Optimal. Leaf size=181 \[ -\frac {13 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac {49 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {49 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {13 \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.41, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4264, 3816, 4019, 3787, 3771, 2641, 3768, 2639} \[ -\frac {13 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac {49 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {49 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {13 \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 2641
Rule 3768
Rule 3771
Rule 3787
Rule 3816
Rule 4019
Rule 4264
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{\frac {9}{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 {9}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{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 {\sec ^{\frac {5}{2}}(c+d x) \left (\frac {5 a}{2}-\frac {11}{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 {7}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(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 {\sec ^{\frac {3}{2}}(c+d x) \left (12 a^2-\frac {41}{2} a^2 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {13 \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(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 \sqrt {\sec (c+d x)} \left (\frac {65 a^3}{4}-\frac {147}{4} a^3 \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {13 \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left (13 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx}{12 a^3}+\frac {\left (49 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx}{20 a^3}\\ &=\frac {49 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {13 \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac {13 \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{12 a^3}-\frac {\left (49 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3}\\ &=-\frac {13 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac {49 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {13 \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac {49 \int \sqrt {\cos (c+d x)} \, dx}{20 a^3}\\ &=-\frac {49 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {13 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac {49 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {13 \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.91, size = 372, normalized size = 2.06 \[ \frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \left (1284 \cos \left (\frac {1}{2} (c-d x)\right )+921 \cos \left (\frac {1}{2} (3 c+d x)\right )+1243 \cos \left (\frac {1}{2} (c+3 d x)\right )+374 \cos \left (\frac {1}{2} (5 c+3 d x)\right )+670 \cos \left (\frac {1}{2} (3 c+5 d x)\right )+65 \cos \left (\frac {1}{2} (7 c+5 d x)\right )+147 \cos \left (\frac {1}{2} (5 c+7 d x)\right )\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right )}{16 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {4 i \sqrt {2} e^{-i (c+d x)} \left (147 \left (-1+e^{2 i c}\right ) \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i (c+d x)}\right )-65 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-e^{2 i (c+d x)}\right )+147 \left (1+e^{2 i (c+d x)}\right )\right ) \sec ^3(c+d x)}{\left (-1+e^{2 i c}\right ) d \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{15 a^3 (\sec (c+d x)+1)^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\cos \left (d x + c\right )}}{a^{3} \cos \left (d x + c\right )^{5} \sec \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{5} \sec \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right )^{5} \sec \left (d x + c\right ) + a^{3} \cos \left (d x + c\right )^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 4.09, size = 555, normalized size = 3.07 \[ -\frac {-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 )}\, \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}\, \left (65 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-147 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \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}\, \left (65 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-147 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right ) \left (\sin ^{2}\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 )}\, \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}\, \left (65 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-147 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+588 \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 )-1634 \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 ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1488 \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 ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-439 \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 ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\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} \]
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 = -0.01 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^{9/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]
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]
________________________________________________________________________________________