Optimal. Leaf size=89 \[ \frac {a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^4 d}-\frac {\left (a^2-b^2\right ) \sin (c+d x)}{b^3 d}+\frac {a \sin ^2(c+d x)}{2 b^2 d}-\frac {\sin ^3(c+d x)}{3 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2837, 12, 772} \[ -\frac {\left (a^2-b^2\right ) \sin (c+d x)}{b^3 d}+\frac {a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^4 d}+\frac {a \sin ^2(c+d x)}{2 b^2 d}-\frac {\sin ^3(c+d x)}{3 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 772
Rule 2837
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x \left (b^2-x^2\right )}{b (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x \left (b^2-x^2\right )}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a^2 \left (1-\frac {b^2}{a^2}\right )+a x-x^2+\frac {a^3-a b^2}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=\frac {a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^4 d}-\frac {\left (a^2-b^2\right ) \sin (c+d x)}{b^3 d}+\frac {a \sin ^2(c+d x)}{2 b^2 d}-\frac {\sin ^3(c+d x)}{3 b d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 79, normalized size = 0.89 \[ \frac {6 b \left (b^2-a^2\right ) \sin (c+d x)+6 a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))+3 a b^2 \sin ^2(c+d x)-2 b^3 \sin ^3(c+d x)}{6 b^4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.69, size = 78, normalized size = 0.88 \[ -\frac {3 \, a b^{2} \cos \left (d x + c\right )^{2} - 6 \, {\left (a^{3} - a b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \, {\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b + 2 \, b^{3}\right )} \sin \left (d x + c\right )}{6 \, b^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 85, normalized size = 0.96 \[ -\frac {\frac {2 \, b^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{2} + 6 \, a^{2} \sin \left (d x + c\right ) - 6 \, b^{2} \sin \left (d x + c\right )}{b^{3}} - \frac {6 \, {\left (a^{3} - a b^{2}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{4}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.21, size = 106, normalized size = 1.19 \[ -\frac {\sin ^{3}\left (d x +c \right )}{3 b d}+\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{2 b^{2} d}-\frac {a^{2} \sin \left (d x +c \right )}{b^{3} d}+\frac {\sin \left (d x +c \right )}{b d}+\frac {a^{3} \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{4} d}-\frac {a \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 79, normalized size = 0.89 \[ -\frac {\frac {2 \, b^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{2} + 6 \, {\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}{b^{3}} - \frac {6 \, {\left (a^{3} - a b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{4}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 78, normalized size = 0.88 \[ \frac {\sin \left (c+d\,x\right )\,\left (\frac {1}{b}-\frac {a^2}{b^3}\right )-\frac {{\sin \left (c+d\,x\right )}^3}{3\,b}+\frac {a\,{\sin \left (c+d\,x\right )}^2}{2\,b^2}-\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (a\,b^2-a^3\right )}{b^4}}{d} \]
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]
________________________________________________________________________________________