Optimal. Leaf size=51 \[ \frac {B (a-a \sin (c+d x))^4}{4 a^6 d}-\frac {(A+B) (a-a \sin (c+d x))^3}{3 a^5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2836, 43} \[ \frac {B (a-a \sin (c+d x))^4}{4 a^6 d}-\frac {(A+B) (a-a \sin (c+d x))^3}{3 a^5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 \left (A+\frac {B x}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left ((A+B) (a-x)^2-\frac {B (a-x)^3}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=-\frac {(A+B) (a-a \sin (c+d x))^3}{3 a^5 d}+\frac {B (a-a \sin (c+d x))^4}{4 a^6 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 34, normalized size = 0.67 \[ \frac {(\sin (c+d x)-1)^3 (4 A+3 B \sin (c+d x)+B)}{12 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.82, size = 64, normalized size = 1.25 \[ \frac {3 \, B \cos \left (d x + c\right )^{4} + 12 \, {\left (A - B\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left ({\left (A - 2 \, B\right )} \cos \left (d x + c\right )^{2} - 4 \, A + 2 \, B\right )} \sin \left (d x + c\right )}{12 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 73, normalized size = 1.43 \[ \frac {3 \, B \sin \left (d x + c\right )^{4} + 4 \, A \sin \left (d x + c\right )^{3} - 8 \, B \sin \left (d x + c\right )^{3} - 12 \, A \sin \left (d x + c\right )^{2} + 6 \, B \sin \left (d x + c\right )^{2} + 12 \, A \sin \left (d x + c\right )}{12 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.59, size = 58, normalized size = 1.14 \[ \frac {\frac {B \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (A -2 B \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (-2 A +B \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}+A \sin \left (d x +c \right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 61, normalized size = 1.20 \[ \frac {3 \, B \sin \left (d x + c\right )^{4} + 4 \, {\left (A - 2 \, B\right )} \sin \left (d x + c\right )^{3} - 6 \, {\left (2 \, A - B\right )} \sin \left (d x + c\right )^{2} + 12 \, A \sin \left (d x + c\right )}{12 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 9.12, size = 68, normalized size = 1.33 \[ \frac {\frac {{\sin \left (c+d\,x\right )}^3\,\left (A-2\,B\right )}{3\,a^2}+\frac {B\,{\sin \left (c+d\,x\right )}^4}{4\,a^2}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (2\,A-B\right )}{2\,a^2}+\frac {A\,\sin \left (c+d\,x\right )}{a^2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 53.39, size = 1182, normalized size = 23.18 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________