Optimal. Leaf size=95 \[ -\frac {64 a^3 \cos ^7(c+d x)}{693 d (a+a \sin (c+d x))^{7/2}}-\frac {16 a^2 \cos ^7(c+d x)}{99 d (a+a \sin (c+d x))^{5/2}}-\frac {2 a \cos ^7(c+d x)}{11 d (a+a \sin (c+d x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2753, 2752}
\begin {gather*} -\frac {64 a^3 \cos ^7(c+d x)}{693 d (a \sin (c+d x)+a)^{7/2}}-\frac {16 a^2 \cos ^7(c+d x)}{99 d (a \sin (c+d x)+a)^{5/2}}-\frac {2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2752
Rule 2753
Rubi steps
\begin {align*} \int \frac {\cos ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx &=-\frac {2 a \cos ^7(c+d x)}{11 d (a+a \sin (c+d x))^{3/2}}+\frac {1}{11} (8 a) \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {16 a^2 \cos ^7(c+d x)}{99 d (a+a \sin (c+d x))^{5/2}}-\frac {2 a \cos ^7(c+d x)}{11 d (a+a \sin (c+d x))^{3/2}}+\frac {1}{99} \left (32 a^2\right ) \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx\\ &=-\frac {64 a^3 \cos ^7(c+d x)}{693 d (a+a \sin (c+d x))^{7/2}}-\frac {16 a^2 \cos ^7(c+d x)}{99 d (a+a \sin (c+d x))^{5/2}}-\frac {2 a \cos ^7(c+d x)}{11 d (a+a \sin (c+d x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.16, size = 59, normalized size = 0.62 \begin {gather*} -\frac {2 \cos ^7(c+d x) \left (151+182 \sin (c+d x)+63 \sin ^2(c+d x)\right )}{693 d (1+\sin (c+d x))^3 \sqrt {a (1+\sin (c+d x))}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.39, size = 64, normalized size = 0.67
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) \left (\sin \left (d x +c \right )-1\right )^{4} \left (63 \left (\sin ^{2}\left (d x +c \right )\right )+182 \sin \left (d x +c \right )+151\right )}{693 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 155, normalized size = 1.63 \begin {gather*} \frac {2 \, {\left (63 \, \cos \left (d x + c\right )^{6} - 7 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{3} + 32 \, \cos \left (d x + c\right )^{2} + {\left (63 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{3} + 96 \, \cos \left (d x + c\right )^{2} + 128 \, \cos \left (d x + c\right ) + 256\right )} \sin \left (d x + c\right ) - 128 \, \cos \left (d x + c\right ) - 256\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{693 \, {\left (a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{6}{\left (c + d x \right )}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 6.49, size = 84, normalized size = 0.88 \begin {gather*} \frac {64 \, \sqrt {2} {\left (63 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 154 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 99 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}\right )}}{693 \, a d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^6}{\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________