Optimal. Leaf size=321 \[ -\frac {\left (8 a^2-5 a b-b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^5}-\frac {\left (8 a^2+5 a b-b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^5}+\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 d \left (a^2-b^2\right )^3}-\frac {a^5}{2 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}-\frac {a^4 \left (a^2+5 b^2\right )}{d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))}+\frac {a^3 \left (a^4+13 a^2 b^2+10 b^4\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^5}-\frac {\sec ^2(c+d x) \left (8 a^3 \left (a^2+5 b^2\right )-b \left (27 a^4+22 a^2 b^2-b^4\right ) \sin (c+d x)\right )}{8 d \left (a^2-b^2\right )^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.88, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2721, 1647, 1629} \[ -\frac {a^5}{2 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}-\frac {a^4 \left (a^2+5 b^2\right )}{d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))}+\frac {a^3 \left (13 a^2 b^2+a^4+10 b^4\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^5}-\frac {\left (8 a^2-5 a b-b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^5}-\frac {\left (8 a^2+5 a b-b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^5}+\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 d \left (a^2-b^2\right )^3}-\frac {\sec ^2(c+d x) \left (8 a^3 \left (a^2+5 b^2\right )-b \left (22 a^2 b^2+27 a^4-b^4\right ) \sin (c+d x)\right )}{8 d \left (a^2-b^2\right )^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1629
Rule 1647
Rule 2721
Rubi steps
\begin {align*} \int \frac {\tan ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^5}{(a+x)^3 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 \left (a^2-b^2\right )^3 d}+\frac {\operatorname {Subst}\left (\int \frac {\frac {a^3 b^6 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3}-\frac {a^2 b^4 \left (4 a^4+3 a^2 b^2-3 b^4\right ) x}{\left (a^2-b^2\right )^3}-\frac {a b^6 \left (23 a^2-3 b^2\right ) x^2}{\left (a^2-b^2\right )^3}-\frac {b^2 \left (4 a^6-12 a^4 b^2+21 a^2 b^4-b^6\right ) x^3}{\left (a^2-b^2\right )^3}}{(a+x)^3 \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 b^2 d}\\ &=\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 \left (a^2-b^2\right )^3 d}-\frac {\sec ^2(c+d x) \left (8 a^3 \left (a^2+5 b^2\right )-b \left (27 a^4+22 a^2 b^2-b^4\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^4 d}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {a^3 b^6 \left (21 a^4+26 a^2 b^2+b^4\right )}{\left (a^2-b^2\right )^4}+\frac {a^2 b^4 \left (8 a^6+a^4 b^2-54 a^2 b^4-3 b^6\right ) x}{\left (a^2-b^2\right )^4}+\frac {a b^6 \left (65 a^4-14 a^2 b^2-3 b^4\right ) x^2}{\left (a^2-b^2\right )^4}+\frac {b^6 \left (27 a^4+22 a^2 b^2-b^4\right ) x^3}{\left (a^2-b^2\right )^4}}{(a+x)^3 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 b^4 d}\\ &=\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 \left (a^2-b^2\right )^3 d}-\frac {\sec ^2(c+d x) \left (8 a^3 \left (a^2+5 b^2\right )-b \left (27 a^4+22 a^2 b^2-b^4\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^4 d}+\frac {\operatorname {Subst}\left (\int \left (-\frac {b^4 \left (-8 a^2+5 a b+b^2\right )}{2 (a+b)^5 (b-x)}+\frac {8 a^5 b^4}{\left (a^2-b^2\right )^3 (a+x)^3}+\frac {8 a^4 b^4 \left (a^2+5 b^2\right )}{\left (a^2-b^2\right )^4 (a+x)^2}+\frac {8 a^3 b^4 \left (a^4+13 a^2 b^2+10 b^4\right )}{\left (a^2-b^2\right )^5 (a+x)}-\frac {b^4 \left (8 a^2+5 a b-b^2\right )}{2 (a-b)^5 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{8 b^4 d}\\ &=-\frac {\left (8 a^2-5 a b-b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^5 d}-\frac {\left (8 a^2+5 a b-b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^5 d}+\frac {a^3 \left (a^4+13 a^2 b^2+10 b^4\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^5 d}-\frac {a^5}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}-\frac {a^4 \left (a^2+5 b^2\right )}{\left (a^2-b^2\right )^4 d (a+b \sin (c+d x))}+\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 \left (a^2-b^2\right )^3 d}-\frac {\sec ^2(c+d x) \left (8 a^3 \left (a^2+5 b^2\right )-b \left (27 a^4+22 a^2 b^2-b^4\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 6.34, size = 304, normalized size = 0.95 \[ -\frac {\left (8 a^2-5 a