Optimal. Leaf size=200 \[ \frac {a^{5/2} (20 A+19 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{4 d}+\frac {a^3 (4 A-9 B) \sin (c+d x)}{4 d \sqrt {\cos (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (4 A+7 B) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{4 d \sqrt {\cos (c+d x)}}+\frac {a B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d \sqrt {\cos (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.63, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2955, 4018, 4015, 3801, 215} \[ \frac {a^3 (4 A-9 B) \sin (c+d x)}{4 d \sqrt {\cos (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (4 A+7 B) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{4 d \sqrt {\cos (c+d x)}}+\frac {a^{5/2} (20 A+19 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{4 d}+\frac {a B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 215
Rule 2955
Rule 3801
Rule 4015
Rule 4018
Rubi steps
\begin {align*} \int \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {a B (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d \sqrt {\cos (c+d x)}}+\frac {1}{2} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \sec (c+d x))^{3/2} \left (\frac {1}{2} a (4 A-B)+\frac {1}{2} a (4 A+7 B) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {a^2 (4 A+7 B) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d \sqrt {\cos (c+d x)}}+\frac {a B (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d \sqrt {\cos (c+d x)}}+\frac {1}{2} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)} \left (\frac {1}{4} a^2 (4 A-9 B)+\frac {1}{4} a^2 (20 A+19 B) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {a^3 (4 A-9 B) \sin (c+d x)}{4 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (4 A+7 B) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d \sqrt {\cos (c+d x)}}+\frac {a B (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d \sqrt {\cos (c+d x)}}+\frac {1}{8} \left (a^2 (20 A+19 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^3 (4 A-9 B) \sin (c+d x)}{4 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (4 A+7 B) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d \sqrt {\cos (c+d x)}}+\frac {a B (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d \sqrt {\cos (c+d x)}}-\frac {\left (a^2 (20 A+19 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\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 )}{4 d}\\ &=\frac {a^{5/2} (20 A+19 B) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{4 d}+\frac {a^3 (4 A-9 B) \sin (c+d x)}{4 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (4 A+7 B) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d \sqrt {\cos (c+d x)}}+\frac {a B (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{2 d \sqrt {\cos (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.95, size = 173, normalized size = 0.86 \[ \frac {a^3 \sin (c+d x) \sqrt {\cos (c+d x)} (A+B \sec (c+d x)) \left (\sqrt {1-\sec (c+d x)} \left ((4 A+11 B) \sec (c+d x)+8 A+2 B \sec ^2(c+d x)\right )+20 A \sqrt {\sec (c+d x)} \sin ^{-1}\left (\sqrt {1-\sec (c+d x)}\right )-19 B \sqrt {\sec (c+d x)} \sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right )}{4 d \sqrt {1-\sec (c+d x)} \sqrt {a (\sec (c+d x)+1)} (A \cos (c+d x)+B)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.55, size = 453, normalized size = 2.26 \[ \left [\frac {4 \, {\left (8 \, A a^{2} \cos \left (d x + c\right )^{2} + {\left (4 \, A + 11 \, B\right )} a^{2} \cos \left (d x + c\right ) + 2 \, B a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + {\left ({\left (20 \, A + 19 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + {\left (20 \, A + 19 \, B\right )} a^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 4 \, \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} {\left (\cos \left (d x + c\right ) - 2\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right )^{2} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right )}{16 \, {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )}}, \frac {2 \, {\left (8 \, A a^{2} \cos \left (d x + c\right )^{2} + {\left (4 \, A + 11 \, B\right )} a^{2} \cos \left (d x + c\right ) + 2 \, B a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + {\left ({\left (20 \, A + 19 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + {\left (20 \, A + 19 \, B\right )} a^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - 2 \, a}\right )}{8 \, {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\cos \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 2.25, size = 376, normalized size = 1.88 \[ -\frac {a^{2} \left (-1+\cos \left (d x +c \right )\right ) \left (16 A \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}-20 A \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1-\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}+20 A \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}-19 B \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1-\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}+19 B \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}+8 A \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}+22 B \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}+4 B \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{8 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{\frac {3}{2}} \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}} \]
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 \[ \int \sqrt {\cos \left (c+d\,x\right )}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
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]
________________________________________________________________________________________