 A+135 B) \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (292 A+345 B) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {4 a^3 (292 A+345 B) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (4 A+3 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 0.86, size = 126, normalized size = 0.55 \[ \frac {a^2 \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} ((1396 A+1215 B) \cos (c+d x)+2 (803 A+870 B) \cos (2 (c+d x))+292 A \cos (3 (c+d x))+292 A \cos (4 (c+d x))+1454 A+345 B \cos (3 (c+d x))+345 B \cos (4 (c+d x))+1395 B)}{630 d \cos ^{\frac {9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 135, normalized size = 0.59 \[ \frac {2 \, {\left (2 \, {\left (292 \, A + 345 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + {\left (292 \, A + 345 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (73 \, A + 60 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 5 \, {\left (26 \, A + 9 \, B\right )} a^{2} \cos \left (d x + c\right ) + 35 \, A a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}} \]
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 = 0.28, size = 133, normalized size = 0.58 \[ -\frac {2 a^{2} \left (-1+\cos \left (d x +c \right )\right ) \left (584 A \left (\cos ^{4}\left (d x +c \right )\right )+690 B \left (\cos ^{4}\left (d x +c \right )\right )+292 A \left (\cos ^{3}\left (d x +c \right )\right )+345 B \left (\cos ^{3}\left (d x +c \right )\right )+219 A \left (\cos ^{2}\left (d x +c \right )\right )+180 B \left (\cos ^{2}\left (d x +c \right )\right )+130 A \cos \left (d x +c \right )+45 B \cos \left (d x +c \right )+35 A \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{315 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.96, size = 533, normalized size = 2.34 \[ \frac {8 \, {\left (\frac {15 \, {\left (\frac {21 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {56 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {36 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {8 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}\right )} B {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{{\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}} {\left (\frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 1\right )}} + \frac {{\left (\frac {315 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {945 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1449 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {1287 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {572 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {104 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}\right )} A {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3}}{{\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {11}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {11}{2}} {\left (\frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1\right )}}\right )}}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.24, size = 647, normalized size = 2.84 \[ \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 {a^2\,\left (292\,A+345\,B\right )\,4{}\mathrm {i}}{315\,d}-\frac {a^2\,{\mathrm {e}}^{c\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,\left (2\,A+5\,B\right )\,4{}\mathrm {i}}{3\,d}+\frac {a^2\,{\mathrm {e}}^{c\,6{}\mathrm {i}+d\,x\,6{}\mathrm {i}}\,\left (2\,A+5\,B\right )\,4{}\mathrm {i}}{3\,d}+\frac {a^2\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\left (24\,A+25\,B\right )\,4{}\mathrm {i}}{5\,d}-\frac {a^2\,{\mathrm {e}}^{c\,5{}\mathrm {i}+d\,x\,5{}\mathrm {i}}\,\left (24\,A+25\,B\right )\,4{}\mathrm {i}}{5\,d}+\frac {a^2\,{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,\left (146\,A+155\,B\right )\,4{}\mathrm {i}}{35\,d}-\frac {a^2\,{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\left (146\,A+155\,B\right )\,4{}\mathrm {i}}{35\,d}-\frac {a^2\,{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}\,\left (292\,A+345\,B\right )\,4{}\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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