Optimal. Leaf size=104 \[ \frac {1}{16 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {3}{16 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {\tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac {a}{12 d (a \sin (c+d x)+a)^3}-\frac {1}{8 d (a \sin (c+d x)+a)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2667, 44, 206} \[ \frac {1}{16 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {3}{16 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {\tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac {a}{12 d (a \sin (c+d x)+a)^3}-\frac {1}{8 d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 206
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \left (\frac {1}{16 a^4 (a-x)^2}+\frac {1}{4 a^2 (a+x)^4}+\frac {1}{4 a^3 (a+x)^3}+\frac {3}{16 a^4 (a+x)^2}+\frac {1}{4 a^4 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a}{12 d (a+a \sin (c+d x))^3}-\frac {1}{8 d (a+a \sin (c+d x))^2}+\frac {1}{16 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {3}{16 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{4 a d}\\ &=\frac {\tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac {a}{12 d (a+a \sin (c+d x))^3}-\frac {1}{8 d (a+a \sin (c+d x))^2}+\frac {1}{16 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {3}{16 d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 85, normalized size = 0.82 \[ -\frac {\sec ^2(c+d x) \left (-3 \sin ^3(c+d x)-6 \sin ^2(c+d x)-\sin (c+d x)+3 (\sin (c+d x)-1) (\sin (c+d x)+1)^3 \tanh ^{-1}(\sin (c+d x))+4\right )}{12 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.73, size = 178, normalized size = 1.71 \[ \frac {12 \, \cos \left (d x + c\right )^{2} + 3 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 4}{24 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.81, size = 106, normalized size = 1.02 \[ \frac {\frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2}} - \frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2}} + \frac {3 \, {\left (2 \, \sin \left (d x + c\right ) - 3\right )}}{a^{2} {\left (\sin \left (d x + c\right ) - 1\right )}} - \frac {11 \, \sin \left (d x + c\right )^{3} + 42 \, \sin \left (d x + c\right )^{2} + 57 \, \sin \left (d x + c\right ) + 30}{a^{2} {\left (\sin \left (d x + c\right ) + 1\right )}^{3}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.23, size = 108, normalized size = 1.04 \[ -\frac {1}{16 a^{2} d \left (\sin \left (d x +c \right )-1\right )}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{8 a^{2} d}-\frac {1}{12 a^{2} d \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {1}{8 a^{2} d \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3}{16 a^{2} d \left (1+\sin \left (d x +c \right )\right )}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{8 a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.37, size = 108, normalized size = 1.04 \[ -\frac {\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} + 6 \, \sin \left (d x + c\right )^{2} + \sin \left (d x + c\right ) - 4\right )}}{a^{2} \sin \left (d x + c\right )^{4} + 2 \, a^{2} \sin \left (d x + c\right )^{3} - 2 \, a^{2} \sin \left (d x + c\right ) - a^{2}} - \frac {3 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {3 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 93, normalized size = 0.89 \[ \frac {\frac {{\sin \left (c+d\,x\right )}^3}{4}+\frac {{\sin \left (c+d\,x\right )}^2}{2}+\frac {\sin \left (c+d\,x\right )}{12}-\frac {1}{3}}{d\,\left (-a^2\,{\sin \left (c+d\,x\right )}^4-2\,a^2\,{\sin \left (c+d\,x\right )}^3+2\,a^2\,\sin \left (c+d\,x\right )+a^2\right )}+\frac {\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{4\,a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sec ^{3}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________