Integrand size = 23, antiderivative size = 288 \[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {2 \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{7/3} d}+\frac {2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\sin (c+d x)}{b^2 d}-\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )} \]
[Out]
Time = 0.39 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3302, 1872, 1901, 1874, 31, 648, 631, 210, 642} \[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {\sin (c+d x) \left (a^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)-b^2\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}-\frac {2 \left (3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{7/3} d}-\frac {\left (-3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{7/3} d}+\frac {2 \left (-3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\sin (c+d x)}{b^2 d} \]
[In]
[Out]
Rule 31
Rule 210
Rule 631
Rule 642
Rule 648
Rule 1872
Rule 1874
Rule 1901
Rule 3302
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{\left (a+b x^3\right )^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-a^2-2 b^2-6 a b x+3 a b x^3}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{3 a b^2 d} \\ & = -\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}-\frac {\text {Subst}\left (\int \left (3 a-\frac {2 \left (2 a^2+b^2+3 a b x\right )}{a+b x^3}\right ) \, dx,x,\sin (c+d x)\right )}{3 a b^2 d} \\ & = -\frac {\sin (c+d x)}{b^2 d}-\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac {2 \text {Subst}\left (\int \frac {2 a^2+b^2+3 a b x}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{3 a b^2 d} \\ & = -\frac {\sin (c+d x)}{b^2 d}-\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac {2 \text {Subst}\left (\int \frac {\sqrt [3]{a} \left (3 a^{4/3} b+2 \sqrt [3]{b} \left (2 a^2+b^2\right )\right )+\sqrt [3]{b} \left (3 a^{4/3} b-\sqrt [3]{b} \left (2 a^2+b^2\right )\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}+\frac {\left (2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} b^2 d} \\ & = \frac {2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\sin (c+d x)}{b^2 d}-\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}-\frac {\left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}+\frac {\left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{3 a^{4/3} b^2 d} \\ & = \frac {2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\sin (c+d x)}{b^2 d}-\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac {\left (2 \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )}{3 a^{5/3} b^{7/3} d} \\ & = -\frac {2 \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{5/3} b^{7/3} d}+\frac {2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac {\sin (c+d x)}{b^2 d}-\frac {\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 3.13 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.52 \[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\frac {4 \sqrt {3} \left (a^2-b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3} b^{7/3}}+\frac {6 \sqrt [3]{-1} \left (2 \sqrt [3]{-1} a^{2/3}+3 b^{2/3}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)\right )}{\sqrt [3]{a} b^{7/3}}+\frac {6 \left (2 a^{2/3}-3 b^{2/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{\sqrt [3]{a} b^{7/3}}-\frac {4 \left (a^2-b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{a^{5/3} b^{7/3}}-\frac {6 \sqrt [3]{-1} \left (2 a^{2/3}+3 \sqrt [3]{-1} b^{2/3}\right ) \log \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)\right )}{\sqrt [3]{a} b^{7/3}}+\frac {2 \left (a^2-b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{a^{5/3} b^{7/3}}-\frac {18 \sin (c+d x)}{b^2}-\frac {27 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},2,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{a b}+\frac {18}{b \left (a+b \sin ^3(c+d x)\right )}+\frac {6 \left (1-\frac {a^2}{b^2}\right ) \sin (c+d x)}{a \left (a+b \sin ^3(c+d x)\right )}}{18 d} \]
[In]
[Out]
Time = 2.44 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(-\frac {\frac {\sin \left (d x +c \right )}{b^{2}}-\frac {\frac {-\left (\sin ^{2}\left (d x +c \right )\right ) b -\frac {\left (a^{2}-b^{2}\right ) \sin \left (d x +c \right )}{3 a}+b}{a +b \left (\sin ^{3}\left (d x +c \right )\right )}+\frac {\frac {2 \left (2 a^{2}+b^{2}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+2 a b \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a}}{b^{2}}}{d}\) | \(311\) |
