Optimal. Leaf size=113 \[ \frac {\left (a^2-b^2\right ) (A b-a B)}{b^4 d (a+b \sin (c+d x))}+\frac {\left (-3 a^2 B+2 a A b+b^2 B\right ) \log (a+b \sin (c+d x))}{b^4 d}-\frac {(A b-2 a B) \sin (c+d x)}{b^3 d}-\frac {B \sin ^2(c+d x)}{2 b^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 113, 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 = {2837, 772} \[ \frac {\left (a^2-b^2\right ) (A b-a B)}{b^4 d (a+b \sin (c+d x))}+\frac {\left (-3 a^2 B+2 a A b+b^2 B\right ) \log (a+b \sin (c+d x))}{b^4 d}-\frac {(A b-2 a B) \sin (c+d x)}{b^3 d}-\frac {B \sin ^2(c+d x)}{2 b^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 772
Rule 2837
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (A+\frac {B x}{b}\right ) \left (b^2-x^2\right )}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {-A b+2 a B}{b}-\frac {B x}{b}+\frac {\left (-a^2+b^2\right ) (A b-a B)}{b (a+x)^2}+\frac {2 a A b-3 a^2 B+b^2 B}{b (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac {\left (2 a A b-3 a^2 B+b^2 B\right ) \log (a+b \sin (c+d x))}{b^4 d}-\frac {(A b-2 a B) \sin (c+d x)}{b^3 d}-\frac {B \sin ^2(c+d x)}{2 b^2 d}+\frac {\left (a^2-b^2\right ) (A b-a B)}{b^4 d (a+b \sin (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.54, size = 111, normalized size = 0.98 \[ \frac {\frac {B \left (b^2-a^2\right ) \log (a+b \sin (c+d x))}{b}+\left (A-\frac {a B}{b}\right ) \left (\frac {(a-b) (a+b)}{a+b \sin (c+d x)}+2 a \log (a+b \sin (c+d x))-b \sin (c+d x)\right )+a B \sin (c+d x)-\frac {1}{2} b B \sin ^2(c+d x)}{b^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 178, normalized size = 1.58 \[ -\frac {4 \, B a^{3} - 4 \, A a^{2} b - 11 \, B a b^{2} + 8 \, A b^{3} + 2 \, {\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (3 \, B a^{3} - 2 \, A a^{2} b - B a b^{2} + {\left (3 \, B a^{2} b - 2 \, A a b^{2} - B b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - {\left (2 \, B b^{3} \cos \left (d x + c\right )^{2} + 8 \, B a^{2} b - 4 \, A a b^{2} - B b^{3}\right )} \sin \left (d x + c\right )}{4 \, {\left (b^{5} d \sin \left (d x + c\right ) + a b^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.23, size = 188, normalized size = 1.66 \[ -\frac {\frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{2} {\left (B - \frac {2 \, {\left (3 \, B a b - A b^{2}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} b}\right )}}{b^{4}} - \frac {2 \, {\left (3 \, B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (\frac {{\left | b \sin \left (d x + c\right ) + a \right |}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2} {\left | b \right |}}\right )}{b^{4}} + \frac {2 \, {\left (\frac {B a^{3} b^{2}}{b \sin \left (d x + c\right ) + a} - \frac {A a^{2} b^{3}}{b \sin \left (d x + c\right ) + a} - \frac {B a b^{4}}{b \sin \left (d x + c\right ) + a} + \frac {A b^{5}}{b \sin \left (d x + c\right ) + a}\right )}}{b^{6}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.66, size = 202, normalized size = 1.79 \[ -\frac {B \left (\sin ^{2}\left (d x +c \right )\right )}{2 b^{2} d}-\frac {A \sin \left (d x +c \right )}{d \,b^{2}}+\frac {2 B a \sin \left (d x +c \right )}{d \,b^{3}}+\frac {2 \ln \left (a +b \sin \left (d x +c \right )\right ) A a}{d \,b^{3}}-\frac {3 \ln \left (a +b \sin \left (d x +c \right )\right ) B \,a^{2}}{d \,b^{4}}+\frac {B \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{2} d}+\frac {A \,a^{2}}{d \,b^{3} \left (a +b \sin \left (d x +c \right )\right )}-\frac {A}{d b \left (a +b \sin \left (d x +c \right )\right )}-\frac {B \,a^{3}}{d \,b^{4} \left (a +b \sin \left (d x +c \right )\right )}+\frac {a B}{d \,b^{2} \left (a +b \sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 118, normalized size = 1.04 \[ -\frac {\frac {2 \, {\left (B a^{3} - A a^{2} b - B a b^{2} + A b^{3}\right )}}{b^{5} \sin \left (d x + c\right ) + a b^{4}} + \frac {B b \sin \left (d x + c\right )^{2} - 2 \, {\left (2 \, B a - A b\right )} \sin \left (d x + c\right )}{b^{3}} + \frac {2 \, {\left (3 \, B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 12.10, size = 128, normalized size = 1.13 \[ \frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (-3\,B\,a^2+2\,A\,a\,b+B\,b^2\right )}{b^4\,d}-\frac {B\,a^3-A\,a^2\,b-B\,a\,b^2+A\,b^3}{b\,d\,\left (\sin \left (c+d\,x\right )\,b^4+a\,b^3\right )}-\frac {\sin \left (c+d\,x\right )\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{d}-\frac {B\,{\sin \left (c+d\,x\right )}^2}{2\,b^2\,d} \]
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]
________________________________________________________________________________________