Integrand size = 21, antiderivative size = 333 \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {3 \left (a^4-24 a^2 b^2+40 b^4\right ) x}{8 a^7}-\frac {3 b \left (2 a^4-11 a^2 b^2+10 b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^7 \sqrt {a-b} \sqrt {a+b} d}+\frac {b \left (13 a^2-30 b^2\right ) \sin (c+d x)}{2 a^6 d}-\frac {3 \left (7 a^2-20 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^5 d}+\frac {\left (3 a^2-10 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^4 b d}-\frac {\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac {\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac {\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))} \]
[Out]
Time = 1.26 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3957, 2970, 3128, 3102, 2814, 2738, 214} \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\left (2 a^2-7 b^2\right ) \sin (c+d x) \cos ^4(c+d x)}{2 a^2 b^2 d (a \cos (c+d x)+b)}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^4(c+d x)}{2 a^2 b d (a \cos (c+d x)+b)^2}+\frac {b \left (13 a^2-30 b^2\right ) \sin (c+d x)}{2 a^6 d}-\frac {3 \left (7 a^2-20 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 a^5 d}+\frac {\left (3 a^2-10 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{2 a^4 b d}-\frac {\left (4 a^2-15 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a^3 b^2 d}-\frac {3 b \left (2 a^4-11 a^2 b^2+10 b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^7 d \sqrt {a-b} \sqrt {a+b}}+\frac {3 x \left (a^4-24 a^2 b^2+40 b^4\right )}{8 a^7} \]
[In]
[Out]
Rule 214
Rule 2738
Rule 2814
Rule 2970
Rule 3102
Rule 3128
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos ^3(c+d x) \sin ^4(c+d x)}{(-b-a \cos (c+d x))^3} \, dx \\ & = -\frac {\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac {\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))}+\frac {\int \frac {\cos ^3(c+d x) \left (-6 \left (a^2-4 b^2\right )-a b \cos (c+d x)+2 \left (4 a^2-15 b^2\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{2 a^2 b^2} \\ & = -\frac {\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac {\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac {\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))}-\frac {\int \frac {\cos ^2(c+d x) \left (-6 b \left (4 a^2-15 b^2\right )-6 a b^2 \cos (c+d x)+12 b \left (3 a^2-10 b^2\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{8 a^3 b^2} \\ & = \frac {\left (3 a^2-10 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^4 b d}-\frac {\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac {\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac {\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))}+\frac {\int \frac {\cos (c+d x) \left (-24 b^2 \left (3 a^2-10 b^2\right )-30 a b^3 \cos (c+d x)+18 b^2 \left (7 a^2-20 b^2\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{24 a^4 b^2} \\ & = -\frac {3 \left (7 a^2-20 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^5 d}+\frac {\left (3 a^2-10 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^4 b d}-\frac {\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac {\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac {\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))}-\frac {\int \frac {-18 b^3 \left (7 a^2-20 b^2\right )+6 a b^2 \left (3 a^2-20 b^2\right ) \cos (c+d x)+24 b^3 \left (13 a^2-30 b^2\right ) \cos ^2(c+d x)}{-b-a \cos (c+d x)} \, dx}{48 a^5 b^2} \\ & = \frac {b \left (13 a^2-30 b^2\right ) \sin (c+d x)}{2 a^6 d}-\frac {3 \left (7 a^2-20 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^5 d}+\frac {\left (3 a^2-10 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^4 b d}-\frac {\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac {\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac {\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))}+\frac {\int \frac {18 a b^3 \left (7 a^2-20 b^2\right )-18 b^2 \left (a^4-24 a^2 b^2+40 b^4\right ) \cos (c+d x)}{-b-a \cos (c+d x)} \, dx}{48 a^6 b^2} \\ & = \frac {3 \left (a^4-24 a^2 b^2+40 b^4\right ) x}{8 a^7}+\frac {b \left (13 a^2-30 b^2\right ) \sin (c+d x)}{2 a^6 d}-\frac {3 \left (7 a^2-20 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^5 d}+\frac {\left (3 a^2-10 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^4 b d}-\frac {\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac {\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac {\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))}+\frac {\left (3 b \left (2 a^4-11 a^2 b^2+10 b^4\right )\right ) \int \frac {1}{-b-a \cos (c+d x)} \, dx}{2 a^7} \\ & = \frac {3 \left (a^4-24 a^2 b^2+40 b^4\right ) x}{8 a^7}+\frac {b \left (13 a^2-30 b^2\right ) \sin (c+d x)}{2 a^6 d}-\frac {3 \left (7 a^2-20 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^5 d}+\frac {\left (3 a^2-10 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^4 b d}-\frac {\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac {\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac {\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))}+\frac {\left (3 b \left (2 a^4-11 a^2 b^2+10 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{-a-b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^7 d} \\ & = \frac {3 \left (a^4-24 a^2 b^2+40 b^4\right ) x}{8 a^7}-\frac {3 b \left (2 a^4-11 a^2 b^2+10 b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^7 \sqrt {a-b} \sqrt {a+b} d}+\frac {b \left (13 a^2-30 b^2\right ) \sin (c+d x)}{2 a^6 d}-\frac {3 \left (7 a^2-20 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^5 d}+\frac {\left (3 a^2-10 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^4 b d}-\frac {\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac {\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac {\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1178\) vs. \(2(333)=666\).
