Optimal. Leaf size=333 \[ -\frac {(283 A-163 B+75 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {(157 A-85 B+45 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{80 a^2 d \sqrt {a \cos (c+d x)+a}}-\frac {(787 A-475 B+195 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{240 a^2 d \sqrt {a \cos (c+d x)+a}}+\frac {(2671 A-1495 B+735 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{240 a^2 d \sqrt {a \cos (c+d x)+a}}-\frac {(21 A-13 B+5 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac {(A-B+C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 1.24, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {4221, 3041, 2978, 2984, 12, 2782, 205} \[ \frac {(157 A-85 B+45 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{80 a^2 d \sqrt {a \cos (c+d x)+a}}-\frac {(787 A-475 B+195 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{240 a^2 d \sqrt {a \cos (c+d x)+a}}+\frac {(2671 A-1495 B+735 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{240 a^2 d \sqrt {a \cos (c+d x)+a}}-\frac {(283 A-163 B+75 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {(21 A-13 B+5 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac {(A-B+C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 2782
Rule 2978
Rule 2984
Rule 3041
Rule 4221
Rubi steps
\begin {align*} \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx\\ &=-\frac {(A-B+C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{2} a (13 A-5 B+5 C)-4 a (A-B) \cos (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac {(A-B+C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {(21 A-13 B+5 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{4} a^2 (157 A-85 B+45 C)-\frac {3}{2} a^2 (21 A-13 B+5 C) \cos (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{8 a^4}\\ &=-\frac {(A-B+C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {(21 A-13 B+5 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac {(157 A-85 B+45 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{80 a^2 d \sqrt {a+a \cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{8} a^3 (787 A-475 B+195 C)+\frac {1}{2} a^3 (157 A-85 B+45 C) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{20 a^5}\\ &=-\frac {(787 A-475 B+195 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{240 a^2 d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {(21 A-13 B+5 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac {(157 A-85 B+45 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{80 a^2 d \sqrt {a+a \cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{16} a^4 (2671 A-1495 B+735 C)-\frac {1}{8} a^4 (787 A-475 B+195 C) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{30 a^6}\\ &=\frac {(2671 A-1495 B+735 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{240 a^2 d \sqrt {a+a \cos (c+d x)}}-\frac {(787 A-475 B+195 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{240 a^2 d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {(21 A-13 B+5 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac {(157 A-85 B+45 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{80 a^2 d \sqrt {a+a \cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int -\frac {15 a^5 (283 A-163 B+75 C)}{32 \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{15 a^7}\\ &=\frac {(2671 A-1495 B+735 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{240 a^2 d \sqrt {a+a \cos (c+d x)}}-\frac {(787 A-475 B+195 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{240 a^2 d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {(21 A-13 B+5 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac {(157 A-85 B+45 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{80 a^2 d \sqrt {a+a \cos (c+d x)}}-\frac {\left ((283 A-163 B+75 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{32 a^2}\\ &=\frac {(2671 A-1495 B+735 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{240 a^2 d \sqrt {a+a \cos (c+d x)}}-\frac {(787 A-475 B+195 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{240 a^2 d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {(21 A-13 B+5 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac {(157 A-85 B+45 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{80 a^2 d \sqrt {a+a \cos (c+d x)}}+\frac {\left ((283 A-163 B+75 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{2 a^2+a x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{16 a d}\\ &=-\frac {(283 A-163 B+75 C) \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{16 \sqrt {2} a^{5/2} d}+\frac {(2671 A-1495 B+735 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{240 a^2 d \sqrt {a+a \cos (c+d x)}}-\frac {(787 A-475 B+195 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{240 a^2 d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B+C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac {(21 A-13 B+5 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac {(157 A-85 B+45 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{80 a^2 d \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 27.80, size = 7162, normalized size = 21.51 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.51, size = 287, normalized size = 0.86 \[ \frac {15 \, \sqrt {2} {\left ({\left (283 \, A - 163 \, B + 75 \, C\right )} \cos \left (d x + c\right )^{5} + 3 \, {\left (283 \, A - 163 \, B + 75 \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (283 \, A - 163 \, B + 75 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (283 \, A - 163 \, B + 75 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) + \frac {2 \, {\left ({\left (2671 \, A - 1495 \, B + 735 \, C\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (911 \, A - 503 \, B + 255 \, C\right )} \cos \left (d x + c\right )^{3} + 32 \, {\left (49 \, A - 25 \, B + 15 \, C\right )} \cos \left (d x + c\right )^{2} - 160 \, {\left (A - B\right )} \cos \left (d x + c\right ) + 96 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{480 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {7}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.51, size = 1045, normalized size = 3.14 \[ \text {result too large to display} \]
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 [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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