Optimal. Leaf size=80 \[ \frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}-\frac{C \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \sqrt{b \cos (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0328293, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {17, 3013} \[ \frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}-\frac{C \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 17
Rule 3013
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{5/2}} \, dx &=\frac{\sqrt{\cos (c+d x)} \int \cos (c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx}{b^2 \sqrt{b \cos (c+d x)}}\\ &=-\frac{\sqrt{\cos (c+d x)} \operatorname{Subst}\left (\int \left (A+C-C x^2\right ) \, dx,x,-\sin (c+d x)\right )}{b^2 d \sqrt{b \cos (c+d x)}}\\ &=\frac{(A+C) \sqrt{\cos (c+d x)} \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}-\frac{C \sqrt{\cos (c+d x)} \sin ^3(c+d x)}{3 b^2 d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0619518, size = 55, normalized size = 0.69 \[ \frac{\sin (c+d x) \sqrt{\cos (c+d x)} (6 A+C \cos (2 (c+d x))+5 C)}{6 b^2 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.256, size = 47, normalized size = 0.6 \begin{align*}{\frac{ \left ( C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+3\,A+2\,C \right ) \sin \left ( dx+c \right ) }{3\,d} \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}} \left ( b\cos \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.69352, size = 77, normalized size = 0.96 \begin{align*} \frac{\frac{C{\left (\sin \left (3 \, d x + 3 \, c\right ) + 9 \, \sin \left (\frac{1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )}}{b^{\frac{5}{2}}} + \frac{12 \, A \sin \left (d x + c\right )}{b^{\frac{5}{2}}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.64481, size = 131, normalized size = 1.64 \begin{align*} \frac{{\left (C \cos \left (d x + c\right )^{2} + 3 \, A + 2 \, C\right )} \sqrt{b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{3 \, b^{3} d \sqrt{\cos \left (d x + c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac{7}{2}}}{\left (b \cos \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]