Optimal. Leaf size=241 \[ \frac {(5 A-177 B) \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {2 B \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{a^{7/2} d}+\frac {(5 A-49 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{64 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}+\frac {(5 A-17 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.77, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2977, 2982, 2782, 205, 2774, 216} \[ \frac {(5 A-49 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{64 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac {(5 A-177 B) \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {2 B \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{a^{7/2} d}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}+\frac {(5 A-17 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 216
Rule 2774
Rule 2782
Rule 2977
Rule 2982
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx &=\frac {(A-B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (\frac {5}{2} a (A-B)+6 a B \cos (c+d x)\right )}{(a+a \cos (c+d x))^{5/2}} \, dx}{6 a^2}\\ &=\frac {(A-B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(5 A-17 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}+\frac {\int \frac {\sqrt {\cos (c+d x)} \left (\frac {3}{4} a^2 (5 A-17 B)+24 a^2 B \cos (c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx}{24 a^4}\\ &=\frac {(A-B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(5 A-17 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}+\frac {(5 A-49 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {\int \frac {\frac {3}{8} a^3 (5 A-49 B)+48 a^3 B \cos (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{48 a^6}\\ &=\frac {(A-B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(5 A-17 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}+\frac {(5 A-49 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {(5 A-177 B) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{128 a^3}+\frac {B \int \frac {\sqrt {a+a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx}{a^4}\\ &=\frac {(A-B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(5 A-17 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}+\frac {(5 A-49 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac {(5 A-177 B) \operatorname {Subst}\left (\int \frac {1}{2 a^2+a x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{64 a^2 d}-\frac {(2 B) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}}} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^4 d}\\ &=\frac {2 B \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^{7/2} d}+\frac {(5 A-177 B) \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {(A-B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {(5 A-17 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}+\frac {(5 A-49 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 3.24, size = 266, normalized size = 1.10 \[ \frac {\cos ^7\left (\frac {1}{2} (c+d x)\right ) \left (\frac {1}{4} \sqrt {\cos (c+d x)} \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) (4 (25 A-181 B) \cos (c+d x)+(67 A-247 B) \cos (2 (c+d x))+97 A-541 B)-\frac {3 i \sqrt {2} e^{\frac {1}{2} i (c+d x)} \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \left (-\sqrt {2} (5 A-177 B) \tanh ^{-1}\left (\frac {1-e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )+128 B \sinh ^{-1}\left (e^{i (c+d x)}\right )-128 B \tanh ^{-1}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right )}{\sqrt {1+e^{2 i (c+d x)}}}\right )}{48 d (a (\cos (c+d x)+1))^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 24.59, size = 327, normalized size = 1.36 \[ -\frac {3 \, \sqrt {2} {\left ({\left (5 \, A - 177 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (5 \, A - 177 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (5 \, A - 177 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (5 \, A - 177 \, B\right )} \cos \left (d x + c\right ) + 5 \, A - 177 \, B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - 2 \, {\left ({\left (67 \, A - 247 \, B\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (25 \, A - 181 \, B\right )} \cos \left (d x + c\right ) + 15 \, A - 147 \, B\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 768 \, {\left (B \cos \left (d x + c\right )^{4} + 4 \, B \cos \left (d x + c\right )^{3} + 6 \, B \cos \left (d x + c\right )^{2} + 4 \, B \cos \left (d x + c\right ) + B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right )}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {5}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.37, size = 703, normalized size = 2.92 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {5}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^{5/2}\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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]
________________________________________________________________________________________