Optimal. Leaf size=155 \[ \frac {E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {4 \sin (c+d x)}{15 a d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2}+\frac {\sin (c+d x)}{6 d \sqrt {\cos (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.25, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4349, 3901,
4104, 4105, 3872, 3856, 2719, 2720} \begin {gather*} \frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac {E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {\sin (c+d x)}{6 d \sqrt {\cos (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}-\frac {4 \sin (c+d x)}{15 a d \sqrt {\cos (c+d x)} (a \sec (c+d x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2719
Rule 2720
Rule 3856
Rule 3872
Rule 3901
Rule 4104
Rule 4105
Rule 4349
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)} \left (\frac {a}{2}-\frac {7}{2} a \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {4 \sin (c+d x)}{15 a d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-2 a^2-\frac {9}{2} a^2 \sec (c+d x)}{\sqrt {\sec (c+d x)} (a+a \sec (c+d x))} \, dx}{15 a^4}\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {4 \sin (c+d x)}{15 a d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2}+\frac {\sin (c+d x)}{6 d \sqrt {\cos (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {3 a^3}{4}-\frac {5}{4} a^3 \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{15 a^6}\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {4 \sin (c+d x)}{15 a d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2}+\frac {\sin (c+d x)}{6 d \sqrt {\cos (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx}{12 a^3}\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {4 \sin (c+d x)}{15 a d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2}+\frac {\sin (c+d x)}{6 d \sqrt {\cos (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\int \sqrt {\cos (c+d x)} \, dx}{20 a^3}+\frac {\int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{12 a^3}\\ &=\frac {E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {4 \sin (c+d x)}{15 a d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2}+\frac {\sin (c+d x)}{6 d \sqrt {\cos (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 1.58, size = 342, normalized size = 2.21 \begin {gather*} \frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (-\frac {\left (4 \cos \left (\frac {1}{2} (c-d x)\right )+26 \cos \left (\frac {1}{2} (3 c+d x)\right )+10 \cos \left (\frac {1}{2} (c+3 d x)\right )+5 \cos \left (\frac {1}{2} (5 c+3 d x)\right )+3 \cos \left (\frac {1}{2} (3 c+5 d x)\right )\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right )}{8 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 i \sqrt {2} e^{-i (c+d x)} \left (3 \left (1+e^{2 i (c+d x)}\right )+3 \left (-1+e^{2 i c}\right ) \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i (c+d x)}\right )-5 e^{i (c+d x)} \left (-1+e^{2 i c}\right ) \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-e^{2 i (c+d x)}\right )\right ) \sec ^3(c+d x)}{d \left (-1+e^{2 i c}\right ) \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{15 a^3 (1+\sec (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 270, normalized size = 1.74
method | result | size |
default | \(\frac {\sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (12 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-22 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3\right )}{60 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(270\) |
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 = 0.71, size = 344, normalized size = 2.22 \begin {gather*} -\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) - 5\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 5 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 3 i \, \sqrt {2} \cos \left (d x + c\right ) + i \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 3 i \, \sqrt {2} \cos \left (d x + c\right ) - i \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 3 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 3 i \, \sqrt {2} \cos \left (d x + c\right ) - i \, \sqrt {2}\right )} {\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 ) + 3 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 3 i \, \sqrt {2} \cos \left (d x + c\right ) + i \, \sqrt {2}\right )} {\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 )}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [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]
________________________________________________________________________________________
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 = -0.01 \begin {gather*} \int \frac {1}{{\cos \left (c+d\,x\right )}^{5/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________