Time = 6.86 (sec) , antiderivative size = 1178, normalized size of antiderivative = 3.54 \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {-\frac {6 \left (8 (c+d x)+\frac {2 b \left (15 a^4-20 a^2 b^2+8 b^4\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {a b \left (3 a^2-4 b^2\right ) \sin (c+d x)}{(a-b) (a+b) (b+a \cos (c+d x))^2}-\frac {3 a \left (2 a^4-7 a^2 b^2+4 b^4\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (b+a \cos (c+d x))}\right )}{a^3}+\frac {6 \left (\frac {6 a b \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {\left (b \left (a^2+2 b^2\right )+a \left (2 a^2+b^2\right ) \cos (c+d x)\right ) \sin (c+d x)}{(b+a \cos (c+d x))^2}\right )}{(a-b)^2 (a+b)^2}-\frac {2 \left (-24 \left (a^2-8 b^2\right ) (c+d x)+\frac {6 b \left (-35 a^6+140 a^4 b^2-168 a^2 b^4+64 b^6\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-96 a b \sin (c+d x)+\frac {a b \left (-5 a^4+20 a^2 b^2-16 b^4\right ) \sin (c+d x)}{(a-b) (a+b) (b+a \cos (c+d x))^2}+\frac {a \left (10 a^6-115 a^4 b^2+220 a^2 b^4-112 b^6\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (b+a \cos (c+d x))}+8 a^2 \sin (2 (c+d x))\right )}{a^5}+\frac {\frac {12 b \left (105 a^8-840 a^6 b^2+2016 a^4 b^4-1920 a^2 b^6+640 b^8\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {48 a^{10} c-960 a^8 b^2 c+1776 a^6 b^4 c+2976 a^4 b^6 c-7680 a^2 b^8 c+3840 b^{10} c+48 a^{10} d x-960 a^8 b^2 d x+1776 a^6 b^4 d x+2976 a^4 b^6 d x-7680 a^2 b^8 d x+3840 b^{10} d x+192 a b \left (a^2-b^2\right )^2 \left (a^4-20 a^2 b^2+40 b^4\right ) (c+d x) \cos (c+d x)+48 \left (a^3-a b^2\right )^2 \left (a^4-20 a^2 b^2+40 b^4\right ) (c+d x) \cos (2 (c+d x))+114 a^9 b \sin (c+d x)+788 a^7 b^3 \sin (c+d x)-5696 a^5 b^5 \sin (c+d x)+8640 a^3 b^7 \sin (c+d x)-3840 a b^9 \sin (c+d x)-36 a^{10} \sin (2 (c+d x))+1221 a^8 b^2 \sin (2 (c+d x))-5182 a^6 b^4 \sin (2 (c+d x))+6880 a^4 b^6 \sin (2 (c+d x))-2880 a^2 b^8 \sin (2 (c+d x))+120 a^9 b \sin (3 (c+d x))-560 a^7 b^3 \sin (3 (c+d x))+760 a^5 b^5 \sin (3 (c+d x))-320 a^3 b^7 \sin (3 (c+d x))-8 a^{10} \sin (4 (c+d x))+56 a^8 b^2 \sin (4 (c+d x))-88 a^6 b^4 \sin (4 (c+d x))+40 a^4 b^6 \sin (4 (c+d x))-8 a^9 b \sin (5 (c+d x))+16 a^7 b^3 \sin (5 (c+d x))-8 a^5 b^5 \sin (5 (c+d x))+2 a^{10} \sin (6 (c+d x))-4 a^8 b^2 \sin (6 (c+d x))+2 a^6 b^4 \sin (6 (c+d x))}{\left (a^2-b^2\right )^2 (b+a \cos (c+d x))^2}}{a^7}}{256 d} \]
[In]
[Out]
Time = 2.45 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.18
method | result | size |
derivativedivides | \(\frac {\frac {\frac {2 \left (\left (\frac {3}{8} a^{4}+3 a^{3} b -3 a^{2} b^{2}-10 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (13 a^{3} b -3 a^{2} b^{2}-30 a \,b^{3}+\frac {11}{8} a^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {11}{8} a^{4}+3 a^{2} b^{2}+13 a^{3} b -30 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (3 a^{3} b -10 a \,b^{3}-\frac {3}{8} a^{4}+3 a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {3 \left (a^{4}-24 a^{2} b^{2}+40 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{7}}+\frac {2 b \left (\frac {\left (\frac {5}{2} a^{3} b^{2}-5 a \,b^{4}-3 a^{4} b +\frac {11}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {5}{2} a^{3} b^{2}-5 a \,b^{4}+3 a^{4} b -\frac {11}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {3 \left (2 a^{4}-11 a^{2} b^{2}+10 b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{7}}}{d}\) | \(392\) |
