Optimal. Leaf size=78 \[ \frac {\cos ^2(a+b x)^{3/4} \csc ^{p-1}(a+b x) \, _2F_1\left (\frac {7}{4},\frac {1-p}{2};\frac {3-p}{2};\sin ^2(a+b x)\right )}{b d (1-p) (d \cos (a+b x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2587, 2577} \[ \frac {\cos ^2(a+b x)^{3/4} \csc ^{p-1}(a+b x) \, _2F_1\left (\frac {7}{4},\frac {1-p}{2};\frac {3-p}{2};\sin ^2(a+b x)\right )}{b d (1-p) (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2577
Rule 2587
Rubi steps
\begin {align*} \int \frac {\csc ^p(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx &=\left (\csc ^p(a+b x) \sin ^p(a+b x)\right ) \int \frac {\sin ^{-p}(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx\\ &=\frac {\cos ^2(a+b x)^{3/4} \csc ^{-1+p}(a+b x) \, _2F_1\left (\frac {7}{4},\frac {1-p}{2};\frac {3-p}{2};\sin ^2(a+b x)\right )}{b d (1-p) (d \cos (a+b x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.28, size = 70, normalized size = 0.90 \[ \frac {2 \sin ^2(a+b x)^{\frac {p-1}{2}} \csc ^{p-1}(a+b x) \, _2F_1\left (-\frac {3}{4},\frac {p+1}{2};\frac {1}{4};\cos ^2(a+b x)\right )}{3 b d (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \cos \left (b x + a\right )} \csc \left (b x + a\right )^{p}}{d^{3} \cos \left (b x + a\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b x + a\right )^{p}}{\left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{p}\left (b x +a \right )}{\left (d \cos \left (b x +a \right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b x + a\right )^{p}}{\left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^p}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{5/2}} \,d x \]
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]
________________________________________________________________________________________