Optimal. Leaf size=215 \[ \frac {4 a^2 (5 A+6 B+7 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {4 a^2 (8 A+9 B+12 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a^2 (19 A+27 B+21 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{105 d}+\frac {4 a^2 (5 A+6 B+7 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 (4 A+9 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{63 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.55, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {4112, 3045, 2976, 2968, 3023, 2748, 2639, 2635, 2641} \[ \frac {4 a^2 (5 A+6 B+7 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {4 a^2 (8 A+9 B+12 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a^2 (19 A+27 B+21 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{105 d}+\frac {4 a^2 (5 A+6 B+7 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 (4 A+9 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{63 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 2968
Rule 2976
Rule 3023
Rule 3045
Rule 4112
Rubi steps
\begin {align*} \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^2 \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^2 \left (\frac {3}{2} a (A+3 C)+\frac {1}{2} a (4 A+9 B) \cos (c+d x)\right ) \, dx}{9 a}\\ &=\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (4 A+9 B) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {4 \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x)) \left (\frac {3}{4} a^2 (11 A+9 B+21 C)+\frac {3}{4} a^2 (19 A+27 B+21 C) \cos (c+d x)\right ) \, dx}{63 a}\\ &=\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (4 A+9 B) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {4 \int \sqrt {\cos (c+d x)} \left (\frac {3}{4} a^3 (11 A+9 B+21 C)+\left (\frac {3}{4} a^3 (11 A+9 B+21 C)+\frac {3}{4} a^3 (19 A+27 B+21 C)\right ) \cos (c+d x)+\frac {3}{4} a^3 (19 A+27 B+21 C) \cos ^2(c+d x)\right ) \, dx}{63 a}\\ &=\frac {2 a^2 (19 A+27 B+21 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (4 A+9 B) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {8 \int \sqrt {\cos (c+d x)} \left (\frac {21}{4} a^3 (8 A+9 B+12 C)+\frac {45}{4} a^3 (5 A+6 B+7 C) \cos (c+d x)\right ) \, dx}{315 a}\\ &=\frac {2 a^2 (19 A+27 B+21 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (4 A+9 B) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {1}{7} \left (2 a^2 (5 A+6 B+7 C)\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{15} \left (2 a^2 (8 A+9 B+12 C)\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {4 a^2 (8 A+9 B+12 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^2 (5 A+6 B+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a^2 (19 A+27 B+21 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (4 A+9 B) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {1}{21} \left (2 a^2 (5 A+6 B+7 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {4 a^2 (8 A+9 B+12 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^2 (5 A+6 B+7 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {4 a^2 (5 A+6 B+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a^2 (19 A+27 B+21 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (4 A+9 B) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{63 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 6.39, size = 1699, normalized size = 7.90 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C a^{2} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{4} + {\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{3} + {\left (A + 2 \, B + C\right )} a^{2} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} + {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right ) + A a^{2} \cos \left (d x + c\right )^{4}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 5.49, size = 514, normalized size = 2.39 \[ -\frac {4 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{2} \left (-560 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (1840 A +360 B \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-2368 A -1044 B -252 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (1568 A +1134 B +672 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-387 A -351 B -273 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+75 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-168 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+90 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-189 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+105 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-252 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{315 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 [B] time = 5.30, size = 369, normalized size = 1.72 \[ \frac {2\,C\,a^2\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {2\,B\,a^2\,\left (\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )+\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\right )}{3\,d}+\frac {2\,C\,a^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}-\frac {2\,A\,a^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,A\,a^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,a^2\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,B\,a^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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]
________________________________________________________________________________________