Optimal. Leaf size=79 \[ -\frac {2 (b \csc (e+f x))^m \, _2F_1\left (-\frac {1}{4},\frac {1}{4} (-1-2 m);\frac {1}{4} (3-2 m);\sin ^2(e+f x)\right )}{d f (1+2 m) \sqrt [4]{\cos ^2(e+f x)} \sqrt {d \tan (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2698, 2682,
2657} \begin {gather*} -\frac {2 (b \csc (e+f x))^m \, _2F_1\left (-\frac {1}{4},\frac {1}{4} (-2 m-1);\frac {1}{4} (3-2 m);\sin ^2(e+f x)\right )}{d f (2 m+1) \sqrt [4]{\cos ^2(e+f x)} \sqrt {d \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2657
Rule 2682
Rule 2698
Rubi steps
\begin {align*} \int \frac {(b \csc (e+f x))^m}{(d \tan (e+f x))^{3/2}} \, dx &=\left ((b \csc (e+f x))^m \left (\frac {\sin (e+f x)}{b}\right )^m\right ) \int \frac {\left (\frac {\sin (e+f x)}{b}\right )^{-m}}{(d \tan (e+f x))^{3/2}} \, dx\\ &=\frac {\left ((b \csc (e+f x))^m \left (\frac {\sin (e+f x)}{b}\right )^{\frac {1}{2}+m}\right ) \int \cos ^{\frac {3}{2}}(e+f x) \left (\frac {\sin (e+f x)}{b}\right )^{-\frac {3}{2}-m} \, dx}{b d \sqrt {\cos (e+f x)} \sqrt {d \tan (e+f x)}}\\ &=-\frac {2 (b \csc (e+f x))^m \, _2F_1\left (-\frac {1}{4},\frac {1}{4} (-1-2 m);\frac {1}{4} (3-2 m);\sin ^2(e+f x)\right )}{d f (1+2 m) \sqrt [4]{\cos ^2(e+f x)} \sqrt {d \tan (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 3.77, size = 87, normalized size = 1.10 \begin {gather*} -\frac {2 (b \csc (e+f x))^m \, _2F_1\left (\frac {1}{4} (-1-2 m),1-\frac {m}{2};\frac {1}{4} (3-2 m);-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{-m/2}}{d f (1+2 m) \sqrt {d \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.25, size = 0, normalized size = 0.00 \[\int \frac {\left (b \csc \left (f x +e \right )\right )^{m}}{\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \csc {\left (e + f x \right )}\right )^{m}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, 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 {{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^m}{{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________