Optimal. Leaf size=271 \[ \frac {4 a^3 (7 A+9 B+5 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (13 A+21 B+35 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}-\frac {4 a^3 (41 A+42 B-35 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (6 A+7 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (7 A+9 B+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.42, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4171, 4102,
4082, 3872, 3856, 2719, 2720} \begin {gather*} -\frac {4 a^3 (41 A+42 B-35 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{105 d}+\frac {2 (7 A+9 B+5 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{15 d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (13 A+21 B+35 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {4 a^3 (7 A+9 B+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 (6 A+7 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{35 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{7 d \sec ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2719
Rule 2720
Rule 3856
Rule 3872
Rule 4082
Rule 4102
Rule 4171
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx &=\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \sec (c+d x))^3 \left (\frac {1}{2} a (6 A+7 B)-\frac {1}{2} a (A-7 C) \sec (c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{7 a}\\ &=\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (6 A+7 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \sec (c+d x))^2 \left (\frac {7}{4} a^2 (7 A+9 B+5 C)-\frac {1}{4} a^2 (11 A+7 B-35 C) \sec (c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{35 a}\\ &=\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (6 A+7 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (7 A+9 B+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {8 \int \frac {(a+a \sec (c+d x)) \left (\frac {1}{4} a^3 (106 A+147 B+140 C)-\frac {1}{4} a^3 (41 A+42 B-35 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{105 a}\\ &=-\frac {4 a^3 (41 A+42 B-35 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (6 A+7 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (7 A+9 B+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {16 \int \frac {\frac {21}{8} a^4 (7 A+9 B+5 C)+\frac {5}{8} a^4 (13 A+21 B+35 C) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{105 a}\\ &=-\frac {4 a^3 (41 A+42 B-35 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (6 A+7 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (7 A+9 B+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {1}{5} \left (2 a^3 (7 A+9 B+5 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (2 a^3 (13 A+21 B+35 C)\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=-\frac {4 a^3 (41 A+42 B-35 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (6 A+7 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (7 A+9 B+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {1}{5} \left (2 a^3 (7 A+9 B+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (2 a^3 (13 A+21 B+35 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {4 a^3 (7 A+9 B+5 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (13 A+21 B+35 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}-\frac {4 a^3 (41 A+42 B-35 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (6 A+7 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (7 A+9 B+5 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 3.23, size = 266, normalized size = 0.98 \begin {gather*} \frac {a^3 e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (2352 i A \cos (c+d x)+3024 i B \cos (c+d x)+1680 i C \cos (c+d x)+80 (13 A+21 B+35 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-112 i (7 A+9 B+5 C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )+126 A \sin (c+d x)+42 B \sin (c+d x)+840 C \sin (c+d x)+550 A \sin (2 (c+d x))+420 B \sin (2 (c+d x))+140 C \sin (2 (c+d x))+126 A \sin (3 (c+d x))+42 B \sin (3 (c+d x))+15 A \sin (4 (c+d x))\right )}{420 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(726\) vs.
\(2(295)=590\).
time = 0.10, size = 727, normalized size = 2.68
method | result | size |
default | \(-\frac {4 a^{3} \left (120 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \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 )}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \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 )}\, \left (36 A +7 B \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+14 \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 )}\, \left (43 A +21 B +5 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 )-2 \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 )}\, \left (104 A +63 B +70 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 )+65 A \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 )}\, \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 )-147 A \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 )}\, \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 B \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 )}\, \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 {-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 )}\, \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 )+175 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 ) \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 )}-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}\, \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 )}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{105 \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}\) | \(727\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.85, size = 232, normalized size = 0.86 \begin {gather*} -\frac {2 \, {\left (5 i \, \sqrt {2} {\left (13 \, A + 21 \, B + 35 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (13 \, A + 21 \, B + 35 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 i \, \sqrt {2} {\left (7 \, A + 9 \, B + 5 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, \sqrt {2} {\left (7 \, A + 9 \, B + 5 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (15 \, A a^{3} \cos \left (d x + c\right )^{3} + 21 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{2} + 5 \, {\left (26 \, A + 21 \, B + 7 \, C\right )} a^{3} \cos \left (d x + c\right ) + 105 \, C a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{105 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{3} \left (\int \frac {A}{\sec ^{\frac {7}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 A}{\sec ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 A}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {A}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {B}{\sec ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 B}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 B}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int B \sqrt {\sec {\left (c + d x \right )}}\, dx + \int \frac {C}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 C}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int 3 C \sqrt {\sec {\left (c + d x \right )}}\, dx + \int C \sec ^{\frac {3}{2}}{\left (c + d x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________