Optimal. Leaf size=77 \[ \frac {2 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b c^2 \sqrt {c \sin (a+b x)}}-\frac {2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2636, 2642, 2641} \[ \frac {2 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b c^2 \sqrt {c \sin (a+b x)}}-\frac {2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2636
Rule 2641
Rule 2642
Rubi steps
\begin {align*} \int \frac {1}{(c \sin (a+b x))^{5/2}} \, dx &=-\frac {2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}}+\frac {\int \frac {1}{\sqrt {c \sin (a+b x)}} \, dx}{3 c^2}\\ &=-\frac {2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}}+\frac {\sqrt {\sin (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{3 c^2 \sqrt {c \sin (a+b x)}}\\ &=-\frac {2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}}+\frac {2 F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{3 b c^2 \sqrt {c \sin (a+b x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 55, normalized size = 0.71 \[ -\frac {2 \left (\cos (a+b x)+\sin ^{\frac {3}{2}}(a+b x) F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{3 b c (c \sin (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {c \sin \left (b x + a\right )}}{{\left (c^{3} \cos \left (b x + a\right )^{2} - c^{3}\right )} \sin \left (b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 105, normalized size = 1.36 \[ -\frac {\sqrt {-\sin \left (b x +a \right )+1}\, \sqrt {2 \sin \left (b x +a \right )+2}\, \left (\sin ^{\frac {5}{2}}\left (b x +a \right )\right ) \EllipticF \left (\sqrt {-\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \left (\sin ^{3}\left (b x +a \right )\right )+2 \sin \left (b x +a \right )}{3 c^{2} \sin \left (b x +a \right )^{2} \cos \left (b x +a \right ) \sqrt {c \sin \left (b x +a \right )}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (c\,\sin \left (a+b\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \sin {\left (a + b x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________