Optimal. Leaf size=104 \[ \frac {\cos ^2(e+f x)^{\frac {1}{2} (1+n p)} (d \csc (e+f x))^m \, _2F_1\left (\frac {1}{2} (1+n p),\frac {1}{2} (1-m+n p);\frac {1}{2} (3-m+n p);\sin ^2(e+f x)\right ) \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1-m+n p)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3740, 2698,
2682, 2657} \begin {gather*} \frac {\tan (e+f x) (d \csc (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (n p+1)} \left (b (c \tan (e+f x))^n\right )^p \, _2F_1\left (\frac {1}{2} (n p+1),\frac {1}{2} (-m+n p+1);\frac {1}{2} (-m+n p+3);\sin ^2(e+f x)\right )}{f (-m+n p+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2657
Rule 2682
Rule 2698
Rule 3740
Rubi steps
\begin {align*} \int (d \csc (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx &=\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int (d \csc (e+f x))^m (c \tan (e+f x))^{n p} \, dx\\ &=\left ((d \csc (e+f x))^m \left (\frac {\sin (e+f x)}{d}\right )^m (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{-m} (c \tan (e+f x))^{n p} \, dx\\ &=\frac {\left (\cos ^{n p}(e+f x) (d \csc (e+f x))^{1+m} \sin (e+f x) \left (\frac {\sin (e+f x)}{d}\right )^{m-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int \cos ^{-n p}(e+f x) \left (\frac {\sin (e+f x)}{d}\right )^{-m+n p} \, dx}{d}\\ &=\frac {\cos ^2(e+f x)^{\frac {1}{2} (1+n p)} (d \csc (e+f x))^{1+m} \, _2F_1\left (\frac {1}{2} (1+n p),\frac {1}{2} (1-m+n p);\frac {1}{2} (3-m+n p);\sin ^2(e+f x)\right ) \sin (e+f x) \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{d f (1-m+n p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 1.46, size = 319, normalized size = 3.07 \begin {gather*} -\frac {d (-3+m-n p) F_1\left (\frac {1}{2} (1-m+n p);n p,1-m;\frac {1}{2} (3-m+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (d \csc (e+f x))^{-1+m} \left (b (c \tan (e+f x))^n\right )^p}{f (-1+m-n p) \left ((-3+m-n p) F_1\left (\frac {1}{2} (1-m+n p);n p,1-m;\frac {1}{2} (3-m+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 \left ((-1+m) F_1\left (\frac {1}{2} (3-m+n p);n p,2-m;\frac {1}{2} (5-m+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+n p F_1\left (\frac {1}{2} (3-m+n p);1+n p,1-m;\frac {1}{2} (5-m+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.35, size = 0, normalized size = 0.00 \[\int \left (d \csc \left (f x +e \right )\right )^{m} \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx\]
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 \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \left (d \csc {\left (e + f x \right )}\right )^{m}\, 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 {\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^m\,{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________