Optimal. Leaf size=88 \[ -\frac {4 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^4 d}+\frac {\sin ^{n+1}(c+d x)}{a^4 d (n+1)}+\frac {4 \sin ^{n+1}(c+d x)}{d \left (a^4 \sin (c+d x)+a^4\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2836, 89, 80, 64} \[ -\frac {4 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^4 d}+\frac {\sin ^{n+1}(c+d x)}{a^4 d (n+1)}+\frac {4 \sin ^{n+1}(c+d x)}{d \left (a^4 \sin (c+d x)+a^4\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 64
Rule 80
Rule 89
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 \left (\frac {x}{a}\right )^n}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {4 \sin ^{1+n}(c+d x)}{d \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {(a (3+4 n)-x) \left (\frac {x}{a}\right )^n}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\sin ^{1+n}(c+d x)}{a^4 d (1+n)}+\frac {4 \sin ^{1+n}(c+d x)}{d \left (a^4+a^4 \sin (c+d x)\right )}-\frac {(4 (1+n)) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{a}\right )^n}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\sin ^{1+n}(c+d x)}{a^4 d (1+n)}-\frac {4 \, _2F_1(1,1+n;2+n;-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^4 d}+\frac {4 \sin ^{1+n}(c+d x)}{d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 72, normalized size = 0.82 \[ \frac {\sin ^{n+1}(c+d x) (-4 (n+1) (\sin (c+d x)+1) \, _2F_1(1,n+1;n+2;-\sin (c+d x))+\sin (c+d x)+4 n+5)}{a^4 d (n+1) (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{5}}{a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{5}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 12.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )}{\left (a +a \sin \left (d x +c \right )\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{5}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^5\,{\sin \left (c+d\,x\right )}^n}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]
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]
________________________________________________________________________________________