Optimal. Leaf size=218 \[ \frac {5 a^{5/2} (8 A+5 C) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{8 d}+\frac {a^3 (24 A-49 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{24 d \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (24 A+31 C) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}{24 d}+\frac {5 a C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{12 d}+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.66, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {4089, 4018, 4015, 3801, 215} \[ \frac {a^3 (24 A-49 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{24 d \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (24 A+31 C) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}{24 d}+\frac {5 a^{5/2} (8 A+5 C) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{8 d}+\frac {5 a C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{12 d}+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 215
Rule 3801
Rule 4015
Rule 4018
Rule 4089
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx &=\frac {C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {\int \frac {(a+a \sec (c+d x))^{5/2} \left (\frac {1}{2} a (6 A-C)+\frac {5}{2} a C \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{3 a}\\ &=\frac {5 a C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {\int \frac {(a+a \sec (c+d x))^{3/2} \left (\frac {3}{4} a^2 (8 A-3 C)+\frac {1}{4} a^2 (24 A+31 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{6 a}\\ &=\frac {a^2 (24 A+31 C) \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {5 a C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {\int \frac {\sqrt {a+a \sec (c+d x)} \left (\frac {1}{8} a^3 (24 A-49 C)+\frac {15}{8} a^3 (8 A+5 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{6 a}\\ &=\frac {a^3 (24 A-49 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+31 C) \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {5 a C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {1}{16} \left (5 a^2 (8 A+5 C)\right ) \int \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^3 (24 A-49 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+31 C) \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {5 a C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}-\frac {\left (5 a^2 (8 A+5 C)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a}}} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 d}\\ &=\frac {5 a^{5/2} (8 A+5 C) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 d}+\frac {a^3 (24 A-49 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+31 C) \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {5 a C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 6.83, size = 411, normalized size = 1.89 \[ \frac {5 (8 A+5 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {\sec ^2(c+d x)-1} (a (\sec (c+d x)+1))^{5/2} \left (\log \left (\sec ^{\frac {3}{2}}(c+d x)+\sqrt {\sec (c+d x)+1} \sqrt {\sec ^2(c+d x)-1}+\sqrt {\sec (c+d x)}\right )-\log (\sec (c+d x)+1)\right ) \left (A+C \sec ^2(c+d x)\right )}{4 d \left (1-\cos ^2(c+d x)\right ) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)}+\frac {(a (\sec (c+d x)+1))^{5/2} \sqrt {(\cos (c+d x)+1) \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \left (\frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (49 C \sin \left (\frac {d x}{2}\right )-24 A \sin \left (\frac {d x}{2}\right )\right )}{12 d}-\frac {\tan \left (\frac {c}{2}\right ) \sec (c) (24 A \cos (c)-75 C \cos (c)-26 C)}{12 d}+\frac {4 A \sin (c) \cos (d x)}{d}+\frac {4 A \cos (c) \sin (d x)}{d}+\frac {2 C \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac {\sec (c) \sec (c+d x) (4 C \sin (c)+13 C \sin (d x))}{6 d}\right )}{\sec ^{\frac {3}{2}}(c+d x) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.56, size = 494, normalized size = 2.27 \[ \left [\frac {15 \, {\left ({\left (8 \, A + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + {\left (8 \, A + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - \frac {4 \, {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + \frac {4 \, {\left (48 \, A a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, A + 25 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 34 \, C a^{2} \cos \left (d x + c\right ) + 8 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{96 \, {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )}}, \frac {15 \, {\left ({\left (8 \, A + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + {\left (8 \, A + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - 2 \, a}\right ) + \frac {2 \, {\left (48 \, A a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, A + 25 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 34 \, C a^{2} \cos \left (d x + c\right ) + 8 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{48 \, {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {\sec \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 2.74, size = 399, normalized size = 1.83 \[ -\frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (120 A \sqrt {2}\, \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}-120 A \sqrt {2}\, \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1-\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right )+75 C \sqrt {2}\, \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}-75 C \sqrt {2}\, \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1-\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right )+192 A \left (\cos ^{4}\left (d x +c \right )\right )-96 A \left (\cos ^{3}\left (d x +c \right )\right )+300 C \left (\cos ^{3}\left (d x +c \right )\right )-96 A \left (\cos ^{2}\left (d x +c \right )\right )-164 C \left (\cos ^{2}\left (d x +c \right )\right )-104 C \cos \left (d x +c \right )-32 C \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )}}\, a^{2}}{96 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}} \]
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 \[ \int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]
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]
________________________________________________________________________________________