default | \(\frac {\frac {\frac {2 \left (\left (\frac {3}{8} a^{4}+3 a^{3} b -3 a^{2} b^{2}-10 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (13 a^{3} b -3 a^{2} b^{2}-30 a \,b^{3}+\frac {11}{8} a^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {11}{8} a^{4}+3 a^{2} b^{2}+13 a^{3} b -30 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (3 a^{3} b -10 a \,b^{3}-\frac {3}{8} a^{4}+3 a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {3 \left (a^{4}-24 a^{2} b^{2}+40 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{7}}+\frac {2 b \left (\frac {\left (\frac {5}{2} a^{3} b^{2}-5 a \,b^{4}-3 a^{4} b +\frac {11}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {5}{2} a^{3} b^{2}-5 a \,b^{4}+3 a^{4} b -\frac {11}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {3 \left (2 a^{4}-11 a^{2} b^{2}+10 b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{7}}}{d}\) | \(392\) |
risch | \(\frac {3 x}{8 a^{3}}-\frac {9 x \,b^{2}}{a^{5}}+\frac {15 x \,b^{4}}{a^{7}}-\frac {i b \,{\mathrm e}^{-3 i \left (d x +c \right )}}{8 a^{4} d}-\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} b^{2}}{4 a^{5} d}-\frac {15 i b \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{4} d}-\frac {i b^{2} \left (-7 a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+12 a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-6 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+22 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-17 a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}+32 b^{3} a \,{\mathrm e}^{i \left (d x +c \right )}-6 a^{4}+11 a^{2} b^{2}\right )}{a^{7} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{2}}-\frac {5 i b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{a^{6} d}+\frac {15 i b \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{4} d}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {i b \,{\mathrm e}^{3 i \left (d x +c \right )}}{8 a^{4} d}+\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} b^{2}}{4 a^{5} d}+\frac {5 i b^{3} {\mathrm e}^{i \left (d x +c \right )}}{a^{6} d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{3}}-\frac {33 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{2 \sqrt {a^{2}-b^{2}}\, d \,a^{5}}+\frac {15 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{7}}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{3}}+\frac {33 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{2 \sqrt {a^{2}-b^{2}}\, d \,a^{5}}-\frac {15 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{7}}+\frac {\sin \left (4 d x +4 c \right )}{32 d \,a^{3}}\) | \(843\) |
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 1041, normalized size of antiderivative = 3.13 \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\int \frac {\sin ^{4}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.42 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.75 \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (a^{4} - 24 \, a^{2} b^{2} + 40 \, b^{4}\right )} {\left (d x + c\right )}}{a^{7}} - \frac {24 \, {\left (2 \, a^{4} b - 11 \, a^{2} b^{3} + 10 \, b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} a^{7}} - \frac {8 \, {\left (6 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 11 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 10 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (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 )^{2} - a - b\right )}^{2} a^{6}} + \frac {2 \, {\left (3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 80 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 104 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 11 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 104 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 80 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{6}}}{8 \, d} \]
[In]
[Out]
Time = 18.29 (sec) , antiderivative size = 3255, normalized size of antiderivative = 9.77 \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
[In]
[Out]