Optimal. Leaf size=400 \[ \frac {8 a^2 \tanh ^{-1}(\sin (c+d x))}{b^6 d}-\frac {4 a^2}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}+\frac {2 \left (a^2+b^2\right ) \tanh ^{-1}(\sin (c+d x))}{b^6 d}+\frac {6 a \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {2 \left (a^2+b^2\right )}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}+\frac {3 a \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 b^4 d \sqrt {a^2+b^2}}-\frac {a^2+b^2}{3 b^3 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {4 a^3 \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d \sqrt {a^2+b^2}}-\frac {4 a \sec (c+d x)}{b^5 d}+\frac {3 a (b \cos (c+d x)-a \sin (c+d x))}{2 b^4 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac {\tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 b^4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.80, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3106, 3094, 3770, 3074, 206, 3076, 3768, 3104} \[ \frac {8 a^2 \tanh ^{-1}(\sin (c+d x))}{b^6 d}+\frac {2 \left (a^2+b^2\right ) \tanh ^{-1}(\sin (c+d x))}{b^6 d}-\frac {4 a^2}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}-\frac {2 \left (a^2+b^2\right )}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}-\frac {a^2+b^2}{3 b^3 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {4 a^3 \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d \sqrt {a^2+b^2}}+\frac {6 a \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}+\frac {3 a \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 b^4 d \sqrt {a^2+b^2}}-\frac {4 a \sec (c+d x)}{b^5 d}+\frac {3 a (b \cos (c+d x)-a \sin (c+d x))}{2 b^4 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac {\tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 b^4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 3074
Rule 3076
Rule 3094
Rule 3104
Rule 3106
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx &=\frac {\int \frac {\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx}{b^2}-\frac {(2 a) \int \frac {\sec ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx}{b^2}+\frac {\left (a^2+b^2\right ) \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx}{b^2}\\ &=-\frac {a^2+b^2}{3 b^3 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {\int \sec ^3(c+d x) \, dx}{b^4}-2 \frac {(2 a) \int \frac {\sec ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^4}+\frac {\left (4 a^2\right ) \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx}{b^4}+2 \frac {\left (a^2+b^2\right ) \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx}{b^4}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx}{b^4}-\frac {\left (2 a \left (a^2+b^2\right )\right ) \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx}{b^4}\\ &=-\frac {a^2+b^2}{3 b^3 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {3 a (b \cos (c+d x)-a \sin (c+d x))}{2 b^4 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac {4 a^2}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}+\frac {\sec (c+d x) \tan (c+d x)}{2 b^4 d}+\frac {\left (4 a^2\right ) \int \sec (c+d x) \, dx}{b^6}-\frac {\left (4 a^3\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^6}+\frac {\int \sec (c+d x) \, dx}{2 b^4}-\frac {a \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{2 b^4}-\frac {a \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^4}+2 \left (-\frac {a^2+b^2}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}+\frac {\left (a^2+b^2\right ) \int \sec (c+d x) \, dx}{b^6}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^6}\right )-2 \left (\frac {2 a \sec (c+d x)}{b^5 d}-\frac {\left (2 a^2\right ) \int \sec (c+d x) \, dx}{b^6}+\frac {\left (2 a \left (a^2+b^2\right )\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^6}\right )\\ &=\frac {4 a^2 \tanh ^{-1}(\sin (c+d x))}{b^6 d}+\frac {\tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac {a^2+b^2}{3 b^3 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {3 a (b \cos (c+d x)-a \sin (c+d x))}{2 b^4 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac {4 a^2}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}+\frac {\sec (c+d x) \tan (c+d x)}{2 b^4 d}+\frac {\left (4 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{b^6 d}+\frac {a \operatorname {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{2 b^4 d}+\frac {a \operatorname {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{b^4 d}+2 \left (\frac {\left (a^2+b^2\right ) \tanh ^{-1}(\sin (c+d x))}{b^6 d}-\frac {a^2+b^2}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}+\frac {\left (a \left (a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{b^6 d}\right )-2 \left (-\frac {2 a^2 \tanh ^{-1}(\sin (c+d x))}{b^6 d}+\frac {2 a \sec (c+d x)}{b^5 d}-\frac {\left (2 a \left (a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{b^6 d}\right )\\ &=\frac {4 a^2 \tanh ^{-1}(\sin (c+d x))}{b^6 d}+\frac {\tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac {4 a^3 \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 \sqrt {a^2+b^2} d}+\frac {3 a \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 b^4 \sqrt {a^2+b^2} d}-2 \left (-\frac {2 a^2 \tanh ^{-1}(\sin (c+d x))}{b^6 d}-\frac {2 a \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}+\frac {2 a \sec (c+d x)}{b^5 d}\right )-\frac {a^2+b^2}{3 b^3 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {3 a (b \cos (c+d x)-a \sin (c+d x))}{2 b^4 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac {4 a^2}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}+2 \left (\frac {\left (a^2+b^2\right ) \tanh ^{-1}(\sin (c+d x))}{b^6 d}+\frac {a \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^2+b^2}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}\right )+\frac {\sec (c+d x) \tan (c+d x)}{2 b^4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 3.47, size = 538, normalized size = 1.34 \[ -\frac {\sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (18 b^2 \left (a^2+b^2\right ) \sin (c+d x) (a \cos (c+d x)+b \sin (c+d x))+6 b \left (12 a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2+30 \left (4 a^2+b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^3-30 \left (4 a^2+b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^3+\frac {60 a \left (4 a^2+3 b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^3 \tanh ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )-b}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}+4 b^3 \left (a^2+b^2\right )-\frac {3 b^2 (a \cos (c+d x)+b \sin (c+d x))^3}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {3 b^2 (a \cos (c+d x)+b \sin (c+d x))^3}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+48 a b (a \cos (c+d x)+b \sin (c+d x))^3+\frac {48 a b \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^3}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {48 a b \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^3}{\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}\right )}{12 b^6 d (a+b \tan (c+d x))^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.81, size = 820, normalized size = 2.05 \[ \frac {6 \, a^{2} b^{5} + 6 \, b^{7} - 30 \, {\left (4 \, a^{6} b - 3 \, a^{4} b^{3} - 8 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{4} - 20 \, {\left (11 \, a^{4} b^{3} + 13 \, a^{2} b^{5} + 2 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left ({\left (4 \, a^{6} - 9 \, a^{4} b^{2} - 9 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{5} + 3 \, {\left (4 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left ({\left (12 \, a^{5} b + 5 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) + 15 \, {\left ({\left (4 \, a^{7} - 7 \, a^{5} b^{2} - 14 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \cos \left (d x + c\right )^{5} + 3 \, {\left (4 \, a^{5} b^{2} + 5 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left ({\left (12 \, a^{6} b + 11 \, a^{4} b^{3} - 2 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{4} b^{3} + 5 \, a^{2} b^{5} + b^{7}\right )} \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 ({\left (4 \, a^{7} - 7 \, a^{5} b^{2} - 14 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \cos \left (d x + c\right )^{5} + 3 \, {\left (4 \, a^{5} b^{2} + 5 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left ({\left (12 \, a^{6} b + 11 \, a^{4} b^{3} - 2 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{4} b^{3} + 5 \, a^{2} b^{5} + b^{7}\right )} \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 (10 \, {\left (a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left ({\left (a^{5} b^{6} - 2 \, a^{3} b^{8} - 3 \, a b^{10}\right )} d \cos \left (d x + c\right )^{5} + 3 \, {\left (a^{3} b^{8} + a b^{10}\right )} d \cos \left (d x + c\right )^{3} + {\left ({\left (3 \, a^{4} b^{7} + 2 \, a^{2} b^{9} - b^{11}\right )} d \cos \left (d x + c\right )^{4} + {\left (a^{2} b^{9} + b^{11}\right )} d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 3.10, size = 548, normalized size = 1.37 \[ \frac {\frac {15 \, {\left (4 \, a^{2} + b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{6}} - \frac {15 \, {\left (4 \, a^{2} + b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{6}} + \frac {15 \, {\left (4 \, a^{3} + 3 \, a b^{2}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{6}} + \frac {6 \, {\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} b^{5}} + \frac {2 \, {\left (27 \, a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 117 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 12 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 216 \, a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 114 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 300 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 54 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 189 \, a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 30 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a^{7} + 5 \, a^{5} b^{2} + 2 \, a^{3} b^{4}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}^{3} a^{3} b^{5}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.47, size = 1255, normalized size = 3.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.48, size = 936, normalized size = 2.34 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.59, size = 1961, normalized size = 4.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{3}{\left (c + d x \right )}}{\left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________