Optimal. Leaf size=339 \[ \frac {e^{11/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d}-\frac {e^{11/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} a^2 d}+\frac {e^{11/2} \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a^2 d}-\frac {e^{11/2} \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a^2 d}+\frac {2 e^6 \sqrt {\sin (2 c+2 d x)} \sec (c+d x) F\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{3 a^2 d \sqrt {e \tan (c+d x)}}+\frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.46, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 17, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3888, 3886, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2611, 2614, 2573, 2641, 2607, 32} \[ \frac {e^{11/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d}-\frac {e^{11/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} a^2 d}+\frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d}+\frac {e^{11/2} \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a^2 d}-\frac {e^{11/2} \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^6 \sqrt {\sin (2 c+2 d x)} \sec (c+d x) F\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{3 a^2 d \sqrt {e \tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 204
Rule 211
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2573
Rule 2607
Rule 2611
Rule 2614
Rule 2641
Rule 3473
Rule 3476
Rule 3886
Rule 3888
Rubi steps
\begin {align*} \int \frac {(e \tan (c+d x))^{11/2}}{(a+a \sec (c+d x))^2} \, dx &=\frac {e^4 \int (-a+a \sec (c+d x))^2 (e \tan (c+d x))^{3/2} \, dx}{a^4}\\ &=\frac {e^4 \int \left (a^2 (e \tan (c+d x))^{3/2}-2 a^2 \sec (c+d x) (e \tan (c+d x))^{3/2}+a^2 \sec ^2(c+d x) (e \tan (c+d x))^{3/2}\right ) \, dx}{a^4}\\ &=\frac {e^4 \int (e \tan (c+d x))^{3/2} \, dx}{a^2}+\frac {e^4 \int \sec ^2(c+d x) (e \tan (c+d x))^{3/2} \, dx}{a^2}-\frac {\left (2 e^4\right ) \int \sec (c+d x) (e \tan (c+d x))^{3/2} \, dx}{a^2}\\ &=\frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {e^4 \operatorname {Subst}\left (\int (e x)^{3/2} \, dx,x,\tan (c+d x)\right )}{a^2 d}+\frac {\left (2 e^6\right ) \int \frac {\sec (c+d x)}{\sqrt {e \tan (c+d x)}} \, dx}{3 a^2}-\frac {e^6 \int \frac {1}{\sqrt {e \tan (c+d x)}} \, dx}{a^2}\\ &=\frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d}-\frac {e^7 \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (e^2+x^2\right )} \, dx,x,e \tan (c+d x)\right )}{a^2 d}+\frac {\left (2 e^6 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}} \, dx}{3 a^2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}}\\ &=\frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d}-\frac {\left (2 e^7\right ) \operatorname {Subst}\left (\int \frac {1}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a^2 d}+\frac {\left (2 e^6 \sec (c+d x) \sqrt {\sin (2 c+2 d x)}\right ) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}} \, dx}{3 a^2 \sqrt {e \tan (c+d x)}}\\ &=\frac {2 e^6 F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a^2 d \sqrt {e \tan (c+d x)}}+\frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d}-\frac {e^6 \operatorname {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a^2 d}-\frac {e^6 \operatorname {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a^2 d}\\ &=\frac {2 e^6 F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a^2 d \sqrt {e \tan (c+d x)}}+\frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d}+\frac {e^{11/2} \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d}+\frac {e^{11/2} \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d}-\frac {e^6 \operatorname {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 a^2 d}-\frac {e^6 \operatorname {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 a^2 d}\\ &=\frac {e^{11/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d}-\frac {e^{11/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d}+\frac {2 e^6 F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a^2 d \sqrt {e \tan (c+d x)}}+\frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d}-\frac {e^{11/2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d}+\frac {e^{11/2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d}\\ &=\frac {e^{11/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d}-\frac {e^{11/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d}+\frac {e^{11/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d}-\frac {e^{11/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d}+\frac {2 e^6 F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a^2 d \sqrt {e \tan (c+d x)}}+\frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 74.21, size = 0, normalized size = 0.00 \[ \int \frac {(e \tan (c+d x))^{11/2}}{(a+a \sec (c+d x))^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \tan \left (d x + c\right )\right )^{\frac {11}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 1.69, size = 721, normalized size = 2.13 \[ \frac {\left (-1+\cos \left (d x +c \right )\right ) \left (-15 i \sin \left (d x +c \right ) \EllipticPi \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )+15 i \sin \left (d x +c \right ) \EllipticPi \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )+15 \sin \left (d x +c \right ) \EllipticPi \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )+15 \sin \left (d x +c \right ) \EllipticPi \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )-50 \sin \left (d x +c \right ) \EllipticF \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )+24 \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right )-44 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}+26 \cos \left (d x +c \right ) \sqrt {2}-6 \sqrt {2}\right ) \left (\frac {e \sin \left (d x +c \right )}{\cos \left (d x +c \right )}\right )^{\frac {11}{2}} \left (\cos ^{3}\left (d x +c \right )\right ) \left (1+\cos \left (d x +c \right )\right )^{2} \sqrt {2}}{30 a^{2} d \sin \left (d x +c \right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^{11/2}}{a^2\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________