Optimal. Leaf size=269 \[ \frac {i A-B}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}+\frac {4 i A-B}{3 a f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}-\frac {(4 i A-B) \sqrt {a+i a \tan (e+f x)}}{5 a^2 f (c-i c \tan (e+f x))^{5/2}}-\frac {2 (4 i A-B) \sqrt {a+i a \tan (e+f x)}}{15 a^2 c f (c-i c \tan (e+f x))^{3/2}}-\frac {2 (4 i A-B) \sqrt {a+i a \tan (e+f x)}}{15 a^2 c^2 f \sqrt {c-i c \tan (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.22, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3669, 79, 47,
37} \begin {gather*} -\frac {2 (-B+4 i A) \sqrt {a+i a \tan (e+f x)}}{15 a^2 c^2 f \sqrt {c-i c \tan (e+f x)}}-\frac {2 (-B+4 i A) \sqrt {a+i a \tan (e+f x)}}{15 a^2 c f (c-i c \tan (e+f x))^{3/2}}-\frac {(-B+4 i A) \sqrt {a+i a \tan (e+f x)}}{5 a^2 f (c-i c \tan (e+f x))^{5/2}}+\frac {-B+i A}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}+\frac {-B+4 i A}{3 a f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 47
Rule 79
Rule 3669
Rubi steps
\begin {align*} \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {A+B x}{(a+i a x)^{5/2} (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i A-B}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}+\frac {((4 A+i B) c) \text {Subst}\left (\int \frac {1}{(a+i a x)^{3/2} (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{3 f}\\ &=\frac {i A-B}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}+\frac {4 i A-B}{3 a f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}+\frac {((4 A+i B) c) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{a f}\\ &=\frac {i A-B}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}+\frac {4 i A-B}{3 a f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}-\frac {(4 i A-B) \sqrt {a+i a \tan (e+f x)}}{5 a^2 f (c-i c \tan (e+f x))^{5/2}}+\frac {(2 (4 A+i B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{5 a f}\\ &=\frac {i A-B}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}+\frac {4 i A-B}{3 a f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}-\frac {(4 i A-B) \sqrt {a+i a \tan (e+f x)}}{5 a^2 f (c-i c \tan (e+f x))^{5/2}}-\frac {2 (4 i A-B) \sqrt {a+i a \tan (e+f x)}}{15 a^2 c f (c-i c \tan (e+f x))^{3/2}}+\frac {(2 (4 A+i B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{15 a c f}\\ &=\frac {i A-B}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}+\frac {4 i A-B}{3 a f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}-\frac {(4 i A-B) \sqrt {a+i a \tan (e+f x)}}{5 a^2 f (c-i c \tan (e+f x))^{5/2}}-\frac {2 (4 i A-B) \sqrt {a+i a \tan (e+f x)}}{15 a^2 c f (c-i c \tan (e+f x))^{3/2}}-\frac {2 (4 i A-B) \sqrt {a+i a \tan (e+f x)}}{15 a^2 c^2 f \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 4.04, size = 170, normalized size = 0.63 \begin {gather*} \frac {\sec (e+f x) (\cos (3 (e+f x))+i \sin (3 (e+f x))) (-45 A+20 (A+i B) \cos (2 (e+f x))+(A+4 i B) \cos (4 (e+f x))-40 i A \sin (2 (e+f x))+10 B \sin (2 (e+f x))-4 i A \sin (4 (e+f x))+B \sin (4 (e+f x))) \sqrt {c-i c \tan (e+f x)}}{120 a c^3 f (-i+\tan (e+f x)) \sqrt {a+i a \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.40, size = 199, normalized size = 0.74
method | result | size |
derivativedivides | \(\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (8 i A \left (\tan ^{6}\left (f x +e \right )\right )-2 i B \left (\tan ^{5}\left (f x +e \right )\right )-2 B \left (\tan ^{6}\left (f x +e \right )\right )+20 i A \left (\tan ^{4}\left (f x +e \right )\right )-8 A \left (\tan ^{5}\left (f x +e \right )\right )-5 i B \left (\tan ^{3}\left (f x +e \right )\right )-5 B \left (\tan ^{4}\left (f x +e \right )\right )+15 i A \left (\tan ^{2}\left (f x +e \right )\right )-20 A \left (\tan ^{3}\left (f x +e \right )\right )-3 i B \tan \left (f x +e \right )+3 i A -12 A \tan \left (f x +e \right )+3 B \right )}{15 f \,a^{2} c^{3} \left (i-\tan \left (f x +e \right )\right )^{3} \left (i+\tan \left (f x +e \right )\right )^{4}}\) | \(199\) |
default | \(\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (8 i A \left (\tan ^{6}\left (f x +e \right )\right )-2 i B \left (\tan ^{5}\left (f x +e \right )\right )-2 B \left (\tan ^{6}\left (f x +e \right )\right )+20 i A \left (\tan ^{4}\left (f x +e \right )\right )-8 A \left (\tan ^{5}\left (f x +e \right )\right )-5 i B \left (\tan ^{3}\left (f x +e \right )\right )-5 B \left (\tan ^{4}\left (f x +e \right )\right )+15 i A \left (\tan ^{2}\left (f x +e \right )\right )-20 A \left (\tan ^{3}\left (f x +e \right )\right )-3 i B \tan \left (f x +e \right )+3 i A -12 A \tan \left (f x +e \right )+3 B \right )}{15 f \,a^{2} c^{3} \left (i-\tan \left (f x +e \right )\right )^{3} \left (i+\tan \left (f x +e \right )\right )^{4}}\) | \(199\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.24, size = 191, normalized size = 0.71 \begin {gather*} -\frac {{\left (3 \, {\left (i \, A + B\right )} e^{\left (10 i \, f x + 10 i \, e\right )} - {\left (-23 i \, A - 13 \, B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, {\left (11 i \, A + B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + 48 \, {\left (-i \, A - B\right )} e^{\left (5 i \, f x + 5 i \, e\right )} + 30 \, {\left (i \, A + B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 48 \, {\left (-i \, A - B\right )} e^{\left (3 i \, f x + 3 i \, e\right )} + 5 \, {\left (-13 i \, A + 7 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - 5 i \, A + 5 \, B\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-3 i \, f x - 3 i \, e\right )}}{240 \, a^{2} c^{3} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B \tan {\left (e + f x \right )}}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}} \left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 9.99, size = 196, normalized size = 0.73 \begin {gather*} \frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (40\,A\,\sin \left (2\,e+2\,f\,x\right )+A\,\cos \left (2\,e+2\,f\,x\right )\,20{}\mathrm {i}+A\,\cos \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}-20\,B\,\cos \left (2\,e+2\,f\,x\right )-4\,B\,\cos \left (4\,e+4\,f\,x\right )-A\,45{}\mathrm {i}+4\,A\,\sin \left (4\,e+4\,f\,x\right )+B\,\sin \left (2\,e+2\,f\,x\right )\,10{}\mathrm {i}+B\,\sin \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}\right )}{120\,a^2\,c^2\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________