Optimal. Leaf size=89 \[ -\frac{2 c^2 \sin (a+b x) \cos ^{2 m+1}(a+b x) \sqrt{c \cos ^m(a+b x)} \, _2F_1\left (\frac{1}{2},\frac{1}{4} (5 m+2);\frac{1}{4} (5 m+6);\cos ^2(a+b x)\right )}{b (5 m+2) \sqrt{\sin ^2(a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0400767, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3208, 2643} \[ -\frac{2 c^2 \sin (a+b x) \cos ^{2 m+1}(a+b x) \sqrt{c \cos ^m(a+b x)} \, _2F_1\left (\frac{1}{2},\frac{1}{4} (5 m+2);\frac{1}{4} (5 m+6);\cos ^2(a+b x)\right )}{b (5 m+2) \sqrt{\sin ^2(a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3208
Rule 2643
Rubi steps
\begin{align*} \int \left (c \cos ^m(a+b x)\right )^{5/2} \, dx &=\left (c^2 \cos ^{-\frac{m}{2}}(a+b x) \sqrt{c \cos ^m(a+b x)}\right ) \int \cos ^{\frac{5 m}{2}}(a+b x) \, dx\\ &=-\frac{2 c^2 \cos ^{1+2 m}(a+b x) \sqrt{c \cos ^m(a+b x)} \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2+5 m);\frac{1}{4} (6+5 m);\cos ^2(a+b x)\right ) \sin (a+b x)}{b (2+5 m) \sqrt{\sin ^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.172526, size = 74, normalized size = 0.83 \[ -\frac{2 \sqrt{\sin ^2(a+b x)} \cot (a+b x) \left (c \cos ^m(a+b x)\right )^{5/2} \, _2F_1\left (\frac{1}{2},\frac{1}{4} (5 m+2);\frac{1}{4} (5 m+6);\cos ^2(a+b x)\right )}{b (5 m+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.419, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( \cos \left ( bx+a \right ) \right ) ^{m} \right ) ^{{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \cos \left (b x + a\right )^{m}\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 \left (c \cos \left (b x + a\right )^{m}\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]