Optimal. Leaf size=299 \[ -\frac {2 \left (A d^2-B c d+c^2 C\right ) \sqrt {a+b \tan (e+f x)}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-\frac {\sqrt {a-i b} (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (c-i d)^{3/2}}-\frac {\sqrt {a+i b} (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (c+i d)^{3/2}}+\frac {2 \sqrt {b} C \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{3/2} f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 3.33, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {3645, 3655, 6725, 63, 217, 206, 93, 208} \[ -\frac {2 \left (A d^2-B c d+c^2 C\right ) \sqrt {a+b \tan (e+f x)}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-\frac {\sqrt {a-i b} (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (c-i d)^{3/2}}-\frac {\sqrt {a+i b} (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (c+i d)^{3/2}}+\frac {2 \sqrt {b} C \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{3/2} f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 93
Rule 206
Rule 208
Rule 217
Rule 3645
Rule 3655
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 \int \frac {\frac {1}{2} (A d (a c+b d)+(b c-a d) (c C-B d))+\frac {1}{2} d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+\frac {1}{2} b C \left (c^2+d^2\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{d \left (c^2+d^2\right )}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 \operatorname {Subst}\left (\int \frac {\frac {1}{2} (A d (a c+b d)+(b c-a d) (c C-B d))+\frac {1}{2} d ((A-C) (b c-a d)+B (a c+b d)) x+\frac {1}{2} b C \left (c^2+d^2\right ) x^2}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 \operatorname {Subst}\left (\int \left (\frac {b C \left (c^2+d^2\right )}{2 \sqrt {a+b x} \sqrt {c+d x}}+\frac {d (a (A c-c C+B d)-b (B c-(A-C) d))+d (A b c+a B c-b c C-a A d+b B d+a C d) x}{2 \sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {(b C) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{d f}+\frac {\operatorname {Subst}\left (\int \frac {d (a (A c-c C+B d)-b (B c-(A-C) d))+d (A b c+a B c-b c C-a A d+b B d+a C d) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {(2 C) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b \tan (e+f x)}\right )}{d f}+\frac {\operatorname {Subst}\left (\int \left (\frac {-d (A b c+a B c-b c C-a A d+b B d+a C d)+i d (a (A c-c C+B d)-b (B c-(A-C) d))}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {d (A b c+a B c-b c C-a A d+b B d+a C d)+i d (a (A c-c C+B d)-b (B c-(A-C) d))}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=-\frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {((i a+b) (A-i B-C)) \operatorname {Subst}\left (\int \frac {1}{(i+x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (c-i d) f}+\frac {((i a-b) (A+i B-C)) \operatorname {Subst}\left (\int \frac {1}{(i-x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (c+i d) f}+\frac {(2 C) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{d f}\\ &=\frac {2 \sqrt {b} C \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {((i a+b) (A-i B-C)) \operatorname {Subst}\left (\int \frac {1}{-a+i b-(-c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{(c-i d) f}+\frac {((i a-b) (A+i B-C)) \operatorname {Subst}\left (\int \frac {1}{a+i b-(c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{(c+i d) f}\\ &=-\frac {\sqrt {a-i b} (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c-i d)^{3/2} f}-\frac {\sqrt {a+i b} (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c+i d)^{3/2} f}+\frac {2 \sqrt {b} C \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac {2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 35.59, size = 621084, normalized size = 2077.20 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
[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(-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]
________________________________________________________________________________________
maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a +b \tan \left (f x +e \right )}\, \left (A +B \tan \left (f x +e \right )+C \left (\tan ^{2}\left (f x +e \right )\right )\right )}{\left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\sqrt {a+b\,\mathrm {tan}\left (e+f\,x\right )}\,\left (C\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,\mathrm {tan}\left (e+f\,x\right )+A\right )}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + b \tan {\left (e + f x \right )}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________