Optimal. Leaf size=231 \[ \frac{a b^2}{2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}-\frac{2 a b \left (a^2+b^2\right )}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac{a \left (8 a^2 b^2+a^4+3 b^4\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}-\frac{\csc ^2(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}+\frac{(2 a-b) \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac{(2 a+b) \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.620968, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {4397, 2837, 12, 1647, 1629} \[ \frac{a b^2}{2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}-\frac{2 a b \left (a^2+b^2\right )}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac{a \left (8 a^2 b^2+a^4+3 b^4\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}-\frac{\csc ^2(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}+\frac{(2 a-b) \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac{(2 a+b) \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4397
Rule 2837
Rule 12
Rule 1647
Rule 1629
Rubi steps
\begin{align*} \int \frac{\sec (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx &=\int \frac{\cot ^2(c+d x) \csc (c+d x)}{(b+a \cos (c+d x))^3} \, dx\\ &=-\frac{a^3 \operatorname{Subst}\left (\int \frac{x^2}{a^2 (b+x)^3 \left (a^2-x^2\right )^2} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{x^2}{(b+x)^3 \left (a^2-x^2\right )^2} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac{\left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}-\frac{\operatorname{Subst}\left (\int \frac{\frac{a^2 b^4 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3}+\frac{a^2 b^3 \left (7 a^2-3 b^2\right ) x}{\left (a^2-b^2\right )^3}+\frac{a^2 \left (2 a^4-3 a^2 b^2-3 b^4\right ) x^2}{\left (a^2-b^2\right )^3}-\frac{a^2 b \left (3 a^2+b^2\right ) x^3}{\left (a^2-b^2\right )^3}}{(b+x)^3 \left (a^2-x^2\right )} \, dx,x,a \cos (c+d x)\right )}{2 a d}\\ &=-\frac{\left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}-\frac{\operatorname{Subst}\left (\int \left (\frac{a (2 a-b)}{2 (a+b)^4 (a-x)}-\frac{a (2 a+b)}{2 (a-b)^4 (a+x)}+\frac{2 a^2 b^2}{\left (a^2-b^2\right )^2 (b+x)^3}-\frac{4 a^2 b \left (a^2+b^2\right )}{\left (a^2-b^2\right )^3 (b+x)^2}+\frac{2 \left (a^6+8 a^4 b^2+3 a^2 b^4\right )}{\left (a^2-b^2\right )^4 (b+x)}\right ) \, dx,x,a \cos (c+d x)\right )}{2 a d}\\ &=\frac{a b^2}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}-\frac{2 a b \left (a^2+b^2\right )}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}-\frac{\left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}+\frac{(2 a-b) \log (1-\cos (c+d x))}{4 (a+b)^4 d}+\frac{(2 a+b) \log (1+\cos (c+d x))}{4 (a-b)^4 d}-\frac{a \left (a^4+8 a^2 b^2+3 b^4\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d}\\ \end{align*}
Mathematica [C] time = 6.31286, size = 703, normalized size = 3.04 \[ \frac{2 i \left (8 a^3 b^2+a^5+3 a b^4\right ) (c+d x) \tan ^3(c+d x) (a \cos (c+d x)+b)^3}{d (a-b)^4 (a+b)^4 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{\left (-8 a^3 b^2-a^5-3 a b^4\right ) \tan ^3(c+d x) (a \cos (c+d x)+b)^3 \log (a \cos (c+d x)+b)}{d \left (b^2-a^2\right )^4 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{a b^2 \tan ^3(c+d x) (a \cos (c+d x)+b)}{2 d (b-a)^2 (a+b)^2 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{i (2 a+b) \tan ^{-1}(\tan (c+d x)) \tan ^3(c+d x) (a \cos (c+d x)+b)^3}{2 d (b-a)^4 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{i (2 a-b) \tan ^{-1}(\tan (c+d x)) \tan ^3(c+d x) (a \cos (c+d x)+b)^3}{2 d (a+b)^4 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{2 a b (b-i a) (b+i a) \tan ^3(c+d x) (a \cos (c+d x)+b)^2}{d (b-a)^3 (a+b)^3 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{(2 a+b) \tan ^3(c+d x) \log \left (\cos ^2\left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)^3}{4 d (b-a)^4 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{(2 a-b) \tan ^3(c+d x) \log \left (\sin ^2\left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)^3}{4 d (a+b)^4 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{\tan ^3(c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)^3}{8 d (a+b)^3 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{\tan ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)^3}{8 d (b-a)^3 (a \sin (c+d x)+b \tan (c+d x))^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.177, size = 324, normalized size = 1.4 \begin{align*} -{\frac{1}{4\,d \left ( a-b \right ) ^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) }}+{\frac{b\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{4\, \left ( a-b \right ) ^{4}d}}+{\frac{a\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{2\, \left ( a-b \right ) ^{4}d}}+{\frac{1}{4\,d \left ( a+b \right ) ^{3} \left ( -1+\cos \left ( dx+c \right ) \right ) }}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) a}{2\,d \left ( a+b \right ) ^{4}}}-{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) b}{4\,d \left ( a+b \right ) ^{4}}}-{\frac{{a}^{5}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}-8\,{\frac{{a}^{3}{b}^{2}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}-3\,{\frac{a{b}^{4}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}+{\frac{a{b}^{2}}{2\,d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2} \left ( b+a\cos \left ( dx+c \right ) \right ) ^{2}}}-2\,{\frac{{a}^{3}b}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( b+a\cos \left ( dx+c \right ) \right ) }}-2\,{\frac{a{b}^{3}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( b+a\cos \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.19394, size = 813, normalized size = 3.52 \begin{align*} -\frac{\frac{8 \,{\left (a^{5} + 8 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left (a + b - \frac{{\left (a - b\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac{4 \,{\left (2 \, a - b\right )} \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac{a^{6} - 2 \, a^{5} b - a^{4} b^{2} + 4 \, a^{3} b^{3} - a^{2} b^{4} - 2 \, a b^{5} + b^{6} - \frac{2 \,{\left (a^{6} - 20 \, a^{5} b - 11 \, a^{4} b^{2} - 24 \, a^{3} b^{3} - 29 \, a^{2} b^{4} + 4 \, a b^{5} - b^{6}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{{\left (a^{6} - 38 \, a^{5} b + 31 \, a^{4} b^{2} - 52 \, a^{3} b^{3} + 63 \, a^{2} b^{4} - 6 \, a b^{5} + b^{6}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{\frac{{\left (a^{9} + a^{8} b - 4 \, a^{7} b^{2} - 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 6 \, a^{4} b^{5} - 4 \, a^{3} b^{6} - 4 \, a^{2} b^{7} + a b^{8} + b^{9}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{2 \,{\left (a^{9} - a^{8} b - 4 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} - 6 \, a^{4} b^{5} - 4 \, a^{3} b^{6} + 4 \, a^{2} b^{7} + a b^{8} - b^{9}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{{\left (a^{9} - 3 \, a^{8} b + 8 \, a^{6} b^{3} - 6 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + 8 \, a^{3} b^{6} - 3 \, a b^{8} + b^{9}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac{\sin \left (d x + c\right )^{2}}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.17415, size = 2338, normalized size = 10.12 \begin{align*} \text{result too large to display} \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*} \int \frac{\sec{\left (c + d x \right )}}{\left (a \sin{\left (c + d x \right )} + b \tan{\left (c + d x \right )}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.3857, size = 1081, normalized size = 4.68 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]