Optimal. Leaf size=238 \[ \frac {5 c^{7/2} (-11 B+3 i A) \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{2 \sqrt {2} a^2 f}-\frac {5 c^3 (-11 B+3 i A) \sqrt {c-i c \tan (e+f x)}}{4 a^2 f}-\frac {5 c^2 (-11 B+3 i A) (c-i c \tan (e+f x))^{3/2}}{24 a^2 f}-\frac {c (-11 B+3 i A) (c-i c \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3588, 78, 47, 50, 63, 208} \[ -\frac {5 c^3 (-11 B+3 i A) \sqrt {c-i c \tan (e+f x)}}{4 a^2 f}-\frac {5 c^2 (-11 B+3 i A) (c-i c \tan (e+f x))^{3/2}}{24 a^2 f}+\frac {5 c^{7/2} (-11 B+3 i A) \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{2 \sqrt {2} a^2 f}-\frac {c (-11 B+3 i A) (c-i c \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 50
Rule 63
Rule 78
Rule 208
Rule 3588
Rubi steps
\begin {align*} \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^2} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(A+B x) (c-i c x)^{5/2}}{(a+i a x)^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2}-\frac {((3 A+11 i B) c) \operatorname {Subst}\left (\int \frac {(c-i c x)^{5/2}}{(a+i a x)^2} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=-\frac {(3 i A-11 B) c (c-i c \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (5 (3 A+11 i B) c^2\right ) \operatorname {Subst}\left (\int \frac {(c-i c x)^{3/2}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{16 a f}\\ &=-\frac {5 (3 i A-11 B) c^2 (c-i c \tan (e+f x))^{3/2}}{24 a^2 f}-\frac {(3 i A-11 B) c (c-i c \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (5 (3 A+11 i B) c^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c-i c x}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{8 a f}\\ &=-\frac {5 (3 i A-11 B) c^3 \sqrt {c-i c \tan (e+f x)}}{4 a^2 f}-\frac {5 (3 i A-11 B) c^2 (c-i c \tan (e+f x))^{3/2}}{24 a^2 f}-\frac {(3 i A-11 B) c (c-i c \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (5 (3 A+11 i B) c^4\right ) \operatorname {Subst}\left (\int \frac {1}{(a+i a x) \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{4 a f}\\ &=-\frac {5 (3 i A-11 B) c^3 \sqrt {c-i c \tan (e+f x)}}{4 a^2 f}-\frac {5 (3 i A-11 B) c^2 (c-i c \tan (e+f x))^{3/2}}{24 a^2 f}-\frac {(3 i A-11 B) c (c-i c \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (5 (3 i A-11 B) c^3\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-i c \tan (e+f x)}\right )}{2 a f}\\ &=\frac {5 (3 i A-11 B) c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{2 \sqrt {2} a^2 f}-\frac {5 (3 i A-11 B) c^3 \sqrt {c-i c \tan (e+f x)}}{4 a^2 f}-\frac {5 (3 i A-11 B) c^2 (c-i c \tan (e+f x))^{3/2}}{24 a^2 f}-\frac {(3 i A-11 B) c (c-i c \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.74, size = 454, normalized size = 1.91 \[ \frac {3 \, \sqrt {\frac {1}{2}} {\left (a^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \sqrt {-\frac {{\left (225 \, A^{2} + 1650 i \, A B - 3025 \, B^{2}\right )} c^{7}}{a^{4} f^{2}}} \log \left (\frac {{\left ({\left (15 i \, A - 55 \, B\right )} c^{4} + \sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} f\right )} \sqrt {-\frac {{\left (225 \, A^{2} + 1650 i \, A B - 3025 \, B^{2}\right )} c^{7}}{a^{4} f^{2}}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a^{2} f}\right ) - 3 \, \sqrt {\frac {1}{2}} {\left (a^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \sqrt {-\frac {{\left (225 \, A^{2} + 1650 i \, A B - 3025 \, B^{2}\right )} c^{7}}{a^{4} f^{2}}} \log \left (\frac {{\left ({\left (15 i \, A - 55 \, B\right )} c^{4} - \sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} f\right )} \sqrt {-\frac {{\left (225 \, A^{2} + 1650 i \, A B - 3025 \, B^{2}\right )} c^{7}}{a^{4} f^{2}}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a^{2} f}\right ) + \sqrt {2} {\left ({\left (-45 i \, A + 165 \, B\right )} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-60 i \, A + 220 \, B\right )} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-9 i \, A + 33 \, B\right )} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (6 i \, A - 6 \, B\right )} c^{3}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{12 \, {\left (a^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.57, size = 179, normalized size = 0.75 \[ -\frac {2 i c^{2} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+5 i B c \sqrt {c -i c \tan \left (f x +e \right )}+c A \sqrt {c -i c \tan \left (f x +e \right )}+2 c^{2} \left (\frac {\left (-\frac {17 i B}{8}-\frac {9 A}{8}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}+\left (\frac {15}{4} i B c +\frac {7}{4} c A \right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\left (-c -i c \tan \left (f x +e \right )\right )^{2}}-\frac {5 \left (11 i B +3 A \right ) \sqrt {2}\, \arctanh \left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{16 \sqrt {c}}\right )\right )}{f \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.94, size = 221, normalized size = 0.93 \[ -\frac {i \, {\left (\frac {15 \, \sqrt {2} {\left (3 \, A + 11 i \, B\right )} c^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{\sqrt {2} \sqrt {c} + \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}{a^{2}} - \frac {12 \, {\left ({\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (9 \, A + 17 i \, B\right )} c^{5} - 2 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (7 \, A + 15 i \, B\right )} c^{6}\right )}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} a^{2} - 4 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} a^{2} c + 4 \, a^{2} c^{2}} + \frac {16 \, {\left (i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} B c^{3} + 3 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (A + 5 i \, B\right )} c^{4}\right )}}{a^{2}}\right )}}{24 \, c f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 9.39, size = 349, normalized size = 1.47 \[ \frac {15\,B\,c^5\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}-\frac {17\,B\,c^4\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{2}}{4\,a^2\,c^2\,f+a^2\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2-4\,a^2\,c\,f\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}-\frac {\frac {A\,c^5\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,7{}\mathrm {i}}{a^2\,f}-\frac {A\,c^4\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,9{}\mathrm {i}}{2\,a^2\,f}}{{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2-4\,c\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )+4\,c^2}-\frac {A\,c^3\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,2{}\mathrm {i}}{a^2\,f}+\frac {10\,B\,c^3\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{a^2\,f}+\frac {2\,B\,c^2\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{3\,a^2\,f}-\frac {\sqrt {2}\,A\,{\left (-c\right )}^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,15{}\mathrm {i}}{4\,a^2\,f}+\frac {\sqrt {2}\,B\,c^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,55{}\mathrm {i}}{4\,a^2\,f} \]
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]
________________________________________________________________________________________