default | \(-\frac {\frac {\sin \left (d x +c \right )}{b^{2}}-\frac {\frac {-\left (\sin ^{2}\left (d x +c \right )\right ) b -\frac {\left (a^{2}-b^{2}\right ) \sin \left (d x +c \right )}{3 a}+b}{a +b \left (\sin ^{3}\left (d x +c \right )\right )}+\frac {\frac {2 \left (2 a^{2}+b^{2}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+2 a b \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a}}{b^{2}}}{d}\) | \(311\) |
risch | \(\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 b^{2} d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 b^{2} d}-\frac {2 i \left (3 a b \,{\mathrm e}^{5 i \left (d x +c \right )}+6 a b \,{\mathrm e}^{3 i \left (d x +c \right )}+2 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-2 i b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3 a b \,{\mathrm e}^{i \left (d x +c \right )}-2 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+2 i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 a \,b^{2} d \left (b \,{\mathrm e}^{6 i \left (d x +c \right )}-3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i a \,{\mathrm e}^{3 i \left (d x +c \right )}-b \right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (729 a^{5} b^{7} d^{3} \textit {\_Z}^{3}+\left (648 a^{5} b^{3} d +324 a^{3} b^{5} d \right ) \textit {\_Z} -64 a^{6}+120 a^{4} b^{2}-48 a^{2} b^{4}-8 b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {243 i a^{5} b^{5} d^{2} \textit {\_R}^{2}}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}+\left (\frac {72 i a^{6} b^{2} d}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}+\frac {72 i a^{4} b^{4} d}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}+\frac {18 i a^{2} b^{6} d}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}\right ) \textit {\_R} +\frac {144 i a^{5} b}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}+\frac {72 i a^{3} b^{3}}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}\right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {16 a^{6}}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}-\frac {78 a^{4} b^{2}}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}-\frac {12 a^{2} b^{4}}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}-\frac {2 b^{6}}{16 a^{6}+78 a^{4} b^{2}+12 a^{2} b^{4}+2 b^{6}}\right )\right )\) | \(674\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.55 (sec) , antiderivative size = 3624, normalized size of antiderivative = 12.58 \[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {\frac {3 \, {\left (3 \, a b \sin \left (d x + c\right )^{2} - 3 \, a b + {\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )\right )}}{a b^{3} \sin \left (d x + c\right )^{3} + a^{2} b^{2}} + \frac {9 \, \sin \left (d x + c\right )}{b^{2}} - \frac {2 \, \sqrt {3} {\left (3 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{2} + b^{2}\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (3 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{2} - b^{2}\right )} \log \left (\sin \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, {\left (3 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{2} - b^{2}\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{9 \, d} \]
[In]
[Out]
\[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int { \frac {\cos \left (d x + c\right )^{7}}{{\left (b \sin \left (d x + c\right )^{3} + a\right )}^{2}} \,d x } \]
[In]
[Out]
Time = 0.41 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.33 \[ \int \frac {\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\sum _{k=1}^3\ln \left (\frac {8\,a^2+4\,b^2+{\mathrm {root}\left (729\,a^5\,b^7\,d^3+648\,a^5\,b^3\,d+324\,a^3\,b^5\,d+120\,a^4\,b^2-48\,a^2\,b^4-8\,b^6-64\,a^6,d,k\right )}^2\,a^2\,b^4\,27+12\,a\,b\,\sin \left (c+d\,x\right )+\mathrm {root}\left (729\,a^5\,b^7\,d^3+648\,a^5\,b^3\,d+324\,a^3\,b^5\,d+120\,a^4\,b^2-48\,a^2\,b^4-8\,b^6-64\,a^6,d,k\right )\,b^4\,\sin \left (c+d\,x\right )\,6+\mathrm {root}\left (729\,a^5\,b^7\,d^3+648\,a^5\,b^3\,d+324\,a^3\,b^5\,d+120\,a^4\,b^2-48\,a^2\,b^4-8\,b^6-64\,a^6,d,k\right )\,a^2\,b^2\,\sin \left (c+d\,x\right )\,12}{a\,b^2\,3}\right )\,\mathrm {root}\left (729\,a^5\,b^7\,d^3+648\,a^5\,b^3\,d+324\,a^3\,b^5\,d+120\,a^4\,b^2-48\,a^2\,b^4-8\,b^6-64\,a^6,d,k\right )}{d}-\frac {\sin \left (c+d\,x\right )}{b^2\,d}-\frac {b\,{\sin \left (c+d\,x\right )}^2-b+\frac {\sin \left (c+d\,x\right )\,\left (a^2-b^2\right )}{3\,a}}{d\,\left (b^3\,{\sin \left (c+d\,x\right )}^3+a\,b^2\right )} \]
[In]
[Out]