Optimal. Leaf size=43 \[ \frac {a^3 (A+B)}{d (a-a \sin (c+d x))}+\frac {a^2 B \log (1-\sin (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 43, 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} \[ \frac {a^3 (A+B)}{d (a-a \sin (c+d x))}+\frac {a^2 B \log (1-\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2836
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx &=\frac {a^3 \operatorname {Subst}\left (\int \frac {A+\frac {B x}{a}}{(a-x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \left (\frac {A+B}{(a-x)^2}-\frac {B}{a (a-x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^2 B \log (1-\sin (c+d x))}{d}+\frac {a^3 (A+B)}{d (a-a \sin (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 41, normalized size = 0.95 \[ \frac {a^3 \left (\frac {A+B}{a-a \sin (c+d x)}+\frac {B \log (1-\sin (c+d x))}{a}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.81, size = 55, normalized size = 1.28 \[ -\frac {{\left (A + B\right )} a^{2} - {\left (B a^{2} \sin \left (d x + c\right ) - B a^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{d \sin \left (d x + c\right ) - d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.21, size = 112, normalized size = 2.60 \[ -\frac {B a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 2 \, B a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{2}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.60, size = 189, normalized size = 4.40 \[ \frac {a^{2} A \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {a^{2} A \sin \left (d x +c \right )}{2 d}+\frac {B \,a^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {B \,a^{2} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {a^{2} A}{d \cos \left (d x +c \right )^{2}}+\frac {B \,a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {B \,a^{2} \sin \left (d x +c \right )}{d}-\frac {B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{2} A \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {B \,a^{2}}{2 d \cos \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 37, normalized size = 0.86 \[ \frac {B a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {{\left (A + B\right )} a^{2}}{\sin \left (d x + c\right ) - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 44, normalized size = 1.02 \[ \frac {B\,a^2\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{d}-\frac {A\,a^2+B\,a^2}{d\,\left (\sin \left (c+d\,x\right )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} \left (\int A \sec ^{3}{\left (c + d x \right )}\, dx + \int 2 A \sin {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int A \sin ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sin {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 2 B \sin ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sin ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________