Optimal. Leaf size=258 \[ \frac {e^4 \sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c+d x)}{b}\right )}{8 b^2 d}-\frac {9 e^4 \sin \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{16 b^2 d}+\frac {5 e^4 \sin \left (\frac {5 a}{b}\right ) \text {Ci}\left (\frac {5 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{16 b^2 d}-\frac {e^4 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c+d x)}{b}\right )}{8 b^2 d}+\frac {9 e^4 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{16 b^2 d}-\frac {5 e^4 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{16 b^2 d}-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d \left (a+b \sin ^{-1}(c+d x)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.37, antiderivative size = 254, normalized size of antiderivative = 0.98, number of steps used = 13, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4805, 12, 4631, 3303, 3299, 3302} \[ \frac {e^4 \sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )}{8 b^2 d}-\frac {9 e^4 \sin \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{16 b^2 d}+\frac {5 e^4 \sin \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (\frac {5 a}{b}+5 \sin ^{-1}(c+d x)\right )}{16 b^2 d}-\frac {e^4 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )}{8 b^2 d}+\frac {9 e^4 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{16 b^2 d}-\frac {5 e^4 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 a}{b}+5 \sin ^{-1}(c+d x)\right )}{16 b^2 d}-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d \left (a+b \sin ^{-1}(c+d x)\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 3299
Rule 3302
Rule 3303
Rule 4631
Rule 4805
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^4}{\left (a+b \sin ^{-1}(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^4 x^4}{\left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \operatorname {Subst}\left (\int \frac {x^4}{\left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e^4 \operatorname {Subst}\left (\int \left (-\frac {\sin (x)}{8 (a+b x)}+\frac {9 \sin (3 x)}{16 (a+b x)}-\frac {5 \sin (5 x)}{16 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{b d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {e^4 \operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b d}-\frac {\left (5 e^4\right ) \operatorname {Subst}\left (\int \frac {\sin (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 b d}+\frac {\left (9 e^4\right ) \operatorname {Subst}\left (\int \frac {\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 b d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {\left (e^4 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b d}+\frac {\left (9 e^4 \cos \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 b d}-\frac {\left (5 e^4 \cos \left (\frac {5 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 b d}+\frac {\left (e^4 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b d}-\frac {\left (9 e^4 \sin \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 b d}+\frac {\left (5 e^4 \sin \left (\frac {5 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 b d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e^4 \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right ) \sin \left (\frac {a}{b}\right )}{8 b^2 d}-\frac {9 e^4 \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c+d x)\right ) \sin \left (\frac {3 a}{b}\right )}{16 b^2 d}+\frac {5 e^4 \text {Ci}\left (\frac {5 a}{b}+5 \sin ^{-1}(c+d x)\right ) \sin \left (\frac {5 a}{b}\right )}{16 b^2 d}-\frac {e^4 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )}{8 b^2 d}+\frac {9 e^4 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{16 b^2 d}-\frac {5 e^4 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 a}{b}+5 \sin ^{-1}(c+d x)\right )}{16 b^2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.29, size = 283, normalized size = 1.10 \[ \frac {e^4 \left (16 \left (-3 \sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )+\sin \left (\frac {3 a}{b}\right ) \text {Ci}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )\right )+3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )-\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )\right )\right )+5 \left (10 \sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )-5 \sin \left (\frac {3 a}{b}\right ) \text {Ci}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )\right )+\sin \left (\frac {5 a}{b}\right ) \text {Ci}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )\right )-10 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )+5 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )\right )-\cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )\right )\right )-\frac {16 b \sqrt {1-(c+d x)^2} (c+d x)^4}{a+b \sin ^{-1}(c+d x)}\right )}{16 b^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}{b^{2} \arcsin \left (d x + c\right )^{2} + 2 \, a b \arcsin \left (d x + c\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.47, size = 1374, normalized size = 5.33 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.23, size = 397, normalized size = 1.54 \[ \frac {e^{4} \left (9 \arcsin \left (d x +c \right ) \Si \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b -9 \arcsin \left (d x +c \right ) \Ci \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b -5 \arcsin \left (d x +c \right ) \Si \left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) b +5 \arcsin \left (d x +c \right ) \Ci \left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) b -2 \arcsin \left (d x +c \right ) \Si \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +2 \arcsin \left (d x +c \right ) \Ci \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +9 \Si \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a -9 \Ci \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a -5 \Si \left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) a +5 \Ci \left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) a -2 \Si \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a +2 \Ci \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +3 \cos \left (3 \arcsin \left (d x +c \right )\right ) b -\cos \left (5 \arcsin \left (d x +c \right )\right ) b -2 \sqrt {1-\left (d x +c \right )^{2}}\, b \right )}{16 d \left (a +b \arcsin \left (d x +c \right )\right ) b^{2}} \]
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 \[ \int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{4} \left (\int \frac {c^{4}}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________