b-b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^5}-\frac {\left (8 a^2+5 a b-b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^5}-\frac {a^5}{2 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}-\frac {a^4 \left (a^2+5 b^2\right )}{d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))}+\frac {a^3 \left (a^4+13 a^2 b^2+10 b^4\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^5}-\frac {7 a+b}{16 d (a+b)^4 (1-\sin (c+d x))}-\frac {7 a-b}{16 d (a-b)^4 (\sin (c+d x)+1)}+\frac {1}{16 d (a+b)^3 (1-\sin (c+d x))^2}+\frac {1}{16 d (a-b)^3 (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.35, size = 981, normalized size = 3.06 \[ -\frac {4 \, a^{9} - 16 \, a^{7} b^{2} + 24 \, a^{5} b^{4} - 16 \, a^{3} b^{6} + 4 \, a b^{8} - 4 \, {\left (6 \, a^{9} + 35 \, a^{7} b^{2} - 39 \, a^{5} b^{4} - 3 \, a^{3} b^{6} + a b^{8}\right )} \cos \left (d x + c\right )^{4} - 16 \, {\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} \cos \left (d x + c\right )^{2} - 16 \, {\left ({\left (a^{7} b^{2} + 13 \, a^{5} b^{4} + 10 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (a^{8} b + 13 \, a^{6} b^{3} + 10 \, a^{4} b^{5}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (a^{9} + 14 \, a^{7} b^{2} + 23 \, a^{5} b^{4} + 10 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left ({\left (8 \, a^{7} b^{2} + 45 \, a^{6} b^{3} + 104 \, a^{5} b^{4} + 125 \, a^{4} b^{5} + 80 \, a^{3} b^{6} + 23 \, a^{2} b^{7} - b^{9}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (8 \, a^{8} b + 45 \, a^{7} b^{2} + 104 \, a^{6} b^{3} + 125 \, a^{5} b^{4} + 80 \, a^{4} b^{5} + 23 \, a^{3} b^{6} - a b^{8}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (8 \, a^{9} + 45 \, a^{8} b + 112 \, a^{7} b^{2} + 170 \, a^{6} b^{3} + 184 \, a^{5} b^{4} + 148 \, a^{4} b^{5} + 80 \, a^{3} b^{6} + 22 \, a^{2} b^{7} - b^{9}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (8 \, a^{7} b^{2} - 45 \, a^{6} b^{3} + 104 \, a^{5} b^{4} - 125 \, a^{4} b^{5} + 80 \, a^{3} b^{6} - 23 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (8 \, a^{8} b - 45 \, a^{7} b^{2} + 104 \, a^{6} b^{3} - 125 \, a^{5} b^{4} + 80 \, a^{4} b^{5} - 23 \, a^{3} b^{6} + a b^{8}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (8 \, a^{9} - 45 \, a^{8} b + 112 \, a^{7} b^{2} - 170 \, a^{6} b^{3} + 184 \, a^{5} b^{4} - 148 \, a^{4} b^{5} + 80 \, a^{3} b^{6} - 22 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, a^{8} b - 8 \, a^{6} b^{3} + 12 \, a^{4} b^{5} - 8 \, a^{2} b^{7} + 2 \, b^{9} + {\left (8 \, a^{8} b + 59 \, a^{6} b^{3} - 45 \, a^{4} b^{5} - 23 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{4} - {\left (11 \, a^{8} b - 36 \, a^{6} b^{3} + 42 \, a^{4} b^{5} - 20 \, a^{2} b^{7} + 3 \, b^{9}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left ({\left (a^{10} b^{2} - 5 \, a^{8} b^{4} + 10 \, a^{6} b^{6} - 10 \, a^{4} b^{8} + 5 \, a^{2} b^{10} - b^{12}\right )} d \cos \left (d x + c\right )^{6} - 2 \, {\left (a^{11} b - 5 \, a^{9} b^{3} + 10 \, a^{7} b^{5} - 10 \, a^{5} b^{7} + 5 \, a^{3} b^{9} - a b^{11}\right )} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (a^{12} - 4 \, a^{10} b^{2} + 5 \, a^{8} b^{4} - 5 \, a^{4} b^{8} + 4 \, a^{2} b^{10} - b^{12}\right )} d \cos \left (d x + c\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 6.81, size = 585, normalized size = 1.82 \[ \frac {\frac {16 \, {\left (a^{7} b + 13 \, a^{5} b^{3} + 10 \, a^{3} b^{5}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{10} b - 5 \, a^{8} b^{3} + 10 \, a^{6} b^{5} - 10 \, a^{4} b^{7} + 5 \, a^{2} b^{9} - b^{11}} - \frac {{\left (8 \, a^{2} + 5 \, a b - b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}} - \frac {{\left (8 \, a^{2} - 5 \, a b - b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}} - \frac {2 \, {\left (8 \, a^{6} b \sin \left (d x + c\right )^{5} + 67 \, a^{4} b^{3} \sin \left (d x + c\right )^{5} + 22 \, a^{2} b^{5} \sin \left (d x + c\right )^{5} - b^{7} \sin \left (d x + c\right )^{5} + 12 \, a^{7} \sin \left (d x + c\right )^{4} + 82 \, a^{5} b^{2} \sin \left (d x + c\right )^{4} + 4 \, a^{3} b^{4} \sin \left (d x + c\right )^{4} - 2 \, a b^{6} \sin \left (d x + c\right )^{4} - 5 \, a^{6} b \sin \left (d x + c\right )^{3} - 159 \, a^{4} b^{3} \sin \left (d x + c\right )^{3} - 27 \, a^{2} b^{5} \sin \left (d x + c\right )^{3} - b^{7} \sin \left (d x + c\right )^{3} - 32 \, a^{7} \sin \left (d x + c\right )^{2} - 148 \, a^{5} b^{2} \sin \left (d x + c\right )^{2} - 16 \, a^{3} b^{4} \sin \left (d x + c\right )^{2} + 4 \, a b^{6} \sin \left (d x + c\right )^{2} - a^{6} b \sin \left (d x + c\right ) + 86 \, a^{4} b^{3} \sin \left (d x + c\right ) + 11 \, a^{2} b^{5} \sin \left (d x + c\right ) + 18 \, a^{7} + 72 \, a^{5} b^{2} + 6 \, a^{3} b^{4}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2} - b \sin \left (d x + c\right ) - a\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.33, size = 465, normalized size = 1.45 \[ \frac {1}{16 d \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {b}{16 d \left (a +b \right )^{4} \left (\sin \left (d x +c \right )-1\right )}+\frac {7 a}{16 d \left (a +b \right )^{4} \left (\sin \left (d x +c \right )-1\right )}-\frac {\ln \left (\sin \left (d x +c \right )-1\right ) a^{2}}{2 d \left (a +b \right )^{5}}+\frac {5 \ln \left (\sin \left (d x +c \right )-1\right ) a b}{16 d \left (a +b \right )^{5}}+\frac {\ln \left (\sin \left (d x +c \right )-1\right ) b^{2}}{16 d \left (a +b \right )^{5}}-\frac {a^{5}}{2 d \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{2}}+\frac {a^{7} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \left (a +b \right )^{5} \left (a -b \right )^{5}}+\frac {13 a^{5} \ln \left (a +b \sin \left (d x +c \right )\right ) b^{2}}{d \left (a +b \right )^{5} \left (a -b \right )^{5}}+\frac {10 a^{3} \ln \left (a +b \sin \left (d x +c \right )\right ) b^{4}}{d \left (a +b \right )^{5} \left (a -b \right )^{5}}-\frac {a^{6}}{d \left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )}-\frac {5 a^{4} b^{2}}{d \left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )}+\frac {1}{16 d \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {b}{16 d \left (a -b \right )^{4} \left (1+\sin \left (d x +c \right )\right )}-\frac {7 a}{16 d \left (a -b \right )^{4} \left (1+\sin \left (d x +c \right )\right )}-\frac {\ln \left (1+\sin \left (d x +c \right )\right ) a^{2}}{2 d \left (a -b \right )^{5}}-\frac {5 \ln \left (1+\sin \left (d x +c \right )\right ) a b}{16 d \left (a -b \right )^{5}}+\frac {\ln \left (1+\sin \left (d x +c \right )\right ) b^{2}}{16 d \left (a -b \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.44, size = 730, normalized size = 2.27 \[ \frac {\frac {16 \, {\left (a^{7} + 13 \, a^{5} b^{2} + 10 \, a^{3} b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{10} - 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} - 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} - b^{10}} - \frac {{\left (8 \, a^{2} + 5 \, a b - b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}} - \frac {{\left (8 \, a^{2} - 5 \, a b - b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}} - \frac {2 \, {\left (18 \, a^{7} + 72 \, a^{5} b^{2} + 6 \, a^{3} b^{4} + {\left (8 \, a^{6} b + 67 \, a^{4} b^{3} + 22 \, a^{2} b^{5} - b^{7}\right )} \sin \left (d x + c\right )^{5} + 2 \, {\left (6 \, a^{7} + 41 \, a^{5} b^{2} + 2 \, a^{3} b^{4} - a b^{6}\right )} \sin \left (d x + c\right )^{4} - {\left (5 \, a^{6} b + 159 \, a^{4} b^{3} + 27 \, a^{2} b^{5} + b^{7}\right )} \sin \left (d x + c\right )^{3} - 4 \, {\left (8 \, a^{7} + 37 \, a^{5} b^{2} + 4 \, a^{3} b^{4} - a b^{6}\right )} \sin \left (d x + c\right )^{2} - {\left (a^{6} b - 86 \, a^{4} b^{3} - 11 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )\right )}}{a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right )} \sin \left (d x + c\right )^{6} + 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \sin \left (d x + c\right )^{5} + {\left (a^{10} - 6 \, a^{8} b^{2} + 14 \, a^{6} b^{4} - 16 \, a^{4} b^{6} + 9 \, a^{2} b^{8} - 2 \, b^{10}\right )} \sin \left (d x + c\right )^{4} - 4 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \sin \left (d x + c\right )^{3} - {\left (2 \, a^{10} - 9 \, a^{8} b^{2} + 16 \, a^{6} b^{4} - 14 \, a^{4} b^{6} + 6 \, a^{2} b^{8} - b^{10}\right )} \sin \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \sin \left (d x + c\right )}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 10.74, size = 1229, normalized size = 3.83 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{5}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________