Optimal. Leaf size=224 \[ \frac {a^{5/2} (8 A+19 C) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{4 d}+\frac {a^3 (56 A-27 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{12 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2 (8 A-21 C) \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{12 d}-\frac {a (4 A-3 C) \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.45, antiderivative size = 224, 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 = {4172, 4103,
4100, 3886, 221} \begin {gather*} \frac {a^{5/2} (8 A+19 C) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{4 d}+\frac {a^3 (56 A-27 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{12 d \sqrt {a \sec (c+d x)+a}}-\frac {a^2 (8 A-21 C) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}{12 d}-\frac {a (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{6 d}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{3 d \sqrt {\sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 221
Rule 3886
Rule 4100
Rule 4103
Rule 4172
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx &=\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \int \frac {(a+a \sec (c+d x))^{5/2} \left (\frac {5 a A}{2}-\frac {1}{2} a (4 A-3 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{3 a}\\ &=-\frac {a (4 A-3 C) \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {\int \frac {(a+a \sec (c+d x))^{3/2} \left (\frac {3}{4} a^2 (8 A-C)-\frac {1}{4} a^2 (8 A-21 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{3 a}\\ &=-\frac {a^2 (8 A-21 C) \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{12 d}-\frac {a (4 A-3 C) \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {\int \frac {\sqrt {a+a \sec (c+d x)} \left (\frac {1}{8} a^3 (56 A-27 C)+\frac {3}{8} a^3 (8 A+19 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{3 a}\\ &=\frac {a^3 (56 A-27 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{12 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2 (8 A-21 C) \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{12 d}-\frac {a (4 A-3 C) \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {1}{8} \left (a^2 (8 A+19 C)\right ) \int \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^3 (56 A-27 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{12 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2 (8 A-21 C) \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{12 d}-\frac {a (4 A-3 C) \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}-\frac {\left (a^2 (8 A+19 C)\right ) \text {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 )}{4 d}\\ &=\frac {a^{5/2} (8 A+19 C) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{4 d}+\frac {a^3 (56 A-27 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{12 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2 (8 A-21 C) \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{12 d}-\frac {a (4 A-3 C) \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 6.71, size = 416, normalized size = 1.86 \begin {gather*} \frac {(8 A+19 C) \cos ^3(c+d x) \left (-\log (1+\sec (c+d x))+\log \left (\sqrt {\sec (c+d x)}+\sec ^{\frac {3}{2}}(c+d x)+\sqrt {1+\sec (c+d x)} \sqrt {-1+\sec ^2(c+d x)}\right )\right ) (a (1+\sec (c+d x)))^{5/2} \sqrt {-1+\sec ^2(c+d x)} \left (A+C \sec ^2(c+d x)\right ) \sin (c+d x)}{2 d \left (1-\cos ^2(c+d x)\right ) (A+2 C+A \cos (2 c+2 d x)) (1+\sec (c+d x))^{5/2}}+\frac {\sqrt {(1+\cos (c+d x)) \sec (c+d x)} (a (1+\sec (c+d x)))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \left (\frac {28 A \cos (d x) \sin (c)}{3 d}+\frac {2 A \cos (2 d x) \sin (2 c)}{3 d}+\frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-56 A \sin \left (\frac {d x}{2}\right )+27 C \sin \left (\frac {d x}{2}\right )\right )}{6 d}+\frac {28 A \cos (c) \sin (d x)}{3 d}+\frac {C \sec (c) \sec (c+d x) \sin (d x)}{d}+\frac {2 A \cos (2 c) \sin (2 d x)}{3 d}-\frac {(-6 C+56 A \cos (c)-33 C \cos (c)) \sec (c) \tan \left (\frac {c}{2}\right )}{6 d}\right )}{(A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {3}{2}}(c+d x) (1+\sec (c+d x))^{5/2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 21.99, size = 380, normalized size = 1.70
method | result | size |
default | \(-\frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (24 A \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (-1-\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sqrt {2}+24 A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \sqrt {2}+57 C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (-1-\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sqrt {2}+57 C \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \sqrt {2}+32 A \left (\cos ^{4}\left (d x +c \right )\right )+224 A \left (\cos ^{3}\left (d x +c \right )\right )-256 A \left (\cos ^{2}\left (d x +c \right )\right )+132 C \left (\cos ^{2}\left (d x +c \right )\right )-108 C \cos \left (d x +c \right )-24 C \right ) \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} a^{2}}{48 d \sin \left (d x +c \right )}\) | \(380\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 3421 vs.
\(2 (192) = 384\).
time = 3.45, size = 3421, normalized size = 15.27 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.47, size = 474, normalized size = 2.12 \begin {gather*} \left [\frac {3 \, {\left ({\left (8 \, A + 19 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + {\left (8 \, A + 19 \, C\right )} a^{2} \cos \left (d x + c\right )\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 (8 \, A a^{2} \cos \left (d x + c\right )^{3} + 64 \, A a^{2} \cos \left (d x + c\right )^{2} + 33 \, C a^{2} \cos \left (d x + c\right ) + 6 \, 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 )^{2} + d \cos \left (d x + c\right )\right )}}, \frac {3 \, {\left ({\left (8 \, A + 19 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + {\left (8 \, A + 19 \, C\right )} a^{2} \cos \left (d x + c\right )\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 (8 \, A a^{2} \cos \left (d x + c\right )^{3} + 64 \, A a^{2} \cos \left (d x + c\right )^{2} + 33 \, C a^{2} \cos \left (d x + c\right ) + 6 \, 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 )}}}{24 \, {\left (d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )}}\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 = -0.00 \begin {gather*} \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}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________