Optimal. Leaf size=80 \[ -\frac {2 \cos (a+b x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-m}{4},\frac {6-m}{4},\cos ^2(a+b x)\right ) \sin (a+b x)}{b (2-m) \sqrt {c \cos ^m(a+b x)} \sqrt {\sin ^2(a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, 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 = {3287, 2722}
\begin {gather*} -\frac {2 \sin (a+b x) \cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {2-m}{4};\frac {6-m}{4};\cos ^2(a+b x)\right )}{b (2-m) \sqrt {\sin ^2(a+b x)} \sqrt {c \cos ^m(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2722
Rule 3287
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {c \cos ^m(a+b x)}} \, dx &=\frac {\cos ^{\frac {m}{2}}(a+b x) \int \cos ^{-\frac {m}{2}}(a+b x) \, dx}{\sqrt {c \cos ^m(a+b x)}}\\ &=-\frac {2 \cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {2-m}{4};\frac {6-m}{4};\cos ^2(a+b x)\right ) \sin (a+b x)}{b (2-m) \sqrt {c \cos ^m(a+b x)} \sqrt {\sin ^2(a+b x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 72, normalized size = 0.90 \begin {gather*} \frac {2 \cot (a+b x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-m}{4},\frac {6-m}{4},\cos ^2(a+b x)\right ) \sqrt {\sin ^2(a+b x)}}{b (-2+m) \sqrt {c \cos ^m(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {c \left (\cos ^{m}\left (b x +a \right )\right )}}\, dx\]
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {1}{\sqrt {c \cos ^{m}{\left (a + b x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {1}{\sqrt {c\,{\cos \left (a+b\,x\right )}^m}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________