Optimal. Leaf size=277 \[ \frac {1015 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {193 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{64 a^3 d \sqrt {a \cos (c+d x)+a}}-\frac {629 \sin (c+d x) \sqrt {\sec (c+d x)}}{64 a^3 d \sqrt {a \cos (c+d x)+a}}-\frac {109 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{64 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac {23 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}}-\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.78, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {4222, 2766, 2978, 2984, 12, 2782, 205} \[ \frac {193 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{64 a^3 d \sqrt {a \cos (c+d x)+a}}-\frac {109 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{64 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac {629 \sin (c+d x) \sqrt {\sec (c+d x)}}{64 a^3 d \sqrt {a \cos (c+d x)+a}}+\frac {1015 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}-\frac {23 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}}-\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 205
Rule 2766
Rule 2782
Rule 2978
Rule 2984
Rule 4222
Rubi steps
\begin {align*} \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx\\ &=-\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {15 a}{2}-4 a \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx}{6 a^2}\\ &=-\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {23 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{48 a 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 {189 a^2}{4}-\frac {69}{2} a^2 \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx}{24 a^4}\\ &=-\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {23 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac {109 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{64 a^2 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 {1737 a^3}{8}-\frac {327}{2} a^3 \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{48 a^6}\\ &=-\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {23 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac {109 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {193 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {5661 a^4}{16}+\frac {1737}{8} a^4 \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{72 a^7}\\ &=-\frac {629 \sqrt {\sec (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}-\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {23 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac {109 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {193 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {9135 a^5}{32 \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{36 a^8}\\ &=-\frac {629 \sqrt {\sec (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}-\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {23 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac {109 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {193 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}+\frac {\left (1015 \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}{128 a^3}\\ &=-\frac {629 \sqrt {\sec (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}-\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {23 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac {109 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {193 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}-\frac {\left (1015 \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 )}{64 a^2 d}\\ &=\frac {1015 \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)}}{64 \sqrt {2} a^{7/2} d}-\frac {629 \sqrt {\sec (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}-\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {23 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac {109 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {193 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 8.40, size = 696, normalized size = 2.51 \[ \frac {\left (\frac {1}{1-2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}\right )^{7/2} \cot ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (-7680 \sin ^{14}\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos ^{10}\left (\frac {1}{2} (c+d x)\right ) \, _6F_5\left (2,2,2,2,2,\frac {7}{2};1,1,1,1,\frac {15}{2};\frac {\sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )-1}\right )+19200 \left (6 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )-7\right ) \sin ^{14}\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos ^8\left (\frac {1}{2} (c+d x)\right ) \, _5F_4\left (2,2,2,2,\frac {7}{2};1,1,1,\frac {15}{2};\frac {\sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )-1}\right )+143 \left (1-2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3 \sqrt {\frac {\sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )-1}} \left (\sqrt {\frac {\sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )-1}} \left (704274992 \sin ^{18}\left (\frac {c}{2}+\frac {d x}{2}\right )-5410719584 \sin ^{16}\left (\frac {c}{2}+\frac {d x}{2}\right )+18305254212 \sin ^{14}\left (\frac {c}{2}+\frac {d x}{2}\right )-35736693140 \sin ^{12}\left (\frac {c}{2}+\frac {d x}{2}\right )+44313222590 \sin ^{10}\left (\frac {c}{2}+\frac {d x}{2}\right )-36160322412 \sin ^8\left (\frac {c}{2}+\frac {d x}{2}\right )+19406027859 \sin ^6\left (\frac {c}{2}+\frac {d x}{2}\right )-6601900452 \sin ^4\left (\frac {c}{2}+\frac {d x}{2}\right )+1291549455 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )-110685960\right )+315 \left (1793816 \sin ^{12}\left (\frac {c}{2}+\frac {d x}{2}\right )-8670660 \sin ^{10}\left (\frac {c}{2}+\frac {d x}{2}\right )+17139064 \sin ^8\left (\frac {c}{2}+\frac {d x}{2}\right )-17629526 \sin ^6\left (\frac {c}{2}+\frac {d x}{2}\right )+9953934 \sin ^4\left (\frac {c}{2}+\frac {d x}{2}\right )-2928877 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )+351384\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sqrt {\frac {\sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )-1}}\right )\right )\right )}{3243240 d (a (\cos (c+d x)+1))^{7/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.25, size = 229, normalized size = 0.83 \[ -\frac {3045 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) + \frac {2 \, {\left (1887 \, \cos \left (d x + c\right )^{4} + 5082 \, \cos \left (d x + c\right )^{3} + 4251 \, \cos \left (d x + c\right )^{2} + 896 \, \cos \left (d x + c\right ) - 128\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x + c\right )^{\frac {5}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.25, size = 390, normalized size = 1.41 \[ \frac {\left (-1+\cos \left (d x +c \right )\right ) \left (3045 \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}}+12180 \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+18270 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}}+12180 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}}-1887 \sqrt {2}\, \left (\cos ^{5}\left (d x +c \right )\right )+3045 \sin \left (d x +c \right ) \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}}-3195 \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {2}+831 \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {2}+3355 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}+1024 \cos \left (d x +c \right ) \sqrt {2}-128 \sqrt {2}\right ) \cos \left (d x +c \right ) \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \sqrt {2}}{384 d \sin \left (d x +c \right )^{3} \left (1+\cos \left (d x +c \right )\right )^{2} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________