Optimal. Leaf size=126 \[ \frac{1}{32 d \left (a^3-a^3 \sin (c+d x)\right )}-\frac{1}{8 d \left (a^3 \sin (c+d x)+a^3\right )}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{32 a^3 d}-\frac{a}{16 d (a \sin (c+d x)+a)^4}-\frac{1}{12 d (a \sin (c+d x)+a)^3}-\frac{3}{32 a d (a \sin (c+d x)+a)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0946766, antiderivative size = 126, normalized size of antiderivative = 1., 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} \[ \frac{1}{32 d \left (a^3-a^3 \sin (c+d x)\right )}-\frac{1}{8 d \left (a^3 \sin (c+d x)+a^3\right )}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{32 a^3 d}-\frac{a}{16 d (a \sin (c+d x)+a)^4}-\frac{1}{12 d (a \sin (c+d x)+a)^3}-\frac{3}{32 a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2667
Rule 44
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{(a-x)^2 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{1}{32 a^5 (a-x)^2}+\frac{1}{4 a^2 (a+x)^5}+\frac{1}{4 a^3 (a+x)^4}+\frac{3}{16 a^4 (a+x)^3}+\frac{1}{8 a^5 (a+x)^2}+\frac{5}{32 a^5 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{a}{16 d (a+a \sin (c+d x))^4}-\frac{1}{12 d (a+a \sin (c+d x))^3}-\frac{3}{32 a d (a+a \sin (c+d x))^2}+\frac{1}{32 d \left (a^3-a^3 \sin (c+d x)\right )}-\frac{1}{8 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{32 a^2 d}\\ &=\frac{5 \tanh ^{-1}(\sin (c+d x))}{32 a^3 d}-\frac{a}{16 d (a+a \sin (c+d x))^4}-\frac{1}{12 d (a+a \sin (c+d x))^3}-\frac{3}{32 a d (a+a \sin (c+d x))^2}+\frac{1}{32 d \left (a^3-a^3 \sin (c+d x)\right )}-\frac{1}{8 d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.168287, size = 95, normalized size = 0.75 \[ -\frac{\sec ^2(c+d x) \left (-15 \sin ^4(c+d x)-45 \sin ^3(c+d x)-35 \sin ^2(c+d x)+15 \sin (c+d x)+15 (\sin (c+d x)-1) (\sin (c+d x)+1)^4 \tanh ^{-1}(\sin (c+d x))+32\right )}{96 a^3 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.102, size = 126, normalized size = 1. \begin{align*} -{\frac{1}{32\,d{a}^{3} \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{5\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{64\,d{a}^{3}}}-{\frac{1}{16\,d{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{1}{12\,d{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{3}{32\,d{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{1}{8\,d{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{5\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{64\,d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.959049, size = 197, normalized size = 1.56 \begin{align*} -\frac{\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{4} + 45 \, \sin \left (d x + c\right )^{3} + 35 \, \sin \left (d x + c\right )^{2} - 15 \, \sin \left (d x + c\right ) - 32\right )}}{a^{3} \sin \left (d x + c\right )^{5} + 3 \, a^{3} \sin \left (d x + c\right )^{4} + 2 \, a^{3} \sin \left (d x + c\right )^{3} - 2 \, a^{3} \sin \left (d x + c\right )^{2} - 3 \, a^{3} \sin \left (d x + c\right ) - a^{3}} - \frac{15 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac{15 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3}}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.83161, size = 595, normalized size = 4.72 \begin{align*} -\frac{30 \, \cos \left (d x + c\right )^{4} - 130 \, \cos \left (d x + c\right )^{2} - 15 \,{\left (3 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \,{\left (3 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 30 \,{\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 36}{192 \,{\left (3 \, a^{3} d \cos \left (d x + c\right )^{4} - 4 \, a^{3} d \cos \left (d x + c\right )^{2} +{\left (a^{3} d \cos \left (d x + c\right )^{4} - 4 \, a^{3} d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sec ^{3}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin{\left (c + d x \right )} + 1}\, dx}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19805, size = 157, normalized size = 1.25 \begin{align*} \frac{\frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} - \frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3}} + \frac{12 \,{\left (5 \, \sin \left (d x + c\right ) - 7\right )}}{a^{3}{\left (\sin \left (d x + c\right ) - 1\right )}} - \frac{125 \, \sin \left (d x + c\right )^{4} + 596 \, \sin \left (d x + c\right )^{3} + 1110 \, \sin \left (d x + c\right )^{2} + 996 \, \sin \left (d x + c\right ) + 405}{a^{3}{\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{768 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]