Optimal. Leaf size=314 \[ \frac {2 \left (8 a^2+49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}+\frac {2 a \left (8 a^2+39 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{315 b^2 d}-\frac {2 a \left (8 a^4+31 a^2 b^2-39 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^4+33 a^2 b^2+147 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.52, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2793, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 \left (8 a^2+49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}+\frac {2 a \left (8 a^2+39 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{315 b^2 d}-\frac {2 a \left (31 a^2 b^2+8 a^4-39 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (33 a^2 b^2+8 a^4+147 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 2793
Rule 3023
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+b \cos (c+d x))^{3/2} \, dx &=\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac {2 \int (a+b \cos (c+d x))^{3/2} \left (a+\frac {7}{2} b \cos (c+d x)-2 a \cos ^2(c+d x)\right ) \, dx}{9 b}\\ &=-\frac {8 a (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac {4 \int (a+b \cos (c+d x))^{3/2} \left (-\frac {3 a b}{2}+\frac {1}{4} \left (8 a^2+49 b^2\right ) \cos (c+d x)\right ) \, dx}{63 b^2}\\ &=\frac {2 \left (8 a^2+49 b^2\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac {8 a (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac {8 \int \sqrt {a+b \cos (c+d x)} \left (-\frac {3}{8} b \left (2 a^2-49 b^2\right )+\frac {3}{8} a \left (8 a^2+39 b^2\right ) \cos (c+d x)\right ) \, dx}{315 b^2}\\ &=\frac {2 a \left (8 a^2+39 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}+\frac {2 \left (8 a^2+49 b^2\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac {8 a (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac {16 \int \frac {\frac {3}{8} a b \left (a^2+93 b^2\right )+\frac {3}{16} \left (8 a^4+33 a^2 b^2+147 b^4\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{945 b^2}\\ &=\frac {2 a \left (8 a^2+39 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}+\frac {2 \left (8 a^2+49 b^2\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac {8 a (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}-\frac {\left (a \left (8 a^4+31 a^2 b^2-39 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{315 b^3}+\frac {\left (8 a^4+33 a^2 b^2+147 b^4\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{315 b^3}\\ &=\frac {2 a \left (8 a^2+39 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}+\frac {2 \left (8 a^2+49 b^2\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac {8 a (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac {\left (\left (8 a^4+33 a^2 b^2+147 b^4\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{315 b^3 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a \left (8 a^4+31 a^2 b^2-39 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{315 b^3 \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 \left (8 a^4+33 a^2 b^2+147 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 a \left (8 a^4+31 a^2 b^2-39 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 a \left (8 a^2+39 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}+\frac {2 \left (8 a^2+49 b^2\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac {8 a (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.36, size = 262, normalized size = 0.83 \[ \frac {-8 a \left (8 a^4+31 a^2 b^2-39 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+b \sin (c+d x) \left (-32 a^4+\left (1606 a b^3-8 a^3 b\right ) \cos (c+d x)+916 a^2 b^2+4 \left (53 a^2 b^2+84 b^4\right ) \cos (2 (c+d x))+170 a b^3 \cos (3 (c+d x))+35 b^4 \cos (4 (c+d x))+301 b^4\right )+8 \left (8 a^5+8 a^4 b+33 a^3 b^2+33 a^2 b^3+147 a b^4+147 b^5\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{1260 b^3 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.25, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \cos \left (d x + c\right )^{4} + a \cos \left (d x + c\right )^{3}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.88, size = 995, normalized size = 3.17 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^3\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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]
________________________________________________________________________________________