Optimal. Leaf size=155 \[ -\frac {64 c^4 \tan (e+f x)}{3 a^2 f \sqrt {c-c \sec (e+f x)}}-\frac {16 c^3 \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{3 a^2 f}-\frac {4 c^2 \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{f \left (a^2 \sec (e+f x)+a^2\right )}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f (a \sec (e+f x)+a)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.32, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {3954, 3793, 3792} \[ -\frac {64 c^4 \tan (e+f x)}{3 a^2 f \sqrt {c-c \sec (e+f x)}}-\frac {16 c^3 \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{3 a^2 f}-\frac {4 c^2 \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{f \left (a^2 \sec (e+f x)+a^2\right )}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f (a \sec (e+f x)+a)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3792
Rule 3793
Rule 3954
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{7/2}}{(a+a \sec (e+f x))^2} \, dx &=\frac {2 c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {(2 c) \int \frac {\sec (e+f x) (c-c \sec (e+f x))^{5/2}}{a+a \sec (e+f x)} \, dx}{a}\\ &=-\frac {4 c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {2 c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {\left (8 c^2\right ) \int \sec (e+f x) (c-c \sec (e+f x))^{3/2} \, dx}{a^2}\\ &=-\frac {16 c^3 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{3 a^2 f}-\frac {4 c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {2 c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {\left (32 c^3\right ) \int \sec (e+f x) \sqrt {c-c \sec (e+f x)} \, dx}{3 a^2}\\ &=-\frac {64 c^4 \tan (e+f x)}{3 a^2 f \sqrt {c-c \sec (e+f x)}}-\frac {16 c^3 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{3 a^2 f}-\frac {4 c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {2 c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.94, size = 84, normalized size = 0.54 \[ \frac {c^3 (195 \cos (e+f x)+138 \cos (2 (e+f x))+45 \cos (3 (e+f x))+134) \cot \left (\frac {1}{2} (e+f x)\right ) \sec (e+f x) \sqrt {c-c \sec (e+f x)}}{6 a^2 f (\cos (e+f x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 103, normalized size = 0.66 \[ \frac {2 \, {\left (45 \, c^{3} \cos \left (f x + e\right )^{3} + 69 \, c^{3} \cos \left (f x + e\right )^{2} + 15 \, c^{3} \cos \left (f x + e\right ) - c^{3}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} + a^{2} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 3.38, size = 125, normalized size = 0.81 \[ -\frac {4 \, \sqrt {2} c^{3} {\left (\frac {9 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c + c^{2}}{{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} a^{2}} - \frac {{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} a^{4} c^{2} + 9 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} a^{4} c^{3}}{a^{6} c^{3}}\right )}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.63, size = 85, normalized size = 0.55 \[ -\frac {2 \left (3 \cos \left (f x +e \right )+1\right ) \left (15 \left (\cos ^{2}\left (f x +e \right )\right )+18 \cos \left (f x +e \right )-1\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}}}{3 a^{2} f \sin \left (f x +e \right )^{3} \left (-1+\cos \left (f x +e \right )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.50, size = 188, normalized size = 1.21 \[ -\frac {4 \, {\left (16 \, \sqrt {2} c^{\frac {7}{2}} - \frac {56 \, \sqrt {2} c^{\frac {7}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {70 \, \sqrt {2} c^{\frac {7}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {35 \, \sqrt {2} c^{\frac {7}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {4 \, \sqrt {2} c^{\frac {7}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {\sqrt {2} c^{\frac {7}{2}} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}}\right )}}{3 \, a^{2} f {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.02, size = 188, normalized size = 1.21 \[ \frac {2\,c^3\,\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,138{}\mathrm {i}+{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,195{}\mathrm {i}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,268{}\mathrm {i}+{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,195{}\mathrm {i}+{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,138{}\mathrm {i}+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,45{}\mathrm {i}+45{}\mathrm {i}\right )}{3\,a^2\,f\,{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+1\right )}^3\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}-1\right )} \]
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]
________________________________________________________________________________________