Optimal. Leaf size=189 \[ \frac {\cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {n p}{2};\frac {1}{2} (2+n p);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f \sqrt {\cos ^2(e+f x)}}-\frac {n p \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);\sin ^2(e+f x)\right ) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{a f (1+n p) \sqrt {\cos ^2(e+f x)}}-\frac {\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a+a \sin (e+f x))} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2905, 2848,
2827, 2722} \begin {gather*} \frac {\cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {n p}{2};\frac {1}{2} (n p+2);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f \sqrt {\cos ^2(e+f x)}}-\frac {n p \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f (n p+1) \sqrt {\cos ^2(e+f x)}}-\frac {\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a \sin (e+f x)+a)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2722
Rule 2827
Rule 2848
Rule 2905
Rubi steps
\begin {align*} \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{a+a \sin (e+f x)} \, dx &=\left ((d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int \frac {(d \sin (e+f x))^{n p}}{a+a \sin (e+f x)} \, dx\\ &=-\frac {\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a+a \sin (e+f x))}+\frac {\left (d n p (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{-1+n p} (a-a \sin (e+f x)) \, dx}{a^2}\\ &=-\frac {\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a+a \sin (e+f x))}-\frac {\left (n p (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{n p} \, dx}{a}+\frac {\left (d n p (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{-1+n p} \, dx}{a}\\ &=\frac {\cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {n p}{2};\frac {1}{2} (2+n p);\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{a f \sqrt {\cos ^2(e+f x)}}-\frac {n p \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);\sin ^2(e+f x)\right ) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{a f (1+n p) \sqrt {\cos ^2(e+f x)}}-\frac {\cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (a+a \sin (e+f x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 157, normalized size = 0.83 \begin {gather*} \frac {\cos (e+f x) \sqrt {\cos ^2(e+f x)} \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n \left (-\left ((2+n p) \, _2F_1\left (\frac {3}{2},\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);\sin ^2(e+f x)\right )\right )+(1+n p) \, _2F_1\left (\frac {3}{2},1+\frac {n p}{2};2+\frac {n p}{2};\sin ^2(e+f x)\right ) \sin (e+f x)\right )}{a f (1+n p) (2+n p) (-1+\sin (e+f x)) (1+\sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (c \left (d \sin \left (f x +e \right )\right )^{p}\right )^{n}}{a +a \sin \left (f x +e \right )}\, dx\]
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]
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*} \frac {\int \frac {\left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n}}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \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 (c\,{\left (d\,\sin \left (e+f\,x\right )\right )}^p\right )}^n}{a+a\,\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________