Integrand size = 23, antiderivative size = 239 \[ \int \sec ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx=\frac {\operatorname {AppellF1}\left (\frac {1}{4},2,-p,\frac {5}{4},\sin ^4(e+f x),-\frac {b \sin ^4(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (1+\frac {b \sin ^4(e+f x)}{a}\right )^{-p}}{f}+\frac {2 \operatorname {AppellF1}\left (\frac {3}{4},2,-p,\frac {7}{4},\sin ^4(e+f x),-\frac {b \sin ^4(e+f x)}{a}\right ) \sin ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (1+\frac {b \sin ^4(e+f x)}{a}\right )^{-p}}{3 f}+\frac {\operatorname {AppellF1}\left (\frac {5}{4},2,-p,\frac {9}{4},\sin ^4(e+f x),-\frac {b \sin ^4(e+f x)}{a}\right ) \sin ^5(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (1+\frac {b \sin ^4(e+f x)}{a}\right )^{-p}}{5 f} \]
[Out]
Time = 0.25 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3302, 1254, 441, 440, 525, 524} \[ \int \sec ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx=\frac {\sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (\frac {b \sin ^4(e+f x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{4},2,-p,\frac {5}{4},\sin ^4(e+f x),-\frac {b \sin ^4(e+f x)}{a}\right )}{f}+\frac {\sin ^5(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (\frac {b \sin ^4(e+f x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{4},2,-p,\frac {9}{4},\sin ^4(e+f x),-\frac {b \sin ^4(e+f x)}{a}\right )}{5 f}+\frac {2 \sin ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (\frac {b \sin ^4(e+f x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{4},2,-p,\frac {7}{4},\sin ^4(e+f x),-\frac {b \sin ^4(e+f x)}{a}\right )}{3 f} \]
[In]
[Out]
Rule 440
Rule 441
Rule 524
Rule 525
Rule 1254
Rule 3302
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x^4\right )^p}{\left (1-x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\left (a+b x^4\right )^p}{\left (-1+x^4\right )^2}+\frac {2 x^2 \left (a+b x^4\right )^p}{\left (-1+x^4\right )^2}+\frac {x^4 \left (a+b x^4\right )^p}{\left (-1+x^4\right )^2}\right ) \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {\left (a+b x^4\right )^p}{\left (-1+x^4\right )^2} \, dx,x,\sin (e+f x)\right )}{f}+\frac {\text {Subst}\left (\int \frac {x^4 \left (a+b x^4\right )^p}{\left (-1+x^4\right )^2} \, dx,x,\sin (e+f x)\right )}{f}+\frac {2 \text {Subst}\left (\int \frac {x^2 \left (a+b x^4\right )^p}{\left (-1+x^4\right )^2} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\left (\left (a+b \sin ^4(e+f x)\right )^p \left (1+\frac {b \sin ^4(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^4}{a}\right )^p}{\left (-1+x^4\right )^2} \, dx,x,\sin (e+f x)\right )}{f}+\frac {\left (\left (a+b \sin ^4(e+f x)\right )^p \left (1+\frac {b \sin ^4(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {x^4 \left (1+\frac {b x^4}{a}\right )^p}{\left (-1+x^4\right )^2} \, dx,x,\sin (e+f x)\right )}{f}+\frac {\left (2 \left (a+b \sin ^4(e+f x)\right )^p \left (1+\frac {b \sin ^4(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {x^2 \left (1+\frac {b x^4}{a}\right )^p}{\left (-1+x^4\right )^2} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\operatorname {AppellF1}\left (\frac {1}{4},2,-p,\frac {5}{4},\sin ^4(e+f x),-\frac {b \sin ^4(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (1+\frac {b \sin ^4(e+f x)}{a}\right )^{-p}}{f}+\frac {2 \operatorname {AppellF1}\left (\frac {3}{4},2,-p,\frac {7}{4},\sin ^4(e+f x),-\frac {b \sin ^4(e+f x)}{a}\right ) \sin ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (1+\frac {b \sin ^4(e+f x)}{a}\right )^{-p}}{3 f}+\frac {\operatorname {AppellF1}\left (\frac {5}{4},2,-p,\frac {9}{4},\sin ^4(e+f x),-\frac {b \sin ^4(e+f x)}{a}\right ) \sin ^5(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \left (1+\frac {b \sin ^4(e+f x)}{a}\right )^{-p}}{5 f} \\ \end{align*}
\[ \int \sec ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx=\int \sec ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx \]
[In]
[Out]
\[\int \left (\sec ^{3}\left (f x +e \right )\right ) {\left (a +b \left (\sin ^{4}\left (f x +e \right )\right )\right )}^{p}d x\]
[In]
[Out]
\[ \int \sec ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{4} + a\right )}^{p} \sec \left (f x + e\right )^{3} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sec ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \sec ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{4} + a\right )}^{p} \sec \left (f x + e\right )^{3} \,d x } \]
[In]
[Out]
\[ \int \sec ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{4} + a\right )}^{p} \sec \left (f x + e\right )^{3} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sec ^3(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx=\int \frac {{\left (b\,{\sin \left (e+f\,x\right )}^4+a\right )}^p}{{\cos \left (e+f\,x\right )}^3} \,d x \]
[In]
[Out]